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Claude 4 Opus (T) via @hdevalence.bsky.social
Thinking Summary

The user is asking me to expand on the bullet points in §A.1 of "Scaling and evaluating sparse autoencoders" paper. Let me find that section... Looking at §A.1 Initialization, there are 4 bullet points: 1. We initialize the bias bp​r​e to be the geometric median of a sample set of data points, following Bricken et al. [2023]. 2. We initialize the encoder directions parallel to the respective decoder directions, so that the corresponding latent read/write directions are the same. This is done only at initialization; we do not tie the parameters as in Cunningham et al. [2023]. This strategy is also presented in concurrent work [Conerly et al., 2024]. Directions are chosen uniformly randomly. 3. We scale decoder latent directions to be unit norm at initialization (and also after each training step), following Bricken et al. [2023]. 4. For baseline models we use torch default initialization for encoder magnitudes. For TopK models, we initialized the magnitude of the encoder such that the magnitude of reconstructed vectors match that of the inputs. However, in our ablations we find this has no effect or a weak negative effect (Figure 16). Note that the scaling factor has nontrivial interaction with n, and scales between Θ(1/√k) and Θ(1/k). This scheme has the advantage that is optimal at init in the infinite width limit. We did not try simpler schemes like scaling by Θ(1/√k). I'll need to reference the "Towards Monosemanticity" paper (Bricken et al. 2023) since it's mentioned and has been provided in the context.

Looking at the initialization details in §A.1 of "Scaling and evaluating sparse autoencoders," I'll expand each bullet point with context from the referenced papers:

Pre-bias Initialization (b_pre)

The pre-bias b_pre is initialized to the geometric median of a sample set of data points, following Bricken et al. [2023]. The geometric median is the point that minimizes the sum of distances to all data points in the sample - it's a robust central tendency measure that's less sensitive to outliers than the arithmetic mean. In the context of sparse autoencoders, this bias term serves to center the data before encoding. By subtracting this geometric median from input activations, the autoencoder can focus on learning deviations from a typical activation pattern rather than having to encode the baseline activation level. This is particularly important when working with neural network activations that may have non-zero baseline values across many dimensions.

The "Towards Monosemanticity" paper notes that they "tie the biases applied in the input and output, so the result is equivalent to subtracting a fixed bias from all activations." This centering helps ensure that the sparse features learned by the autoencoder represent meaningful deviations from typical behavior rather than just capturing baseline activation levels.

Parallel Encoder-Decoder Initialization

The encoder directions are initialized parallel to their corresponding decoder directions, meaning W_enc[i] is initialized proportional to W_dec[:,i]^T. This creates an initial configuration where each latent dimension has aligned "read" (encoder) and "write" (decoder) directions. Unlike some approaches that maintain this tie throughout training (parameter tying), this method only uses parallel initialization and then allows the encoder and decoder to diverge during training.

This initialization strategy has several advantages. First, it ensures that at initialization, activating a latent dimension through the encoder will produce a reconstruction in the same direction through the decoder, creating a sensible starting point. Second, it helps avoid initial reconstruction errors that might arise from randomly misaligned encoder-decoder pairs. The directions themselves are chosen uniformly at random on the unit sphere, ensuring good coverage of the activation space while maintaining the encoder-decoder alignment. This approach balances the benefits of structured initialization with the flexibility of allowing the encoder and decoder to specialize differently during training.

Decoder Normalization

The decoder weight vectors (columns of W_dec) are normalized to unit norm both at initialization and after every training step. This normalization serves multiple critical purposes in the sparse autoencoder architecture. From the "Towards Monosemanticity" paper, we learn that this normalization is particularly necessary for L1-regularized autoencoders because "otherwise the L1 loss can be gamed by making the latents arbitrarily small" while scaling up the decoder weights correspondingly.

Even for TopK autoencoders where L1 regularization isn't used, the authors found that decoder normalization still improves reconstruction MSE. This consistent scale for decoder directions makes the magnitude of latent activations directly interpretable - a latent with activation 2.0 contributes twice as much to the reconstruction as one with activation 1.0, regardless of which feature we're discussing. The normalization also helps with optimization dynamics, as noted in the "Towards Monosemanticity" paper's mention of gradient projection to "account for interaction between Adam and decoder normalization."

