Anthropic 论文：叠加现象与玩具模型

It would be very convenient if the individual neurons of artificial neural networks corresponded to cleanly interpretable features of the input. For example, in an “ideal” ImageNet classifier, each neuron would fire only in the presence of a specific visual feature, such as the color red, a left-facing curve, or a dog snout. Empirically, in models we have studied, some of the neurons do cleanly map to features. But it isn't always the case that features correspond so cleanly to neurons, especially in large language models where it actually seems rare for neurons to correspond to clean features. This brings up many questions. Why is it that neurons sometimes align with features and sometimes don't? Why do some models and tasks have many of these clean neurons, while they're vanishingly rare in others?

In this paper, we use toy models — small ReLU networks trained on synthetic data with sparse input features — to investigate how and when models represent more features than they have dimensions. We call this phenomenon superposition . When features are sparse, superposition allows compression beyond what a linear model would do, at the cost of "interference" that requires nonlinear filtering.

Consider a toy model where we train an embedding of five features of varying importanceWhere “importance” is a scalar multiplier on mean squared error loss. in two dimensions, add a ReLU afterwards for filtering, and vary the sparsity of the features. With dense features, the model learns to represent an orthogonal basis of the most important two features (similar to what Principal Component Analysis might give us), and the other three features are not represented. But if we make the features sparse, this changes:

Not only can models store additional features in superposition by tolerating some interference, but we'll show that, at least in certain limited cases, models can perform computation while in superposition. (In particular, we'll show that models can put simple circuits computing the absolute value function in superposition.) This leads us to hypothesize that the neural networks we observe in practice are in some sense noisily simulating larger, highly sparse networks. In other words, it's possible that models we train can be thought of as doing “the same thing as” an imagined much-larger model, representing the exact same features but with no interference.

Feature superposition isn't a novel idea. A number of previous interpretability papers have considered it , and it's very closely related to the long-studied topic of compressed sensing in mathematics , as well as the ideas of distributed, dense, and population codes in neuroscience and deep learning . What, then, is the contribution of this paper?

For interpretability researchers, our main contribution is providing a direct demonstration that superposition occurs in artificial neural networks given a relatively natural setup, suggesting this may also occur in practice. That is, we show a case where interpreting neural networks as having sparse structure in superposition isn't just a useful post-hoc interpretation, but actually the "ground truth" of a model. We offer a theory of when and why this occurs, revealing a phase diagram for superposition. This explains why neurons are sometimes "monosemantic" responding to a single feature, and sometimes "polysemantic" responding to many unrelated features. We also discover that, at least in our toy model, superposition exhibits complex geometric structure.

But our results may also be of broader interest. We find preliminary evidence that superposition may be linked to adversarial examples and grokking, and might also suggest a theory for the performance of mixture of experts models. More broadly, the toy model we investigate has unexpectedly rich structure, exhibiting phase changes, a geometric structure based on uniform polytopes, "energy level"-like jumps during training, and a phenomenon which is qualitatively similar to the fractional quantum Hall effect in physics, among other striking phenomena. We originally investigated the subject to gain understanding of cleanly-interpretable neurons in larger models, but we've found these toy models to be surprisingly interesting in their own right.

In our toy models, we are able to demonstrate that:

Superposition is a real, observed phenomenon; Both monosemantic and polysemantic neurons can form; At least some kinds of computation can be performed in superposition; Whether features are stored in superposition is governed by a phase change; Superposition organizes features into geometric structures such as digons, triangles, pentagons, and tetrahedrons

Our toy models are simple ReLU networks, so it seems fair to say that neural networks exhibit these properties in at least some regimes, but it's very unclear what to generalize to real networks.

In our work, we often think of neural networks as having features of the input represented as directions in activation space. This isn't a trivial claim. It isn't obvious what kind of structure we should expect neural network representations to have. When we say something like "word embeddings have a gender direction" or "vision models have curve detector neurons", one is implicitly making strong claims about the structure of network representations.

Despite this, we believe this kind of "linear representation hypothesis" is supported both by significant empirical findings and theoretical arguments. One might think of this as two separate properties, which we'll explore in more detail shortly:

If we hope to reverse engineer neural networks, we need a property like decomposability. Decomposability is what allows us to reason about the model without fitting the whole thing in our heads! But it's not enough for things to be decomposable: we need to be able to access the decomposition somehow. In order to do this, we need to identify the individual features within a representation. In a linear representation, this corresponds to determining which directions in activation space correspond to which independent features of the input.

Sometimes, identifying feature directions is very easy because features seem to correspond to neurons. For example, many neurons in the early layers of InceptionV1 clearly correspond to features (e.g. curve detector neurons ). Why is it that we sometimes get this extremely helpful property, but in other cases don't? We hypothesize that there are really two countervailing forces driving this:

Privileged Basis: Only some representations have a privileged basis which encourages features to align with basis directions (i.e. to correspond to neurons); Superposition: Linear representations can represent more features than dimensions, using a strategy we call superposition. This can be seen as neural networks simulating larger networks. This pushes features away from corresponding to neurons

Superposition has been hypothesized in previous work , and in some cases, assuming something like superposition has been shown to help find interpretable structure . However, we're not aware of feature superposition having been unambiguously demonstrated to occur in neural networks before ( demonstrates a closely related phenomenon of model superposition). The goal of this paper is to change that, demonstrating superposition and exploring how it interacts with privileged bases. If superposition occurs in networks, it deeply influences what approaches to interpretability research make sense, so unambiguous demonstration seems important.

The goal of this section will be to motivate these ideas and unpack them in detail.

It's worth noting that many of the ideas in this section have close connections to ideas in other lines of interpretability research (especially disentanglement), neuroscience (distributed representations, population codes, etc), compressed sensing, and many other lines of work. This section will focus on articulating our perspective on the problem. We'll discuss these other lines of work in detail in Related Work.

Empirical Phenomena

When we talk about "features" and how they're represented, this is ultimately theory building around several observed empirical phenomena. Before describing how we conceptualize those results, we'll simply describe some of the major results motivating our thinking:

Word Embeddings – A famous result by Mikolov et al. found that word embeddings appear to have directions which correspond to semantic properties, allowing for embedding arithmetic vectors such as V("king") - V("man") + V("woman") = V("queen") (but see ); Latent Spaces – Similar "vector arithmetic" and interpretable direction results have also been found for generative adversarial networks (e.g. ); Interpretable Neurons – There is a significant body of results finding neurons which appear to be interpretable (in RNNs ; in CNNs ; in GANs ), activating in response to some understandable property. This work has faced some skepticism . In response, several papers have aimed to give extremely detailed accounts of a few specific neurons, in the hope of dispositively establishing examples of neurons which truly detect some understandable property (notably Cammarata et al. , but also ); Universality – Many analogous neurons responding to the same properties can be found across networks; Polysemantic Neurons – At the same time, there are also many neurons which appear to not respond to an interpretable property of the input, and in particular, many polysemantic neurons which appear to respond to unrelated mixtures of inputs

As a result, we tend to think of neural network representations as being composed of features which are represented as directions. We'll unpack this idea in the following sections.

What are Features?

Our use of the term "feature" is motivated by the interpretable properties of the input we observe neurons (or word embedding directions) responding to. There's a rich variety of such observed properties!In the context of vision, these have ranged from low-level neurons like curve detectors and high-low frequency detectors, to more complex neurons like oriented dog-head detectors or car detectors, to extremely abstract neurons corresponding to famous people, emotions, geographic regions, and more . In language models, researchers have found word embedding directions such as a male-female or singular-plural direction , low-level neurons disambiguating words that occur in multiple languages, much more abstract neurons, and "action" output neurons that help produce certain words . We'd like to use the term "feature" to encompass all these properties.

But even with that motivation, it turns out to be quite challenging to create a satisfactory definition of a feature. Rather than offer a single definition we're confident about, we consider three potential working definitions:

Features as arbitrary functions. One approach would be to define features as any function of the input (as in ). But this doesn't quite seem to fit our motivations. There's something special about these features that we're observing: they seem to in some sense be fundamental abstractions for reasoning about the data, with the same features forming reliably across models. Features also seem identifiable: cat and car are two features while cat+car and cat-car seem like mixtures of features rather than features in some important sense; Features as interpretable properties. All the features we described are strikingly understandable to humans. One could try to use this for a definition: features are the presence of human understandable "concepts" in the input. But it seems important to allow for features we might not understand. If AlphaFold discovers some important chemical structure for predicting protein folding, it very well might not be something we initially understand!; Neurons in Sufficiently Large Models. A final approach is to define features as properties of the input which a sufficiently large neural network will reliably dedicate a neuron to representing.This definition is trickier than it seems. Specifically, something is a feature if there exists a large enough model size such that it gets a dedicated neuron. This creates a kind "epsilon-delta" like definition. Our present understanding – as we'll see in later sections – is that arbitrarily large models can still have a large fraction of their features be in superposition. However, for any given feature, assuming the feature importance curve isn't flat, it should eventually be given a dedicated neuron. This definition can be helpful in saying that something is a feature – curve detectors are a feature because you find them across a range of models larger than some minimal size – but unhelpful for the much more common case of features we only hypothesize about or observe in superposition. For example, curve detectors appear to reliably occur across sufficiently sophisticated vision models, and so are a feature. For interpretable properties which we presently only observe in polysemantic neurons, the hope is that a sufficiently large model would dedicate a neuron to them. This definition is slightly circular, but avoids the issues with the earlier ones

We've written this paper with the final "neurons in sufficiently large models" definition in mind. But we aren't overly attached to it, and actually think it's probably important to not prematurely attach to a definition.A famous book by Lakatos illustrates the importance of uncertainty about definitions and how important rethinking definitions often is in the context of research.

Features as Directions

We don't think it's a coincidence that neural networks empirically seem to have linear representations. Neural networks are built from linear functions interspersed with non-linearities. In some sense, the linear functions are the vast majority of the computation (for example, as measured in FLOPs). Linear representations are the natural format for neural networks to represent information in! Concretely, there are three major benefits:

Linear representations are the natural outputs of obvious algorithms a layer might implement. If one sets up a neuron to pattern match a particular weight template, it will fire more as a stimulus matches the template better and less as it matches it less well; Linear representations make features "linearly accessible." A typical neural network layer is a linear function followed by a non-linearity. If a feature in the previous layer is represented linearly, a neuron in the next layer can "select it" and have it consistently excite or inhibit that neuron. If a feature were represented non-linearly, the model would not be able to do this in a single step; Statistical Efficiency. Representing features as different directions may allow non-local generalization in models with linear transformations (such as the weights of neural nets), increasing their statistical efficiency relative to models which can only locally generalize. This view is especially advocated in some of Bengio's writing (e.g. ). A more accessible argument can be found in this blog post

It is possible to construct non-linear representations, and retrieve information from them, if you use multiple layers (although even these examples can be seen as linear representations with more exotic features). We provide an example in the appendix. However, our intuition is that non-linear representations are generally inefficient for neural networks.

One might think that a linear representation can only store as many features as it has dimensions, but it turns out this isn't the case! We'll see that the phenomenon we call superposition will allow models to store more features – potentially many more features – in linear representations.

For discussion on how this view of features squares with a conception of features as being multidimensional manifolds, see the appendix “What about Multidimensional Features?”.

Privileged vs Non-privileged Bases

Even if features are encoded as directions, a natural question to ask is which directions? In some cases, it seems useful to consider the basis directions, but in others it doesn't. Why is this?

When researchers study word embeddings, it doesn't make sense to analyze basis directions. There would be no reason to expect a basis dimension to be different from any other possible direction. One way to see this is to imagine applying some random linear transformation M to the word embedding, and apply M^{-1} to the following weights. This would produce an identical model where the basis dimensions are totally different. This is what we mean by a non-privileged basis. Of course, it's possible to study activations without a privileged basis, you just need to identify interesting directions to study somehow, such as creating a gender direction in a word embedding by taking the difference vector between "man" and "woman".

But many neural network layers are not like this. Often, something about the architecture makes the basis directions special, such as applying an activation function. This "breaks the symmetry", making those directions special, and potentially encouraging features to align with the basis dimensions. We call this a privileged basis, and call the basis directions "neurons." Often, these neurons correspond to interpretable features.

From this perspective, it only makes sense to ask if a neuron is interpretable when it is in a privileged basis. In fact, we typically reserve the word "neuron" for basis directions which are in a privileged basis. (See longer discussion here.)

Note that having a privileged basis doesn't guarantee that features will be basis-aligned – we'll see that they often aren't! But it's a minimal condition for the question to even make sense.

The Superposition Hypothesis

Even when there is a privileged basis, it's often the case that neurons are "polysemantic", responding to several unrelated features. One explanation for this is the superposition hypothesis. Roughly, the idea of superposition is that neural networks "want to represent more features than they have neurons", so they exploit a property of high-dimensional spaces to simulate a model with many more neurons.

Several results from mathematics suggest that something like this might be plausible:

Almost Orthogonal Vectors. Although it's only possible to have n orthogonal vectors in an n-dimensional space, it's possible to have \exp(n) many "almost orthogonal" ( 3×), and that the level of vulnerability closely tracks the number of features per dimension (the reciprocal of feature dimensionality).

We're hesitant to speculate about the extent to which superposition is responsible for adversarial examples in practice. There are compelling theories for why adversarial examples occur without reference to superposition (e.g. ). But it is interesting to note that if one wanted to try to argue for a "superposition maximalist stance", it does seem like many interesting phenomena related to adversarial examples can be predicted from superposition. As seen above, superposition can be used to explain why adversarial examples exist. It also predicts that adversarially robust models would have worse performance, since making models robust would require giving up superposition and representing less features. It predicts that more adversarially robust models might be more interpretable (see e.g. ). Finally, it could arguably predict that adversarial examples transfer (see e.g. ) if the arrangement of features in superposition is heavily influenced by which features are correlated or anti-correlated (see earlier results on this). It might be interesting for future work to see how far the hypothesis that superposition is a significant contributor to adversarial examples can be driven.

In addition to observing that superposition can cause models to be vulnerable to adversarial examples, we briefly experimented with adversarial training to see if the relationship could be used in the other direction to reduce superposition. To keep training reasonably efficient, we used the analytic optimal attack against a random feature. We found that this did reduce superposition, but attacks had to be made unreasonably large (80% input L2 norm) to fully eliminate it, which didn't seem satisfying. Perhaps stronger adversarial attacks would work better. We didn't explore this further since the increased cost and complexity of adversarial training made us want to prioritize other lines of attack on superposition first.

Superposition in a Privileged Basis

So far, we've explored superposition in a model without a privileged basis. We can rotate the hidden activations arbitrarily and, as long as we rotate all the weights, have the exact same model behavior. That is, for any ReLU output model with weights W, we could take an arbitrary orthogonal matrix O and consider the model W' = OW. Since (OW)^T(OW) = W^TW, the result would be an identical model!

Models without a privileged basis are elegant, and can be an interesting analogue for certain neural network representations which don't have a privileged basis – word embeddings, or the transformer residual stream. But we'd also (and perhaps primarily) like to understand neural network representations where there are neurons which do impose a privileged basis, such as transformer MLP layers or conv net neurons.

Our goal in this section is to explore the simplest toy model which gives us a privileged basis. There are at least two ways we could do this: we could add an activation function or apply L1 regularization to the hidden layer. We'll focus on adding an activation function, since the representation we are most interested in understanding is hidden layers with neurons, such as the transformer MLP layer.

This gives us the following "ReLU hidden layer" model:

We'll train this model on the same data as before.

Adding a ReLU to the hidden layer radically changes the model from an interpretability perspective. The key thing is that while W in our previous model was challenging to interpret (recall that we visualized W^TW rather than W), W in the ReLU hidden layer model can be directly interpreted, since it connects features to basis-aligned neurons.

We'll discuss this in much more detail shortly, but here's a comparison of weights resulting from a linear hidden layer model and a ReLU hidden layer model:

Recall that we think of basis elements in the input as "features," and basis elements in the middle layer as "neurons". Thus W is a map from features to neurons.

What we see in the above plot is that the features are aligning with neurons in a structured way! Many of the neurons are simply dedicated to representing a feature! (This is the critical property that justifies why neuron-focused interpretability approaches – such as much of the work in the original Circuits thread – can be effective in some circumstances.)

Let's explore this in more detail.

Visualizing Superposition in Terms of Neurons

Having a privileged basis opens up new possibilities for visualizing our models. As we saw above, we can simply inspect W. We can also make a per-neuron stacked bar plot where, for every neuron, we visualize its weights as a stack of rectangles on top of each other:

Each column in the stack plot visualizes one column of W; Each rectangle represents one weight entry, with height corresponding to the absolute value; The color of each rectangle corresponds to the feature it acts on (i.e. which row of W it's in); Negative values go below the x-axis; The order of the rectangles is not significant

This stack plot visualization can be nice as models get bigger. It also makes polysemantic neurons obvious: they simply correspond to having more than one weight.

We'll now visualize a ReLU hidden layer toy model with n=10;~ m=5; I^i = 0.75^i and varying feature sparsity levels. We chose a very small model (only 5 neurons) both for ease of visualization, and to circumvent some issues with this toy model we'll discuss below.

However, we found that these small models were harder to optimize. For each model shown, we trained 1000 models and visualized the one with the lowest loss. Although the typical solutions are often similar to the minimal loss solutions shown, selecting the minimal loss solutions reveals even more structure in how features align with neurons. It also reveals that there are ranges of sparsity values where the optimal solution for all models trained on data with that sparsity has the same weight configurations.

The solutions are visualized below, both visualizing the raw W and a neuron stacked bar plot. We color features in the stacked bar plot based on whether they're in superposition, and color neurons as being monosemantic or polysemantic depending on whether they store more than one feature. Neuron order was chosen by hand (since it's arbitrary).

The most important thing to pay attention to is how there's a shift from monosemantic to polysemantic neurons as sparsity increases. Monosemantic neurons do exist in some regimes! Polysemantic neurons exist in others. And they can both exist in the same model! Moreover, while it's not quite clear how to formalize this, it looks a great deal like there's a neuron-level phase change, mirroring the feature phase changes we saw earlier.

It's also interesting to examine the structure of the polysemantic solutions, which turn out to be surprisingly structured and neuron-aligned. Features typically correspond to sets of neurons (monosemantic neurons might be seen as the special case where features only correspond to singleton sets). There's also structure in how polysemantic neurons are. They transition from monosemantic, to only representing a few features, to gradually representing more. However, it's unclear how much of this is generalizable to real models.

Unfortunately, the toy model described in this section has a significant weakness, which limits the regimes in which it shows interesting results. The issue is that the model doesn't benefit from the ReLU hidden layer – it has no role except limiting how the model can encode information. If given any chance, the model will circumvent it. For example, given a hidden layer bias, the model will set all the biases to be positive, shifting the neurons into a positive regime where they behave linearly. If one removes the bias, but gives the model enough features, it will simulate a bias by averaging over many features. The model will only use the ReLU activation function if absolutely forced, which is a significant mark against studying this toy model.

We'll introduce a model without this issue in the next section, but wanted to study this model as a simpler case study.

Computation in Superposition

So far, we've shown that neural networks can store sparse features in superposition and then recover them. But we actually believe superposition is more powerful than this – we think that neural networks can perform computation entirely in superposition rather than just using it as storage. This model will also give us a more principled way to study a privileged basis where features align with basis dimensions.

To explore this, we consider a new setup where we imagine our input and output layer to be the layers of our hypothetical disentangled model, but have our hidden layer be a smaller layer we're imagining to be the observed model which might use superposition. We'll then try to compute a simple non-linear function and explore whether it can use superposition to do this. Since the model will have (and need to use) the hidden layer non-linearity, we'll also see features align with a privileged basis.

Specifically, we'll have the model compute y=\text{abs}(x). Absolute value is an appealing function to study because there's a very simple way to compute it with ReLU neurons: \text{abs}(x) = \text{ReLU}(x) + \text{ReLU}(-x). This simple structure will make it easy for us to study the geometry of how the hidden layer is leveraged to do computation.

Since this model needs ReLU to compute absolute value, it doesn't have the issues the model in the previous section had with trying to avoid the activation function.