Encoder Magnitude Initialization

For baseline (ReLU) models, the encoder weights use PyTorch's default initialization (typically Kaiming/He initialization for layers followed by ReLU). However, for TopK models, the authors attempted a more sophisticated initialization where encoder magnitudes are scaled so that reconstructed vectors match the magnitude of input vectors. This involves a scaling factor that depends on both the number of active latents k and potentially the total number of latents n, with the scaling somewhere between Θ(1/√k) and Θ(1/k).

The theoretical motivation is that in the infinite-width limit, this initialization scheme would be optimal - the expected reconstruction would perfectly match the input magnitude. However, empirically the authors found this sophisticated initialization provided no benefit or even slightly hurt performance compared to simpler approaches. This highlights an important lesson in deep learning: theoretically motivated initialization schemes don't always translate to practical improvements, possibly because the optimization dynamics quickly move away from the initial point or because finite-width effects dominate. The authors note they didn't explore simpler schemes like uniformly scaling by Θ(1/√k), suggesting this remains an open area for investigation.

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Claude 4 Opus (T) via @hdevalence.bsky.social
Thinking Summary

The user is asking me to summarize the parts of the "ghost grads" notes that are relevant to §A.2 of the sparse autoencoders paper. Looking at §A.2, it describes an auxiliary loss (AuxK) that models reconstruction error using the top-k_aux dead latents, which is similar to "ghost grads". From the Anthropic Circuits Updates, I can see the evolution of ghost grads: January 2024 update: - Ghost grads is presented as an improvement on resampling for dead neurons - They calculate an additional loss term for dead neurons - Use exponential activation function instead of ReLU for dead neurons - Scale output to have L2 norm that's 1/2 of the autoencoder residual - This helps dead neurons get gradient signal to explain the autoencoder residual - Results in very few dead neurons and performs as well or better than neuron resampling - Roughly doubles compute requirements but worth it for large autoencoders where >50% neurons can be dead February 2024 update: - They had a bug that applied ghost grads to all neurons at start of training - The bug actually improved training loss - Ghost grads helped on 1L models but showed little difference on larger models - They don't fully understand why/when ghost grads helps March 2024 update: - With other improvements, ghost grads no longer decreases training loss even on 1L models - Training loss is roughly equal, not justifying the increased FLOPs - Some evidence their implementation causes loss spikes - They've essentially abandoned ghost grads at this point The OpenAI paper's AuxK loss seems to be inspired by ghost grads but is a simpler implementation that just models reconstruction error using dead latents.

Based on the embedded documents, here's a summary of the ghost grads evolution relevant to §A.2:

January 2024 (Initial Ghost Grads):

  • Ghost grads was introduced as an improvement over neuron resampling for handling dead neurons
  • It adds an auxiliary loss term that:
    • Takes dead neurons (not activated for 2M tokens)
    • Replaces ReLU with exponential activation
    • Scales output to L2 norm = 0.5 × autoencoder residual
    • Computes MSE between this and the residual
  • Resulted in very few dead neurons and performed as well/better than resampling
  • Roughly doubled compute requirements

February 2024 (Bug Discovery):

  • Found a bug that applied ghost grads to ALL neurons initially
  • Surprisingly, this bug improved training loss
  • Ghost grads worked well on 1-layer models but showed little benefit on larger models
  • Team expressed uncertainty about why/when it helps

March 2024 (Abandonment):

  • With other optimization improvements, ghost grads no longer improved training loss
  • Didn't justify the 2x compute cost
  • Evidence it caused loss spikes in their setup
  • Effectively abandoned the approach

Relevance to §A.2: The OpenAI paper's AuxK loss appears to be inspired by ghost grads but is a simpler implementation - it just models reconstruction error using top-k_aux dead latents without the exponential activation or other complexities. The paper notes AuxK "occasionally NaNs at large scale," which might relate to the loss spikes Anthropic observed.

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Claude 4 Sonnet (T) via @hdevalence.bsky.social
Thinking Summary

The user is asking about the definitions of L(C) and L(N) from the paper on scaling sparse autoencoders. Let me look through the paper to find these definitions. Looking at section 3.1, I can see two approaches for choosing autoencoder size: 3.1.1 Training to compute-MSE frontier (L(C)) This follows Lindsey et al. (2024) and trains autoencoders to the optimal MSE given the available compute, disregarding convergence. This method was introduced for pre-training language models. The paper finds that MSE follows a power law L(C) of compute. 3.1.2 Training to convergence (L(N)) This looks at training autoencoders to convergence (within some ε). This gives a bound on the best possible reconstruction achievable by their training method if they disregard compute efficiency. So: - L(C) = Loss as a function of Compute - training to optimal MSE given fixed compute budget - L(N) = Loss as a function of Number of latents - training to convergence The paper mentions that in practice, they would ideally train to some intermediate token budget between L(N) and L(C).