Experiment Setup

The input feature vector, x, is still sparse, with each feature x_i having probability S_i of being 0. However, since we want to have the model compute absolute value, we need to allow it to take on non-positive values for this to be a non-trivial task. As a result, if it is non-zero, its value is now sampled uniformly from [-1,1]. The target output y is y=\text{abs}(x).

Following the previous section, we'll consider the "ReLU hidden layer" toy model variant, but no longer tie the two weights to be identical:

h = \text{ReLU}(W_1x) y' = \text{ReLU}(W_2h+b)

The loss is still the mean squared error weighted by feature importances I_i as before.

Basic Results

With this model, it's a bit less straightforward to study how individual features get embedded; because of the ReLU on the hidden layer, we can't just study W_2^TW_1. And because W_2 and W_1 are now learned independently, we can't just study columns of W_1. We believe that with some manipulation we could recover much of the simplicity of the earlier model by considering "positive features" and "negative features" independently, but we're going to focus on another perspective instead.

As we saw in the previous section, having a hidden layer activation function means that it makes sense to visualize the weights in terms of neurons. We can visualize W directly or as a neuron stack plot as we did before. We can also visualize it as a graph, which can sometimes be helpful for understanding computation.

Let's look at what happens when we train a model with n=3 features to perform absolute value on m=6 hidden layer neurons. Without superposition, the model needs two hidden layer neurons to implement absolute value on one feature.

The resulting model – modulo a subtle issue about rescaling input and output weightsNote that there's a degree of freedom for the model in learning W_1: We can rescale any hidden unit by scaling its row of W_1 by \alpha, and its column of W_2 by \alpha^{-1}, and arrive at the same model. For consistency in the visualization, we rescale each hidden unit before visualizing so that the largest-magnitude weight to that neuron from W_1 has magnitude 1. – performs absolute value exactly as one might expect. For each input feature x_i, it constructs a "positive side" neuron \text{ReLU}(x_i) and a "negative side" neuron \text{ReLU}(-x_i). It then adds these together to compute absolute value:

Superposition vs Sparsity

We've seen that – as expected – our toy model can learn to implement absolute value. But can it use superposition to compute absolute value for more features? To test this, we train models with n=100 features and m=40 neurons and a feature importance curve I_i = 0.8^i, varying feature sparsity.These specific values were chosen to illustrate the phenomenon we're interested in: the absolute value model learns more easily when there are more neurons, but we wanted to keep the numbers small enough that it could be easily visualized.

A couple of notes on visualization: Since we're primarily interested in understanding superposition and polysemantic neurons, we'll show a stacked weight plot of the absolute values of weights. The features are colored by superposition. To make the diagrams easier to read, neurons are faintly colored based on how polysemantic they are (as judged by eye based on the plots). Neuron order is sorted by the importance of the largest feature.

Much like we saw in the ReLU hidden layer models, these results demonstrate that activation functions, under the right circumstances, create a privileged basis and cause features to align with basis dimensions. In the dense regime, we end up with each neuron representing a single feature, and we can read feature values directly off of neuron activations.

However, once the features become sufficiently sparse, this model, too, uses superposition to represent more features than it has neurons. This result is notable because it demonstrates the ability of neural networks to perform computation even on data that is represented in superposition.One question you might ask is whether we can quantify the ability of superposition to enable extra computation by examining the loss. Unfortunately, we can't easily do this. Superposition occurs when we change the task, making it sparser. As a result, the losses of models with different amounts of superposition are not comparable – they're measuring the loss on different tasks! Remember that the model is required to use the hidden layer ReLU in order to compute an absolute value; gradient descent manages to find solutions that usefully approximate the computation even when each neuron encodes a mix of multiple features.

Focusing on the intermediate sparsity regimes, we find several additional qualitative behaviors that we find fascinatingly reminiscent of behavior that has been observed in real, full-scale neural networks:

To begin, we find that in some regimes, many of the model's neurons will encode pure features, but a subset of them will be highly polysemantic. This is similar to the phase change we saw earlier in the ReLU output model. However, in that case, the phase change was with respect to features, with more important features not being put in superposition. In this experiment, the neurons don't have any intrinsic importance, but we see that the neurons representing the most important features (on the left) tend to be monosemantic.

We find this to bear a suggestive resemblance to some previous work in vision models, which found some layers that contained "mostly pure" feature neurons, but with some neurons representing additional features on a different scale.

We also note that many neurons appear to be associated with a single "primary" feature – encoded by a relatively large weight – coupled with one or more "secondary" features encoded with smaller-magnitude weights to that neuron. If we were to observe the activations of such a neuron over a range of input examples, we would find that the largest activations of that neuron were all or nearly-all associated with the presence of the "primary" feature, but that the lower-magnitude activations were much more polysemantic.

Intriguingly, that description closely matches what researchers have found in previous work on language models – many neurons appear interpretable when we examine their strongest activations over a dataset, but can be shown on further investigation to activate for other meanings or patterns, often at a lower magnitude. While only suggestive, the ability of our toy model to reproduce these qualitative features of larger neural networks offers an exciting hint that these models are illuminating general phenomena.

The Asymmetric Superposition Motif

If neural networks can perform computation in superposition, a natural question is to ask how exactly they're doing so. What does that look like mechanically, in terms of the weights? In this subsection, we'll (mostly) work through one such model and see an interesting motif of asymmetric superposition. (We use the term "motif" in the sense of the original circuit thread, inspired by its use in systems biology .)

The model we're trying to understand is shown below on the left, visualized as a neuron weight stack plot, with features corresponding to colors. The model is only doing a limited amount of superposition, and many of the weights can be understood as simply implementing absolute value in the expected way.

However, there are a few neurons doing something else…

These other neurons implement two instances of asymmetric superposition and inhibition. Each instance consists of two neurons:

One neuron does asymmetric superposition. In normal superposition, one might store features with equal weights (eg. W=[1,-1]) and then have equal output weights (W=[1,1]). In asymmetric superposition, one stores the features with different magnitudes (eg. W=[2,-\frac{1}{2}]) and then has reciprocal output weights (eg. W=[\frac{1}{2}, 2]). This causes one feature to heavily interfere with the other, but avoid the other interfering with the first!

To avoid the consequences of that interference, the model has another neuron heavily inhibit the feature in the case where there would have been positive interference. This essentially converts positive interference (which could greatly increase the loss) into negative interference (which has limited consequences due to the output ReLU).

There are a few other weights this doesn't explain. (We believe they're effectively small conditional biases.) But this asymmetric superposition and inhibition pattern appears to be the primary story.

The Strategic Picture of Superposition

Although superposition is scientifically interesting, much of our interest comes from a pragmatic motivation: we believe that superposition is deeply connected to the challenge of using interpretability to make claims about the safety of AI systems. In particular, it is a clear challenge to the most promising path we see to be able to say that neural networks won't perform certain harmful behaviors or to catch "unknown unknowns" safety problems. This is because superposition is deeply linked to the ability to identify and enumerate over all features in a model, and the ability to enumerate over all features would be a powerful primitive for making claims about model behavior.

We begin this section by describing how "solving superposition" in a certain sense is equivalent to many strong interpretability properties which might be useful for safety. Next, we'll describe three high level strategies one might take to "solving superposition." Finally, we'll describe a few other additional strategic considerations.

Safety, Interpretability, & "Solving Superposition"

We'd like a way to have confidence that models will never do certain behaviors such as "deliberately deceive" or "manipulate." Today, it's unclear how one might show this, but we believe a promising tool would be the ability to identify and enumerate over all features. The ability to have a universal quantifier over the fundamental units of neural network computation is a significant step towards saying that certain types of circuits don't exist.Ultimately we want to say that a model doesn't implement some class of behaviors. Enumerating over all features makes it easy to say a feature doesn't exist (e.g. "there is no 'deceptive behavior' feature") but that isn't quite what we want. We expect models that need to represent the world to represent unsavory behaviors. But it may be possible to build more subtle claims such as "all 'deceptive behavior' features do not participate in circuits X, Y and Z." It also seems like a powerful tool for addressing "unknown unknowns", since it's a way that one can fully cover network behavior, in a sense.

How does this relate to superposition? It turns out that the ability to enumerate over features is deeply intertwined with superposition. One way to see this is to imagine a neural network with a privileged basis and without superposition (like the monosemantic neurons found in early InceptionV1, e.g. ): features would simply correspond to neurons, and you could enumerate over features by enumerating over neurons. Superposition also makes it harder to find interpretable directions in a model without a privileged basis. Without superposition, one could try to do something like the Gram–Schmidt process, progressively identifying interpretable directions and then removing them to make future features easier to identify. But with superposition, one can't simply remove a direction even if one knows that it is a feature direction. The connection also goes the other way: if one has the ability to enumerate over features, one can perform compressed sensing using the feature directions to (with high probability) "unfold" a superposition model's activations into those of a larger, non-superposition model.

For this reason, we'll call any method that gives us the ability to enumerate over features – and equivalently, unfold activations – a "solution to superposition". Any solution is on the table, from creating models that just don't have superposition, to identifying what directions correspond to features after the fact. We'll discuss the space of possibilities shortly.

We've motivated "solving superposition" in terms of feature enumeration, but it's worth noting that it's equivalent to (or necessary for) many other interpretability properties one might care about:

Decomposing Activation Space. The most fundamental challenge of any interpretability agenda is to defeat the curse of dimensionality. For mechanistic interpretability, this ultimately reduces to whether we can decompose activation space into independently understandable components, analogous to how computer program memory can be decomposed into variables. Identifying features is what allows us to decompose the model in terms of them; Describing Activations in Terms of Pure Features. One of the most obvious casualties of superposition is that we can't describe activations in terms of pure features. When features are relatively basis aligned, we can take an activation – say the activations for a dog head in a vision model – and decompose them into individual underlying features, like a floppy ear, short golden fur, and a snout. (See the "semantic dictionary" interface in Building Blocks .) Solving superposition would allow us to do this for every model; Understanding Weights (ie. Circuit Analysis). Neural network weights can typically only be understood when they're connecting together understandable features. All the circuit analysis seen in the original circuit thread (see especially ), was fundamentally only possible because the weights connected non-polysemantic neurons. We need to solve superposition for this to work in general; Even very basic approaches become perilous with superposition. It isn't just sophisticated approaches to interpretability which are harmed by superposition. Even very basic methods one might consider become unreliable. For example, if one is concerned about language models exhibiting manipulative behavior, one might ask if an input has a significant cosine similarity to the representations of other examples of deceptive behavior. Unfortunately, superposition means that cosine similarity has the potential to be misleading, since unrelated features start to be embedded with positive dot products to each other. However, if we solve superposition, this won't be an issue – either we'll have a model where features align with neurons, or a way to use compressed sensing to lift features to a space where they no longer have positive dot products

Three Ways Out

At a very high level, there seem to be three potential approaches to resolving superposition:

Create models without superposition; Find an overcomplete basis that describes how features are represented in models with superposition; Hybrid approaches in which one changes models, not resolving superposition, but making it easier for a second stage of analysis to find an overcomplete basis that describes it

Our sense is that all of these approaches are possible if one doesn't care about having a competitive model. For example, we believe it's possible to accomplish any of these for the toy models described in this paper. However, as one starts to consider serious neural networks, let alone modern large language models, all of these approaches begin to look very difficult. We'll outline the challenges we see for each approach in the following sections.

With that said, it's worth highlighting one bright spot before we focus on the challenges. You might have believed that superposition was something you could never fully get rid of, but that doesn't seem to be the case. All our results seem to suggest that superposition and polysemanticity are phases with sharp transitions. That is, there may exist a regime for every model where it has no superposition or polysemanticity. The question is largely whether the cost of getting rid of or otherwise resolving superposition is too high.

Approach 1: Creating Models Without Superposition

It's actually quite easy to get rid of superposition in the toy models described in this paper, albeit at the cost of a higher loss. Simply apply an L1 regularization term to the hidden layer activations (i.e. add \lambda ||h||_1 to the loss). This actually has a nice interpretation in terms of killing features below a certain importance threshold, especially if they're not basis aligned. Generalizing this to real neural networks isn't trivial, but we expect it can be done. (This approach would be similar to work attempting to use sparsity to encourage basis-aligned word embeddings .)

However, it seems likely that models are significantly benefitting from superposition. Roughly, the sparser features are, the more features can be squeezed in per neuron. And many features in language models seem very sparse! For example, language models know about individuals with only modest public presences, such as several of the authors of this paper. Presumably we only occur with frequency significantly less than one in a million tokens. As a result, it may be the case that superposition effectively makes models much bigger.

All of this paints a picture where getting rid of superposition may be fairly achievable, but doing so will have a large performance cost. For a model with a fixed number of neurons, superposition helps – potentially a lot.

But this is only true if the constraint is thought of in terms of neurons. That is, a superposition model with n neurons likely has the same performance as a significantly larger monosemantic model with kn neurons. But neurons aren't the fundamental constraint: flops are. In the most common model architectures, flops and neurons have a strict correspondence, but this doesn't have to be the case and it's much less clear that superposition is optimal in the broader space of possibilities.

One family of models which change the flop-neuron relationship are Mixture of Experts (MoE) models (see review ). The intuition is that most neurons are for specialized circumstances and don't need to activate most of the time. For example, German-specific neurons don't need to activate on French text. Harry Potter neurons don't need to activate on scientific papers. So MoE models organize neurons into blocks or experts, which only activate a small fraction of the time. This effectively allows the model to have k times more neurons for a similar flop budget, given the constraint that only 1/k of the neurons activate in a given example and that they must activate in a block. Put another way, MoE models can recover neuron sparsity as free flops, as long as the sparsity is organized in certain ways.

It's unclear how far this can be pushed, especially given difficult engineering constraints. But there's an obvious lower bound, which is likely too optimistic but is interesting to think about: what if models only expended flops on neuron activations, and recovered the compute of all non-activating neurons? In this world, it seems unlikely that superposition would be optimal: you could always split a polysemantic neuron into dedicated neurons for each feature with the same cost, except for the cases where there would have been interference that hurt the model anyways. Our preliminary investigations comparing various types of superposition in terms of "loss reduction per activation frequency" seem to suggest that superposition is not optimal on these terms, although it asymptotically becomes as good as dedicated feature dimensions. Another way to think of this is that superposition exploits a gap between the sparsity of neurons and the sparsity of the underlying features; MoE eats that same gap, and so we should expect MoE models to have less superposition.

To be clear, MoE models are already well studied, and we don't think this changes the capabilities case for them. (If anything, superposition offers a theory for why MoE models have not proven more effective for capabilities when the case for them seems so initially compelling!) But if one's goal is to create competitive models that don't have superposition, MoE models become interesting to think about. We don't necessarily think that they specifically are the right path forward – our goal here has been to use them as an example of why we think it remains plausible there may be ways to build competitive superposition-free models.

Approach 2: Finding an Overcomplete Basis

The opposite strategy of creating a superposition-free model is to take a regular model, which has superposition, and find an overcomplete basis describing how features are embedded after the fact. This appears to be a relatively standard sparse coding (or dictionary learning) problem, where we want to take the activations of neural network layers and find out which directions correspond to features.More formally, given a matrix H \sim [d,m] ~=~[h_0, h_1, …] of hidden layer activations h \sim [m] sampled over d stimuli, if we believe there are n underlying features, we can try to find matrices A\sim [d,n] and B \sim [n,m] such that A is sparse. This approach has been explored by some prior work .

The advantage of this is that we don't need to worry about whether we're damaging model performance. On the other hand, many other things are harder:

It’s no longer easy to know how many features you have to enumerate. A monosemantic model represents a feature per neuron, but when finding an overcomplete basis there’s an additional challenge of identifying how many features to use for it; Solutions are no longer integrated into the surface computational structure. Neural networks can be understood in terms of their surface structure – neurons, attention heads, etc – and virtual structure that implicitly emerge (e.g. virtual attention heads ). A model described by an overcomplete basis has "virtual neurons": there's a further gap between the surface and virtual structure

It's a different, major engineering challenge. Seriously attempting to solve superposition by applying sparse coding to real neural nets suggests a massive sparse coding problem. For truly large language models, one would be starting with something like a millions (neurons) by billions (tokens) matrix and then trying to do an extremely overcomplete factorization, perhaps trying to factor it to be a thousand or more times larger. This is a major engineering challenge which is different from the standard distributed training challenges ML labs are set up for; Interference is no longer pushing in your favor. If you try to train models without superposition, interference between features is pushing the training process to have less superposition. If you instead try to decode superposition after the fact, whatever amount of superposition is "baked in" by the training process and you don't have part of the objective pushing in your favor

Approach 3: Hybrid Approaches

In addition to approaches which address superposition purely at training time, or purely after the fact, it may be possible to take "hybrid approaches" which do a mixture. For example, even if one can't change models without superposition, it may be possible to produce models with less superposition, which are then easier to decode.In particular, it seems like we should expect to be able to reduce superposition at least a little bit with essentially no effect on performance, just by doing something like L1 regularization without any architectural changes. Note that models should have a level of superposition where the derivative of loss with respect to the amount of superposition is zero – otherwise, they'd use more or less superposition. As a result, there should be at least some margin within which we can reduce the amount of superposition without affecting model performance. Alternatively, it may be possible for architecture changes to make finding an overcomplete basis easier or more computationally tractable in large models, separately from trying to reduce superposition.

Additional Considerations

Phase Changes as Cause For Hope. Is totally getting rid of superposition a realistic hope? One could easily imagine a world where it can only be asymptotically reduced, and never fully eliminated. While the results in this paper seem to suggest that superposition is hard to get rid of because it's actually very useful, the upshot of it corresponding to a phase change is that there's a regime where it totally doesn't exist. If we can find a way to push models in the non-superposition regime, it seems likely it can be totally eliminated.

Any superposition-free model would be a powerful tool for research. We believe that most of the research risk is in whether one can make performant superposition free models, rather than whether it's possible to make superposition free models at all. Of course, ultimately, we need to make performant models. But a non-performant superposition free model could still be a very useful research tool for studying superposition in normal models. At present, it's challenging to study superposition in models because we have no ground truth for what the features are. (This is also the reason why the toy models described in this paper can be studied – we do know what the features are!) If we had a superposition-free model, we may be able to use it as a ground truth to study superposition in regular models.

Local bases are not enough. Earlier, when we considered the geometry of non-uniform superposition, we observed that models often form local orthogonal bases, where co-occurring features are orthogonal. This suggests a strategy for locally understanding models on sufficiently narrow sub-distributions. However, if our goal is to eventually make useful statements about the safety of models, we need mechanistic accounts that hold for the full distribution (and off distribution). Local bases seem unlikely to give this to us.

Discussion

To What Extent Does Superposition Exist in Real Models?

Why are we interested in toy models? We believe they are useful proxies for studying the superposition we suspect might exist in real neural networks. But how can we know if they're actually a useful toy model? Our best validation is whether their predictions are consistent with empirical observations regarding polysemanticity. To the best of our knowledge they are. In particular:

Polysemantic neurons exist. Polysemantic neurons form in our third model, just as they are observed in a wide range of neural networks; Neurons are sometimes "cleanly interpretable" and sometimes "polysemantic", often in the same layer. Our third model exhibits both polysemantic and non-polysemantic neurons, often at the same time. This is analogous to how real neural networks often have a mixture of polysemantic and non-polysemantic neurons in the same layer; InceptionV1 has more polysemantic neurons in later layers. Empirically, the fraction of neurons which are polysemantic in InceptionV1 increases with depth. One natural explanation is that as features become higher-level the stimuli they detect become rarer and thus sparser (for example, in vision, a high-level floppy ear feature is less common than a low-level Gabor filter's edge). A major prediction of our model is that superposition and polysemanticity increase as sparsity increases; Early Transformer MLP neurons are extremely polysemantic. Our experience is that neurons in the first MLP layer in Transformer language models are often extremely polysemantic. If the goal of the first MLP layer is to distinguish between different interpretations of the same token (eg. "die" in English vs German vs Dutch vs Afrikaans), such features would be very sparse and our toy model would predict lots of polysemanticity

This doesn't mean that everything about our toy model reflects real neural networks. Our intuition is that some of the phenomena we observe (superposition, monosemantic vs polysemantic neurons, perhaps the relationship to adversarial examples) are likely to generalize, while other phenomena (especially the geometry and learning dynamics results) are much more uncertain.