Looking at Section 3.1 of the paper, here are the definitions:

L(C) - Training to compute-MSE frontier:

  • This trains autoencoders to achieve the optimal MSE given a fixed compute budget, disregarding convergence
  • Following the approach from language model scaling laws (Kaplan et al., 2020; Hoffmann et al., 2022)
  • The loss L(C) follows a power law as a function of compute C
  • Focuses on compute efficiency rather than fully training the model

L(N) - Training to convergence:

  • This trains autoencoders to convergence (within some ε), regardless of compute cost
  • Gives a bound on the best possible reconstruction achievable by their training method
  • The loss L(N) is measured as a function of the number of latents N after full convergence
  • Provides the theoretical best performance but is computationally expensive

The key distinction is:

  • L(C): Optimize for compute efficiency - "What's the best loss I can get with X amount of compute?"
  • L(N): Optimize for final performance - "What's the best loss I can eventually achieve with N latents if I train long enough?"

The paper notes that in practice, they would ideally train to some intermediate token budget between these two extremes. They also find that the optimal learning rate for L(N) is about 4x smaller than for L(C).

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Claude 4 Sonnet (T) via @hdevalence.bsky.social
Thinking Summary

That's a really good observation. Let me think through this: The user is pointing out that in Section A.4 (batch size), the paper mentions that "batch size affects L(C) substantially, we find that the L(N) loss does not depend strongly on batch size when optimization hyperparameters are set appropriately." This is indeed a strong indicator that L(N) is measuring something more fundamental about the autoencoder's capabilities, while L(C) is more of an artifact of the optimization process. If L(C) varies with batch size, that suggests it's capturing optimization dynamics - how efficiently you can use your compute budget depends on your batch size, learning rate schedule, etc. But if L(N) is invariant to batch size (when hyperparameters are set appropriately), that suggests it's measuring the actual representational capacity of the autoencoder architecture. This supports the paper's argument in Section 3.1.1 vs 3.1.2 where they say that L(C) is "arguably unprincipled for autoencoder training" because: 1. The latents are the important artifact (not reconstruction predictions like in language models) 2. Comparing MSE across different n is not fair because larger n has a looser information bottleneck The batch size invariance of L(N) suggests it's measuring something intrinsic to the model architecture and task, rather than optimization artifacts.

Excellent observation! You're absolutely right. The fact that batch size affects L(C) substantially but L(N) remains invariant (when hyperparameters are properly set) is strong evidence that L(N) captures something more fundamental about the autoencoder's representational capacity.

This batch size sensitivity difference suggests:

L(C) captures optimization artifacts:

  • Sensitive to batch size, learning rate schedules, optimization dynamics
  • Measures "how efficiently can I use my compute budget" rather than "how well can this architecture represent the data"
  • More about the optimization process than the model's intrinsic capabilities

L(N) captures intrinsic model properties:

  • Invariant to batch size (when hyperparameters are appropriate)
  • Measures the actual representational limit of the architecture
  • Gets at the fundamental question: "What's the best this model architecture can do on this task?"

This strongly supports the paper's argument that L(N) is the more principled choice for autoencoder evaluation. The authors note that comparing MSE across different n under L(C) isn't fair because "the latents have a looser information bottleneck with larger n, so lower MSE is more easily achieved."

The batch size invariance of L(N) suggests it's measuring something closer to the true scaling laws of representation learning, while L(C) conflates model capacity with optimization efficiency. This makes L(N) much more scientifically meaningful for understanding how autoencoder performance scales with model size.

It's a nice empirical validation of their theoretical argument about which metric is more principled.