Open Questions

This paper has shown that the superposition hypothesis is true in certain toy models. But if anything, we're left with many more questions about it than we had at the start. In this final section, we review some of the questions which strike us as most important: what do we know, and would we like for future work to clarify?

Is there a statistical test for catching superposition?; How can we control whether superposition and polysemanticity occur? Put another way, can we change the phase diagram such that features don't fall into the superposition regime? Pragmatically, this seems like the most important question. L1 regularization of activations, adversarial training, and changing the activation function all seem promising; Are there any models of superposition which have a closed-form solution? Saxe et al. demonstrate that it's possible to create nice closed-form solutions for linear neural networks. We made some progress towards this for the n=2; m=1 ReLU output model (and Tom McGrath makes further progress in his comment), but it would be nice to solve this more generally; How realistic are these toy models? To what extent do they capture the important properties of real models with respect to superposition? How can we tell?; Can we estimate the feature importance curve or feature sparsity curve of real models? If one takes our toy models seriously, the most important properties for understanding the problem are the feature importance and sparsity curves. Is there a way we can estimate them for real models? (Likely, this would involve training models of varying sizes or amounts of regularization, observing the loss and neuron sparsities, and trying to infer something.); Should we expect superposition to go away if we just scale enough? What assumptions about the feature importance curve and sparsity would need to be true for that to be the case? Alternatively, should we expect superposition to remain a constant fraction of represented features, or even to increase as we scale?; Are we measuring the maximally principled things? For example, what is the most principled definition of superposition / polysemanticity?; How important are polysemantic neurons? If X% of the model is interpretable neurons and 1-X% are polysemantic, how much should we believe we understand from understanding the X% interpretable neurons? (See also the "feature packing principle" suggested above.); How many features should we expect to be stored in superposition? This was briefly discussed in the previous section. It seems like results from compressed sensing should be able to give us useful upper-bounds, but it would be nice to have a clearer understanding – and perhaps tighter bounds!; Does the apparent phase change we observe in features/neurons have any connection to phase changes in compressed sensing?; How does superposition relate to non-robust features? An interesting paper by Gabriel Goh (archive.org backup) explores features in a linear model in terms of the principal components of the data. It focuses on a trade off between "usefulness" and "robustness" in the principal component features, but it seems like one could also relate it to the interpretability of features. How much would this perspective change if one believed the superposition hypothesis – could it be that the useful, non-robust features are an artifact of superposition?; To what extent can neural networks "do useful computation" on features in superposition? Is the absolute value problem representative of computation in superposition generally, or idiosyncratic? What class of computation is amenable to being performed in superposition? Does it require a sparse structure to the computation?; How does superposition change if features are not independent? Can superposition pack features more efficiently if they are anti-correlated?; Can models effectively use nonlinear representations? We suspect models will tend not to use them, but further experimentation could provide good evidence. See the appendix on nonlinear compression. For example investigating the representations used by autoencoders with multi-layer encoders and decoders with really small bottlenecks on random uncorrelated data

Related Work

Interpretable Features

Our work is inspired by research exploring the features that naturally occur in neural networks. Many models form at least some interpretable features. Word embeddings have semantic directions (see ). There is evidence of interpretable neurons in RNNs (e.g. ), convolutional neural networks (see generally e.g. ; individual neuron families ), and in some limited cases, transformer language models (see detailed discussion in our previous paper). However this work has also found many "polysemantic" neurons which are not interpretable as a single concept .

Superposition

The earliest reference to superposition in artificial neural networks that we're aware of is Arora et al.'s work , who suggest that the word embeddings of words with multiple different word senses may be superpositions of the vectors for the distinct meanings. Arora extend this idea to there being many sparse "atoms of discourse" in superposition, an idea which was generalized to other kinds of embedding vectors and explored in more detail by Goh .

In parallel with this, investigations of individual neurons in models with privileged bases were beginning to grapple with "polysemantic" neurons which respond to unrelated inputs . A natural hypothesis was that these polysemantic neurons are disambiguated by the combined activation of other neurons. This line of thinking eventually became the "superposition hypothesis" for circuits .

Separate from all of this, Cheung et al. explore a slightly different idea one might describe as "model level" superposition: can neural network parameters represent multiple completely independent models? Their investigation is motivated by catastrophic forgetting, but seems quite related to the questions investigated in this paper. Model level superposition can be seen as feature level superposition for highly correlated sets of features, similar to the "almost orthogonal bases" experiment we considered above.

Disentanglement

The goal of learning disentangled representations arises from Bengio et al.'s influential position paper on representation learning : "we would like our representations to disentangle the factors of variation… to learn representations that separate the various explanatory sources." Since then, a literature has developed motivated by this goal, tending to focus on creating generative models which separate out major factors of variation in their latent spaces. This research touches on questions related to superposition, but is also quite different in a number of ways.

Concretely, disentanglement research often explores whether one can train a VAE or GAN where basis dimensions correspond to the major features one might use to describe the problem (e.g. rotation, lighting, gender… as relevant). Early work often focused on semi-supervised approaches where the features were known in advance, but fully unsupervised approaches started to develop around 2016 .

Put another way, the goal of disentanglement might be described as imposing a strong privileged basis on representations which are rotationally invariant by default. This helps get at ways in which the questions of polysemanticity and superposition are a bit different from disentanglement. Consider that when we deal with neurons, rather than embeddings, we have a privileged basis by default. It varies by model, but many neurons just cleanly respond to features. This means that polysemanticity arises as a kind of anomalous behavior, and superposition arises as a hypothesis for explaining it. The question then isn't how to impose a privileged basis, but how to remove superposition as a fundamental problem to accessing features.

Of course, if the superposition hypothesis is true, there are still a number of connections to disentanglement. On the one hand, it seems likely superposition occurs in the latent spaces of generative models, even though that isn't an area we've investigated. If so, it may be that superposition is a major reason why disentanglement is difficult. Superposition may allow generative models to be much more effective than they would otherwise be without. Put another way, disentanglement often assumes a small number of important latent variables to explain the data. There are clearly examples of such variables, like the orientation of objects – but what if a large number of sparse, rare, individually unimportant features are collectively very important? Superposition would be the natural way for models to represent this.A more subtle issue is that GANs and VAEs often assume that their latent space is Gaussianly distributed. Sparse latent variables are very non-Gaussian, but the central limit theorem means that the superposition of many such variables will gradually look more Gaussian. So the latent spaces of some generative models may in fact force models to use superposition! On the other hand, one could imagine ideas from the disentanglement literature being useful in creating architectures that resist superposition by creating an even more strongly privileged basis.

Compressed Sensing

The toy problems we consider are quite similar to the problems considered in the field of compressed sensing, which is also known as compressive sensing and sparse recovery. However, there are some important differences:

Compressed sensing recovers vectors by solving an optimization problem using general techniques, while our toy model must use a neural network layer. Compressed sensing algorithms are, in principle, much more powerful than our toy model; Compressed sensing works using the number of non-zero entries as the measure of sparsity, while we use the probability that each dimension is zero as the sparsity. These are not wholly unrelated: concentration of measure implies that our vectors have a bounded number of non-zero entries with high probability; Compressed sensing requires that the embedding matrix (usually called the measurement matrix) have a certain “incoherent” structure such as the restricted isometry property or nullspace property . Our toy model learns the embedding matrix, and will often simply ignore many input dimensions to make others easier to recover; Features in our toy model have different "importances", which means the model will often prefer to be able to recover “important” features more accurately, at the cost of not being able to recover “less important” features at all

In general, our toy model is solving a similar problem using less powerful methods than compressed sensing algorithms, especially because the computational model is so much more restricted (to just a single linear transformation and a non-linearity) compared to the arbitrary computation that might be used by a compressed sensing algorithm.

As a result, compressed sensing lower bounds—which give lower bounds on the dimension of the embedding such that recovery is still possible—can be interpreted as giving an upper bound on the amount of superposition in our toy model. In particular, in various compressed sensing settings, one can recover an n-dimensional k-sparse vector from an m dimensional projection if and only if m = \Omega(k \log (n/k)) . While the connection is not entirely straightforward, we apply one such result to the toy model in the appendix.

At first, this bound appears to allow a number of features that is exponential in m to be packed into the m-dimensional embedding space. However, in our setting, the integer k for which all vectors have at most k non-zero entries is determined by the fixed density parameter S as k = O((1 - S)n). As a result, our bound is actually m = \Omega(-n (1 - S) \log(1 - S)). Therefore, the number of features is linear in m but modulated by the sparsity. Note that this has a nice information-theoretic interpretation: \log(1 - S) is the surprisal of a given dimension being non-zero, and is multiplied by the expected number of non-zeros. This is good news if we are hoping to eliminate superposition as a phenomenon! However, these bounds also allow for the amount of superposition to increase dramatically with sparsity – hopefully this is an artifact of the techniques in the proofs and not an inherent barrier to reducing or eliminating superposition.

A striking parallel between our toy model and compressed sensing is the existence of phase changes.Note that in the compressed sensing case, the phase transition is in the limit as the number of dimensions becomes large – for finite-dimensional spaces, the transition is fast but not discontinuous. In compressed sensing, if one considers a two-dimensional space defined by the sparsity and dimensionality of the vectors, there are sharp phase changes where the vector can almost surely be recovered in one regime and almost surely not in the other . It isn't immediately obvious how to connect these phase changes in compressed sensing – which apply to recovery of the entire vector, rather than one particular component – to the phase changes we observe in features and neurons. But the parallel is suspicious.

Another interesting line of work has tried to build useful sparse recovery algorithms using neural networks . While we find it useful for analysis purposes to view the toy model as a sparse recovery algorithm, so that we may apply sparse recovery lower bounds, we do not expect that the toy model is useful for the problem of sparse recovery. However, there may be an exciting opportunity to relate our understanding of the phenomenon of superposition to these and other techniques.

Sparse Coding and Dictionary Learning

Sparse Coding studies the problem of finding a sparse representation of dense data. One can think of it as being like compressed sensing, except the matrix projecting sparse vectors into the lower dimensional space is also unknown. This topic goes by many different names including sparse coding (most common in neuroscience), dictionary learning (in computer science), and sparse frame design (in mathematics). For a general introduction, we refer readers to a textbook by Michael Elad .

Classic sparse coding algorithms take an expectation-maximization approach (this includes Olshausen et al's early work , the MOD algorithm , and the k-SVD algorithm ). More recently, new methods based on gradient descent and autoencoders have begun building on these ideas .

From our perspective, sparse coding is interesting because it's probably the most natural mathematical formulation of trying to "solve superposition" by discovering which directions correspond to features.Interestingly, this is the reverse of how sparse coding is typically thought of in neuroscience. Neuroscience often thinks of biological neurons as sparse coding their inputs, whereas we're interested in applying it the opposite direction, to find features in superposition over neurons. But can we actually use these methods to solve superposition in practice? Previous work has attempted to use sparse coding to find sparse structure . More recently, research by Sharkey et al following up on the original publication of this paper has had preliminary success in extracting features out of superposition in toy models using a sparse autoencoder. In general, we're only in the very preliminary investigations of using sparse coding and dictionary learning in this way, but the situation seems quite optimistic. See the section Approach 2: Finding an Overcomplete Basis for more discussion.

Theories of Neural Coding and Representation

One challenge in comparing our work with the neuroscience literature is that a “distributed representation” seems to mean different things. Consider an overly-simplified example of a population of neurons, each taking a binary value of active or inactive, and a stimulus set of sixteen items: four shapes, with four colors (example borrowed from ). A “local code” would be one with a “red triangle” neuron, a “red square” neuron, and so on. In what sense could the representation be made more “distributed”? One sense is by representing independent features separately — e.g. four “shape” neurons and four “color” neurons. A second sense is by representing more items than neurons — i.e. using a binary code over four neurons to encode 2^4 = 16 stimuli. In our framework, these senses correspond to decomposability (representing stimuli as compositions of independent features) and superposition (representing more features than neurons, at cost of interference if features co-occur).

Decomposability doesn’t necessarily mean each feature gets its own neuron. Instead, it could be that each feature corresponds to a “direction in activation-space”We haven’t encountered a specific term in the distributed coding literature that corresponds to this hypothesis specifically, although the idea of a “direction in activation-space” is common in the literature, which may be due to ignorance on our part. We call this hypothesis linearity., given scalar “activations” (which in biological neurons would be firing rate). Then, only if there is a privileged basis, “feature neurons” are incentivized to develop. In biological neurons, metabolic considerations are often hypothesized to induce a privileged basis, and thus a “sparse code”. This would be expected if the nervous system’s energy expenditure increases linearly or sublinearly with firing rate.Experimental evidence seems to support this Additionally, neurons are the units by which biological neural networks can implement non-linear transformations, so if a feature needs to be non-linearly transformed, a “feature neuron” is a good way to achieve that.

Any decomposable linear code that uses orthogonal feature vectors is functionally equivalent from the viewpoint of a linear readout. So, a code can both be “maximally distributed” — in the sense that every neuron participates in representing every input, making each neuron extremely polysemantic — and also have no more features than it has dimensions. In this conception, it’s clear that a code can be fully “distributed” and also have no superposition.

A notable difference between our work, and the neuroscience literature we have encountered, is that we consider as a central concept the likelihood that features co-occur with some probability.A related, but different, concept in the neuroscience literature is the “binding problem” in which e.g. a red triangle is a co-occurrence of exactly one shape and exactly one color, which is not a representational challenge, but a binding problem arises if a decomposed code needs to represent simultaneously also a blue square — which shape feature goes with which color feature? Our work does not engage with the binding question, merely treating this as a co-occurrence of “blue”, “red”, “triangle”, and “square”. A “maximally-dense distributed code” makes the most sense in the case where items never co-occur; if the network only needs to represent one item at a time, it can tolerate a very extreme degree of superposition. By contrast, a network that could plausibly need to represent all the items at once can do so without interference between the items if it uses a code with no superposition. One example of high feature co-occurrence could be encoding spatial frequency in a receptive field; these visual neurons need to be able to represent white noise, which has energy at all frequencies. An example of limited co-occurrence could be a motor “reach” task to discrete targets, far enough apart that only one can be reached at a time.

One hypothesis in neuroscience is that highly compressed representations might have an important use in long-range communication between brain areas. Under this theory, sparse representations are used within a brain area to do computation, and then are compressed for transmission across a small number of axons. Our experiments with the absolute value toy model shows that networks can do useful computation even under a code with a moderate degree of superposition. This suggests that all neural codes, not just those used for efficient communication, could plausibly be “compressed” to some degree; the regional code might not necessarily need to be decompressed to a fully sparse one.

It's worth noting that the term "distributed representation" is also used in deep learning, and has the same ambiguities of meaning there. Our sense is that some influential early works (e.g. ) may have primarily meant the "independent features are represented independently" decomposability sense, but we believe that other work intends to suggest something similar to what we call superposition.

Additional Connections

After publishing the original version of this paper, a number of readers generously brought to our attention additional connections to prior work. We don't have a sufficiently deep understanding of this work to offer a detailed review, but we offer a brief overview below:

Vector Symbolic Architectures and Hyperdimensional Computing (see reviews ) are models from theoretical neuroscience of how neural systems can manipulate symbols. Many of the core ideas of how quasi-orthogonal vectors and the "blessings of dimensionality" enable computation are closely related to our notions of superposition; Frames (see review ) are a generalization of the idea of a mathematical basis. The way superposition encodes features in lower dimensional spaces might be seen as frames, at least in some cases. In particular, the "Mercedes-Benz Frame" is equivalent to the triangular geometry superposition we sometimes observe; Although we discuss compressed sensing and sparse coding above, it's worth noting that this only scratches the surface of research on how sparse vectors can be encoded in lower dimensional dense vectors, and there's a large body of additional work not captured by these topics

Comments & Replications

Inspired by the original Circuits Thread and Distill's Discussion Article experiment, the authors invited several external researchers who we had previously discussed our preliminary results with to comment on this work. Their comments are included below.

Replication & Forthcoming Paper

Kshitij Sachan is a research intern at Redwood Research.

Redwood Research has been working on toy models of polysemanticity, inspired by Anthropic's work. We plan to separately publish our results, and during our research we replicated many of the experiments in this paper. Specifically, we replicated all plots in the Demonstrating Superposition and Superposition as a Phase Change sections (visualizations of the relu models with different sparsities and the phase diagrams) as well as the plot in The Geometry of Superposition – Uniform Superposition. We found the phase diagrams look quite different depending on the activation function, suggesting that in this toy model some activation functions induce more polysemanticity than others.

Original Authors' Response: Redwood's further analysis of the superposition phase change significantly advanced our own understanding of the issue – we're very excited for their analysis to be shared with the world. We also appreciate the independent replication of our basic results.

Update: The research by Redwood mentioned in the previous comment, Polysemanticity and Capacity in Neural Networks (Alignment Forum, Arxiv) is out! They study a slightly different toy model, and get some really interesting results. Highlights include analytical traction on understanding a variant of the toy model, understanding superposition in terms of constrained optimization, and analysis of the role different activation functions play.

Replication & Further Results

Tom McGrath is a research scientist at DeepMind.

The results in this paper are an important contribution – they really further our theoretical understanding of a phenomenon that may be central to interpretability research and understanding network representations more generally. It’s surprising that such simple settings can produce these rich phenomena. We’ve reproduced the experiments in the Demonstrating Superposition and Superposition as a Phase Change sections and have a minor additional result to contribute.

It is possible to exactly solve the expected loss for the n=2, m=1 case of the basic ReLU output toy model (ignoring bias terms). The derivation is mathematically simple but somewhat long-winded: the ‘tricks’ are to (1) represent the sparse portion of the input distribution with delta functions, and (2) replace the ReLu with a restriction of the domain of integration:

\int_D \text{ReLU}(f(x)) dx = \int_{D \cap f(x)>0} f(x) dx

Making this substitution renders the integral analytically tractable, which allows us to plot the full loss surface and solve for the loss minima directly. We show some example loss surfaces below:

Although many of these loss surfaces (Figure 1a, 1b) have minima qualitatively similar to one of the network weights used in the section Superposition as a Phase Change, we also find a new phase where W_1\simeq W_2 \simeq \frac{1}{\sqrt{2}}: weights are similar rather than antipodal. This ‘confused feature’ regime occurs when sparsity is low and both features are important (Figure 1c). (This is slightly similar to the behavior described in The Geometry of Superposition – Collapsing of Correlated Features, but occurs without the features being correlated!) Further, although the solutions we find are often qualitatively similar to the weights used in Superposition as a Phase Change, they can be quantitatively different, as Figure 1a shows. The transition from Figure 1a to Figure 1b is continuous: the minima moves smoothly in weight space as the degree of sparsity alters. This explains the ‘blurry’ region around the triple point in the phase diagram.

As Figure 1c shows, some combinations of sparsity and relative feature importance lead to loss surfaces with two minima (once the symmetry (W_1, W_2) \to (-W_1, -W_2) has been accounted for). If this pattern holds for larger values of n and m (and we see no reason why it would not) this could account for the Discrete "Energy Level" Jumps phenomenon as solutions hop between minima. In some cases (e.g. when parameters approach those needed for a phase transition) the global minimum can have a considerably smaller basin of attraction than local minima. The transition between the antipodal and confused-feature solutions appears to be discontinuous.

Original Authors' Response: This closed form analysis of the n=2, m=1 case is fascinating. We hadn't realized that W_1\simeq W_2 \simeq \frac{1}{\sqrt{2}} could be a solution without correlated features! The clarification of the "blurry behavior" and the observation about local minima are also very interesting. More generally, we're very grateful for the independent replication of our core results.

Replication

Jeffrey Wu and Dan Mossing are members of the Alignment team at OpenAI.

We are very excited about these toy models of polysemanticity. This work sits at a rare intersection of being plausibly very important for training more interpretable models and being very simple and elegant. The results have been surprisingly easy to replicate – we have reproduced (with very little fuss) plots similar to those in the Demonstrating Superposition – Basic Results, Geometry – Feature Dimensionality, and Learning Dynamics – Discrete "Energy Level" Jumps sections.

Original Authors' Response: We really appreciate this replication of our basic results. Some of our findings were quite surprising to us, and this gives us more confidence that they aren't the result of an idiosyncratic quirk or bug in our implementations.

Replication

Spencer Becker-Kahn is a senior research scholar at the Future of Humanity Institute and a SERI Machine Learning Alignment Theory Scholar.