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Claude 4 Opus (T) via @hdevalence.bsky.social
Thinking Summary

The user is asking about a specific part of section A.6 in the linked paper. Let me find that section in the content snapshot. Looking at section A.6: "In theory, the bpre lr should be scaled linearly with the norm of the data to make the autoencoder completely invariant to input scale. In practice, we find it to tolerate an extremely wide range of values with little impact on quality." So they're asking about why the pre-bias learning rate (bpre lr) should theoretically scale linearly with the norm of the data to maintain scale invariance. This is about optimization and how gradient updates work. If your input data is scaled by a factor $s$, and you want the autoencoder's behavior to be invariant to this scaling, you need to adjust the learning rates appropriately. Let me think through this: 1. If we scale input $x$ by factor $s$, we get $sx$ 2. The pre-bias term $b_{pre}$ is subtracted from the input: $(x - b_{pre})$ becomes $(sx - b_{pre})$ 3. For scale invariance, we'd want the effective behavior to remain the same 4. The gradient with respect to $b_{pre}$ will scale with the input scale 5. To compensate, the learning rate needs to scale accordingly Let me work through the math more carefully...

Looking at section A.6, they mention that "in theory, the bpreb_{pre} lr should be scaled linearly with the norm of the data to make the autoencoder completely invariant to input scale."

This relates to how gradients flow through the pre-bias term during optimization. Here's the intuition:

In their architecture, the encoder computes: z=TopK(Wenc(x−bpre))z = \text{TopK}(W_{enc}(x - b_{pre}))

If you scale your input data by a factor ss (i.e., x→sxx \rightarrow sx), then:

  • The gradient of the loss with respect to bpreb_{pre} will also scale by approximately ss
  • This is because ∂L∂bpre\frac{\partial \mathcal{L}}{\partial b_{pre}} involves terms that depend on the scale of (x−bpre)(x - b_{pre})

To maintain the same effective optimization dynamics regardless of input scale, you'd want the parameter updates to have the same relative effect. Since the gradient scales with ss, the learning rate should scale with 1/s1/s to cancel this out.

More precisely, if the typical norm of your data is ∥x∥\|x\|, then scaling the learning rate linearly with ∥x∥\|x\| ensures that the optimization remains invariant to rescaling all inputs by a constant factor.

This is a common consideration in deep learning - similar reasoning applies to why we often use normalized initialization schemes and why normalization layers can help with optimization. The interesting note here is that they found it works well in practice even without this careful scaling, suggesting the optimization is fairly robust.

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@hdevalence.bsky.social

Reading §d.2, understand that there's eff gains from sparse computations and why, but it would be helpful to have a mapping between the math formulas used in the definitions of the autoencoders and the specific kernels described in this section ie what are the formulas and for each where does the sparsity come from

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Claude 4 Opus (T) via @hdevalence.bsky.social
Thinking Summary

The user has shared a link to a paper about sparse autoencoders and then asked about Section D.2, which discusses kernel optimizations for computational efficiency. They want to understand the mapping between the mathematical formulas in the autoencoder definitions and the specific kernels described, particularly where the sparsity comes from. Looking at the paper content, I can see: 1. The autoencoder definitions are in Section 2.2 and 2.3: - Encoder: z = TopK(W_enc(x - b_pre)) - Decoder: x̂ = W_dec z + b_pre 2. Section D.2 mentions two main kernels: - DenseSparseMatmul: multiplication between dense and sparse matrix - MatmulAtSparseIndices: multiplication of two dense matrices evaluated at sparse indices 3. Five optimizations are listed: - Decoder forward pass uses DenseSparseMatmul - Decoder gradient uses DenseSparseMatmul - Latent gradient uses MatmulAtSparseIndices - Encoder gradient uses DenseSparseMatmul - Pre-bias gradient uses a trick The user wants to understand how these connect to the math formulas and where sparsity comes from in each case.

Looking at Section D.2, I can help map the mathematical operations to the kernels. Let me break down where the sparsity comes from in each operation:

Forward Pass

Encoder: z=TopK(Wenc(x−bpre))z = \text{TopK}(W_{\text{enc}}(x - b_{\text{pre}}))

  • This produces a sparse vector zz where only kk out of nn latents are non-zero
  • The encoder computation itself is dense (full matrix multiply)

Decoder: x^=Wdecz+bpre\hat{x} = W_{\text{dec}} z + b_{\text{pre}}

  • Uses DenseSparseMatmul: WdecW_{\text{dec}} (dense, d×nd \times n) multiplied by zz (sparse, only kk non-zeros)
  • Only need to compute weighted sum of kk columns of WdecW_{\text{dec}} instead of all nn columns

Backward Pass

Let's denote δx=∂L∂x^\delta_x = \frac{\partial \mathcal{L}}{\partial \hat{x}} as the gradient w.r.t. reconstruction.