After seeing preliminary results, I independently replicated some of the key graphs from Demonstrating Superposition – Basic Results and, using very small toy models, produced a series of plots consistent with the conceptual picture emerging in Geometry – Feature Dimensionality and Superposition and Learning Dynamics.

See also Twitter thread.

Engineering Monosemanticity in Toy Models

Adam Jermyn is an independent researcher focused on AI alignment and interpretability. He was previously a Research Fellow at the Flatiron Institute’s Center for Computational Astrophysics. Evan Hubinger is a Research Fellow at MIRI. Nicholas Schiefer is a member of the technical staff at Anthropic and an author of the original paper.

Inspired by the results in this paper and the previous paper introducing the SoLU activation, we have been investigating whether changes to the model architecture or training process can reduce superposition in toy models. After replicating several of these results independently, we made the following extensions in that direction:

A modified toy model that tries to more accurately represent the non-toy case where sparse features are projected down into a non-sparse input to the model with fewer dimensions; A “many neuron” architecture that gives the model enough capacity to avoid polysemanticity entirely; A training and initialization method that actually makes these toy models monosemantic; A systematic exploration of the polysemantic neurons that tend to appear in these models, which guided our explorations of the training methods

At least in some limits, this suggests there may not be a price to be paid for monosemanticity. Detailed results can be found in our paper, Engineering Monosemanticity in Toy Models (Alignment Forum, ArXiV).

Fractional Dimensionality and "Pressure"

Tom Henighan and Chris Olah are authors of the original paper.

In the "Feature Dimensionality" section, we found that features organized into clean polytopes when there are more features than can be easily represented in the embedding dimensions.

We briefly investigated this further and found that the number of features competing to be represented significantly influences this phenomenon. Cleaner structure often seems to emerge when there is more "pressure" – more features competing to be represented. This is especially true at high sparsity levels. Additionally, training longer seems to also produce cleaner structure.

More investigation would be needed to really understand this phenomenon.

Replication

Marius Hobbhahn is a PhD student at the University of Tuebingen.

I replicated the “Basic Results” in Section 2 (“Demonstrating Superposition”) and all of Section 7 (“Superposition in a Privileged Basis”) of the “Toy Models of Superposition” paper. All of my findings are identical to the ones described in the paper. I replicated most findings in the follow-up “Superposition, Memorization, and Double Descent” paper.

The details of my replication can be found in my write-up “More Findings on Memorization and Double Descent”.

Extracting Features with Sparse Autoencoders

Lee Sharkey, Dan Braun, and Beren Millidge are researchers at Conjecture.

The results from this paper, and the strategic picture it paints, inspired our preliminary follow-up work that aimed to address some of the challenges described in the section titled 'Approach 2: Finding an overcomplete basis'.

Before studying the activations of real neural networks, where we're not sure what the 'ground truth' features are, we studied a toy example. We generated a set of toy ground truth features and created a dataset using sparse combinations of them. We found that a one-layer sparse autoencoder with an L_1 penalty on its hidden activations learned features that were almost identical to the ground truth features. This is encouraging because it indicates that a relatively simple method should be able to recover the features used by neural networks, inasmuch as they have them.

For the toy dataset, we knew how many ground truth features there were. But we ultimately want to count the features used by real neural networks, where the number of features is unknown. We explored three ways to count the features in the toy dataset: a) Counting dead neurons in the autoencoders; b) looking at autoencoder losses; and c) comparing the features learned by autoencoders of different sizes. We found indications that these methods might be suitable to count the number of features in superposition in real neural data.

We also applied our method to real activations from a small language model. Our initial, preliminary investigations led to inconclusive results, possibly resulting from having used autoencoders that were either too small or undertrained. At the time of writing, investigations are ongoing.

Linear Representation in Othello

Neel Nanda is an external researcher in mechanistic interpretability. This is a description of his blog post, Actually, Othello-GPT Has A Linear Emergent World Representation.

I describe a natural experiment of the linear representation hypothesis described in this paper – the idea that features correspond to directions in neural networks.

Background: Martin Wattenberg (an author on this paper) and colleagues recently found that, if you train transformer language model trained to predict the next token in synthetic Othello games (where each move is a randomly chosen legal move), it forms an emergent model of the board state (despite only being trained to predict the next move!). They showed that the state of the board (whether each cell was empty, black or white) could be recovered with high accuracy by a one hidden layer MLP probe. They further showed that you could use the world model to causally intervene on the model’s residual stream. By choosing another board state, and changing the residual stream (with gradient descent) such that the probe indicates that new board state, they caused the model to output legal moves in the new board state, even if the edited board state was impossible to reach via legal Othello play!

Pre-Registered Hypothesis: The probing and causal intervention together provided strong evidence that the model had learned to represent features corresponding to the state of each square on the board. Yet, notably, linear probes were not able to recover the board state. Since linear features should be recoverable with a one-layer probe, and the causal intervention suggests the model both computes and uses the board state, this seemed like significant evidence against the linear representation hypothesis.

However, Chris Olah (an author on this paper) argued the model might still be representing features linearly if it used a different set of features, and that the probe and causal intervention may be picking up on this different set of features. This created an informal pre-registered prediction of the hypothesis which was contrary to the evidence at the time.

Results: I independently came to the same conclusion as Chris and investigated the Othello playing model. I found that the model does form an emergent model of the board state that is linearly represented and can be extracted with a linear probe. But that as the model plays both black and white moves, the model represents the state of a cell as whether it has “my colour” vs “opponent’s colour”. Further, I found circumstantial evidence that these features are used by the model, as we can linearly intervene on the residual stream using the directions given by the probe to edit the represented board state, and the model plays legal moves in the new board state.

I consider these results to be notable, as the paper’s results provided evidence against the linear representation hypothesis and the hypothesis faced genuine risk of falsification. And the hypothesis made non-trivial predictions that were contrary to where the evidence pointed, but these turned out to be true. This is both a proof of concept that there are underlying principles of neural networks which have predictive power about models, and a natural experiment supporting the linear representation hypothesis.

I think that there is further work interpreting the Othello playing model that could test other hypotheses in this paper and our broader conceptual frameworks about neural networks and transformers, such as by looking for monosemantic vs superposed neurons in its MLP layers. The model is both complex enough to be interesting and expose principles of how transformers learn algorithms, yet the algorithmic nature of the task and existence of the probe suggests that finding circuits will be tractable. I elaborate on what I consider promising directions of future work in a follow-up post.

Leverage Score and Feature Dimensionality

Fred Zhang is a PhD student in the Theory Group of the EECS Department at UC Berkeley.

In the Geometry of Superposition section, the paper defines a notion of feature dimensionality,

D_i ~=~ \frac{||W_i||^2}{\sum_j (\hat{W_i} \cdot W_j)^2},

where W_i is the ith feature and \hat{W_i}=W_i / \| W_i \|_2 is its normalized version. For simplicity of notation, we assume that there are n features, each being a d-dimensional real vector.

Following this definition, the paper makes the remark that "Empirically, it seems that the dimensionality of all features add up to the number of embedding dimensions when the features are 'packed efficiently' in some sense." In this comment, I point out a natural, theoretical explanation of this observation. The argument is via the notion of leverage score in matrix approximation. I’ll define it first, then explain how it connects to feature dimensionality.

At a conceptual level, leverage score is a measure of the importance of a row in composing the row space of a matrix. For instance, if a row is orthogonal to all other rows, its leverage score is 1, meaning that it’s maximally important. This is natural, since removing it would decrease the rank of the matrix and completely change the row space. Formally, if W be an n by d matrix (think of it as tall and thin, so n > d), then the leverage score of ith row W_i is

\tau_i = \max_{x : \|x\|_2 = 1} \frac{\langle W_i , x \rangle^2}{\sum_{j=1}^n \langle W_j , x \rangle^2}.

Notice that the denominator term equals \| Wx \|_2^2 . Hence, the leverage score of W_i is the maximum contribution \langle W_i, x\rangle^2 can make towards || Wx ||^2 (over all possible directions x). This is a fractional number between 0 and 1. Also observe that this indeed has the nice property that if W_i is orthogonal to all other rows, \tau_i equals 1, because all except the ith term within the summation of the denominator is 0.

Three quick remarks on this definition:

It coincides with the notion of maximal data dimensionality in a comment by Chris Olah on Data Dimensionality of MNIST in the follow-up work of Superposition, Memorization, and Double Descent; There are other equivalent definitions of leverage score; see section 3.3 of here; Leverage score arises in numerical linear algebra. It is well-known that sampling rows proportional to their leverage score yields an almost square matrix that approximates the original, potentially very tall matrix. For a formal statement, see Theorem 17 of https://arxiv.org/abs/1411.4357

Returning to my main point, another nice fact about leverage score is they sum up to the rank of the matrix. In the tall and thin case above, they sum up to d (if the matrix is full-rank). Given that, it is natural this paper makes the empirical observation that the sum of D_i roughly equals the embedding dimension d, if the vectors are "efficiently packed". Specifically, one formal way of stating that they are packed efficiently is that they are roughly in isotropic position, that is, the variance \langle W_j, x\rangle^2 along any direction x is close to 1. In other words, the covariance structure of the embedding vectors is close to identity. (Geometrically, it means they are well spread out. This would be the case, for example, if the vectors form a regular simplex, just as it happens in the experiments of this paper.) Then in the above definition of leverage score, each summand within the denominator is 1 for any x. So to maximize the numerator, we just take x to be W_i itself—and this precisely recovers the definition of feature dimensionality D_i. Thus, in this well-spread-out case, these two notions are roughly the same, and they both sum up to d.

Code

We provide a notebook to reproduce some of the core diagrams in this article here. (It isn't comprehensive, since we needed to rewrite code for our experiments to run outside our codebase.) We provide a separate notebook for the theoretical phase change diagrams.

Note that the reproductions by other researchers mentioned in comments above were not based on this code, but are instead fully independent replications with clean code from the description in an early draft of this article.

Acknowledgments

We're extremely grateful to a number of colleagues across several organizations for their invaluable support in our writing of this paper.

Jeff Wu, Daniel Mossing, Tom McGrath, and Kshitij Sachan did independent replications of many of our experiments, greatly increasing our confidence in our results. Kshitij Sachan's and Tom McGrath's additional investigations and insightful questions both pushed us to clarify our understanding of the superposition phase change (both as reflected in this paper, and in further understanding which we learned from them not captured here). Buck Shlegeris, Adam Scherlis, and Adam Jermyn shared valuable insights into the mathematical nature of the toy problem and related work. Adam Jermyn also coined the term "virtual neurons."

Gabriel Goh, Neel Nanda, Vladimir Mikulik, and Nick Cammarata gave detailed feedback which improved the paper, in addition to being motivating. Alex Dimakis, Piotr Indyk, Dan Yamins generously took time to discuss these results with us and give advice on how they might connect to their area of expertise. Finally, we benefited from the feedback and comments of James Bradbury, Sebastian Farquhar, Shan Carter, Patrick Mineault, Alex Tamkin, Paul Christiano, Evan Hubinger, Ian McKenzie, and Sid Black. We're additionally grateful to Trenton Bricken and Manjari Narayan for referring us to valuable related work we originally missed. Thanks to Ken Kahn for typo corrections.

Finally, we're very grateful to all our colleagues at Anthropic for their advice and support: Daniela Amodei, Jack Clark, Tom Brown, Ben Mann, Nick Joseph, Danny Hernandez, Amanda Askell, Kamal Ndousse, Andy Jones, Timothy Telleen-Lawton, Anna Chen, Yuntao Bai, Jeffrey Ladish, Deep Ganguli, Liane Lovitt, Nova DasSarma, Jia Yuan Loke, Jackson Kernion, Tom Conerly, Scott Johnston, Jamie Kerr, Sheer El Showk, Stanislav Fort, Rebecca Raible, Saurav Kadavath, Rune Kvist, Jarrah Bloomfield, Eli Tran-Johnson, Rob Gilson, Guro Khundadze, Filipe Dobreira, Ethan Perez, Sam Bowman, Sam Ringer, Sebastian Conybeare, Jeeyoon Hyun, Michael Sellitto, Jared Mueller, Joshua Landau, Cameron McKinnon, Sandipan Kundu, Jasmine Brazilek, Da Yan, Robin Larson, Noemí Mercado, Anna Goldie, Azalia Mirhoseini, Jennifer Zhou, Erick Galankin, James Sully, Dustin Li, James Landis.

Author Contributions

Basic Results – The basic toy model results demonstrating the existence of superposition were done by Nelson Elhage and Chris Olah. Chris suggested the toy model and Nelson ran the experiments.

Phase Change – Chris Olah ran the empirical phase change experiments, with help from Nelson Elhage. Martin Wattenberg introduced the theoretical model where exact losses for specific weight configurations can be computed.

Geometry – The uniform superposition geometry results were discovered by Nelson Elhage and Nicholas Schiefer, with help from Chris Olah. Nelson discovered the original m/||W||_F^2 mysterious "stickiness". Chris introduced the definition of feature dimensionality. Nicholas and Nelson then investigated the polytopes that formed. As for non-uniform superposition, Martin Wattenberg performed the initial investigations of the resulting geometry, focusing on the behavior of correlated features. Chris extended this with an investigation of the role of relative feature importance and sparsity.

Learning Dynamics – Nelson Elhage discovered the "energy level jump" phenomenon, in collaboration with Nicholas Schiefer and Chris Olah. Martin Wattenberg discovered the "geometric transformations" phenomenon.

Adversarial Examples – Chris Olah and Catherine Olsson found evidence of a connection between superposition and adversarial examples.

Superposition with a Privileged Basis / Doing Computation – Chris Olah did the basic investigation of superposition in a privileged basis. Nelson Elhage, with help from Chris, investigated the "absolute value" model which provided a more principled demonstration of superposition and showed that computation could be done while in superposition. Nelson discovered the "asymmetric superposition" motif.

Theory – The theoretical picture articulated over the course of this paper (especially in the "mathematical understanding" section) was developed in conversations between all authors, but especially Chris Olah, Jared Kaplan, Martin Wattenberg, Nelson Elhage, Tristan Hume, Tom Henighan, Catherine Olsson, Nicholas Schiefer, Dawn Drain, Shauna Kravec, Roger Grosse, Robert Lasenby, and Sam McCandlish. Jared introduced the strategy of rewriting the loss by grouping terms with the number of active features. Both Jared and Martin independently noticed the value of investigating the n=2; m=1 case as the simplest case to understand. Nicholas and Dawn clarified our understanding of the connection to compressed sensing.

Strategic Picture – The strategic picture articulated in this paper – What does superposition mean for interpretability and safety? What would a suitable solution be? How might one solve it? – developed in extensive conversations between authors, and in particular Chris Olah, Tristan Hume, Nelson Elhage, Dario Amodei, Jared Kaplan. Nelson Elhage recognized the potential importance of "enumerative safety", further articulated by Dario. Tristan brainstormed extensively about ways one might solve superposition and pushed Chris on this topic.

Writing – The paper was primarily drafted by Chris Olah, with some sections by Nelson Elhage, Tristan Hume, Martin Wattenberg, and Catherine Olsson. All authors contributed to editing, with particularly significant contributions from Zac Hatfield Dodds, Robert Lasenby, Kipply Chen, and Roger Grosse.

Illustration – The paper was primarily illustrated by Chris Olah, with help from Tristan Hume, Nelson Elhage, and Catherine Olsson.

This paper focuses on the assumption that representations are linear. But what if models don’t use linear feature directions to represent information? What might such a thing concretely look like?

Neural networks have nonlinearities that make it theoretically possible to compress information even more compactly than a linear superposition. There are reasons we think models are unlikely to pervasively use nonlinear compression schemes:

The model needs to decompress things before it can compute with them naturally: Most of the computation in the model is linear, so this kind of compression is likely only worth it to save space in the residual stream across many layers before being decompressed to be computed with linearly again; They’re probably difficult to learn: Nonlinear compression schemes may require finely tuned approximations of discontinuities, and for the compression and decompression to line up, and may be difficult for gradient descent to learn; They probably take enough neurons that the benefit over superposition isn’t worth it:

Representing the piecewise linear functions in the simple example with ReLU neurons using the universal function approximation result that each line segment takes two neurons, would require 12 neurons per Z segment, so only starts to beat linear compression at a combined 36 neurons for the compression and decompression; This comparison has only one hidden dimension and dense features, which is somewhat of a degenerate case for superposition. Superposition is much more powerful for compression of sparse features in many dimensions. We suspect in large models the scaling is in favor of superposition, although this is just intuition and it’s possible that scaling nonlinear compression is competitive

Regardless of whether large models end up using nonlinear compression, it should be possible to view directions being used with nonlinear compression as linear feature directions and reverse engineer the computation being used for compression like any other circuit. If this kind of encoding is pervasive throughout the network then it may merit some kind of automated decoding. It shouldn’t pose a fundamental challenge to interpretability unless the model learns a scheme for doing complex computation while staying in a complicated nonlinear representation, which we suspect is unlikely.

To help provide intuition, the simplest example of what a nonlinear compression scheme might look like is compressing two [0,1) dimensions x and y into a single [0,1) dimension t:

t = \frac{\lfloor Zx \rfloor + y}{Z}

This works by quantizing the x dimension using some integer Z such that the floating point precision of t is split between x and y. This particular function needs the discontinuous floor function to compute, and the discontinuous fmod function to invert, but models can’t compute discontinuous functions. However it’s possible to replace the discontinuities with steep linear segments that are only some epsilon value wide.

We can compare the mean squared error loss on random uniform dense values of x and y and see that even with epsilons as large as 0.1 and Z values as small as 3 the nonlinear compression outperforms linear compression such as picking one of the dimensions or using the average:

Connection between compressed sensing lower bounds and the toy model

Here, we formalize the relationship between a compressed sensing lower bound and the toy model.

Let T(x) : \mathbb{R}^n \to \mathbb{R}^n be the complete toy model autoencoder defined by T(x) = \mathrm{ReLU}(W_2 W_1 x - b) for an m \times n matrix W_1 and an n \times m matrix W_2.

We derive the following theorem:

Theorem 1. Suppose that the toy model recovers all x with T(x) such that \|T(x) - x\|_2 \leq \varepsilon for sufficiently small \varepsilon and W_1 has the (\delta, k) restricted isometry property. The inner dimension of the projection matrix W is m = \Omega(k log(n/k)).

We prove this result by framing our toy model as a compressed sensing algorithm. The primary barrier to doing so is that our optimization only searches for vectors that are close in \ell_2 distance to the original vector and may not itself be exactly k-sparse. The following lemma resolves this concern through a denoising step:

Lemma 1. Suppose that we have a toy model T(x) with the properties in Theorem 1. Then there exists a compressed sensing algorithm f(y) : \mathbb{R}^m \to \mathbb{R}^n for the measurement matrix W_1.

Proof. We construct f(y) as follows. First, compute \tilde{x} = \mathrm{ReLU}(W_2 y - b), as in T(x). This produces the vector \tilde{x} = T(x) and so by supposition \|T(x) - x\|_2 \leq \varepsilon. Next, we threshold \tilde{x} to obtain \tilde{x}' by dropping all but its k largest entries. Lastly, we solve the optimization problem: \min_{x'} \|x' - \tilde{x}'\| subject to W_1 x' = y, which is convex because x' and \tilde{x}' have the same support. For sufficiently small \varepsilon (specifically, \varepsilon smaller than the (k + 1)th largest entry in x), both \tilde{x} and the nearest k-sparse vector to x have the same support, and so the convex optimization problem has a unique solution: the nearest k sparse vector to x. Therefore, f is a compressed sensing algorithm for W_1 with approximation factor 1. \qed.

Lastly, we use the deterministic compressed sensing lower bound of Do Ba, Indyk, Price, and Woodruff :

Theorem 2 (Corollary 3.1 in ). Given a k \times n matrix A with the restricted isometry property, a sparse recovery algorithm finds a k-sparse approximation \hat{x} of x \in \mathbb{R}^n from Ax such that

\|x - \hat{x}\|_1 \leq C(k) \min_{x', \|x'\|_0 \leq k} \|x - x'\|_1

for an approximation factor C(k). If C(k) = O(1), then a sparse recovery algorithm exists only if m = \Omega(k \log (n/k)).

Theorem 1 follows directly from Lemma 1 and Theorem 2.