1. Decoder gradient: ∂L∂Wdec=δxzT\frac{\partial \mathcal{L}}{\partial W_{\text{dec}}} = \delta_x z^T

  • Uses DenseSparseMatmul: outer product of dense δx\delta_x with sparse zTz^T
  • Only updates kk columns of the weight gradient

2. Latent gradient: ∂L∂z=WdecTδx\frac{\partial \mathcal{L}}{\partial z} = W_{\text{dec}}^T \delta_x

  • Uses MatmulAtSparseIndices: only compute gradients for the kk active latents
  • No need to compute gradients for zero latents (which would be zeroed anyway by TopK)

3. Encoder gradient: ∂L∂Wenc=∂L∂z(x−bpre)T\frac{\partial \mathcal{L}}{\partial W_{\text{enc}}} = \frac{\partial \mathcal{L}}{\partial z} (x - b_{\text{pre}})^T

  • Uses DenseSparseMatmul: only the kk non-zero rows of ∂L∂z\frac{\partial \mathcal{L}}{\partial z} contribute
  • Results in updates to only kk rows of encoder weights

4. Pre-bias gradient: The trick mentioned involves summing gradients across batch before multiplication, reducing memory bandwidth.

The key insight is that TopK creates a sparse bottleneck where only k≪nk \ll n latents are active, allowing all downstream computations to exploit this sparsity for up to 6x speedup in the limit.

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@hdevalence.bsky.social

Ok. I was unsure about MatmulAtSparseIndices since it wasn't obvious when you would ever want to do that mathematically. But it sounds like the answer is that because of the TopK zeroing, we can know in advance which part of the matrix <> matrix computation we don't need?

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Claude 4 Opus (T) via @hdevalence.bsky.social
Thinking Summary

The user is asking about MatmulAtSparseIndices and when you'd want to mathematically compute only certain entries of a matrix multiplication. They correctly understand that it's because TopK tells us in advance which latents are active. Let me explain this clearly - the key insight is that when computing the gradient w.r.t. z, we have: - $\frac{\partial \mathcal{L}}{\partial z} = W_{\text{dec}}^T \delta_x$ - This would normally be a full matrix-vector multiply producing an n-dimensional vector - But we know that only k entries of z were non-zero (the ones selected by TopK) - The chain rule means gradients for the zero entries don't matter - they'll be zeroed anyway - So we only need to compute the k entries of the gradient vector corresponding to the k active latents This is exactly what MatmulAtSparseIndices does - it computes only specific entries of a matrix multiplication result, not the full result.

Exactly right! The key insight is that TopK gives us the sparsity pattern before we need to compute gradients.

When computing ∂L∂z=WdecTδx\frac{\partial \mathcal{L}}{\partial z} = W_{\text{dec}}^T \delta_x:

  • Normally this would produce an nn-dimensional gradient vector (one entry per latent)
  • But we know from the forward pass that only kk specific latents were non-zero
  • By the chain rule, gradients for the zero latents will be zeroed out anyway when backpropagating through TopK
  • So why compute them at all?

MatmulAtSparseIndices exploits this by only computing the kk entries we actually need:

  • Instead of computing all nn dot products in WdecTδxW_{\text{dec}}^T \delta_x
  • We only compute the kk dot products corresponding to the indices where zz was non-zero
  • This turns an O(n⋅d)O(n \cdot d) operation into O(k⋅d)O(k \cdot d)

It's like having a recipe that says "multiply these two matrices" but adding "...oh, but I only need elements [17, 42, 156, ...] of the result" - so you just compute those specific dot products rather than the full matrix multiply.

This pattern appears specifically because TopK creates a deterministic sparsity pattern during the forward pass that we can exploit during backpropagation. Without that known sparsity structure, you'd have to compute the full matrix product first.

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@hdevalence.bsky.social

Finally, there is a recurring “repetition” feature that is ∼ 20% dense. Its top activations are mostly highly repetitive sequences, such as series of dates, chapter indices, numbers, punctuations, repeated exact phrases, or other repetitive things such as Chess PGN notation. However, like the first-token-position latents, random activations of this latent are typically appear unrelated and uninterpretable.

What is the significance of the last sentence?

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Claude 4 Opus (T) via @hdevalence.bsky.social
Thinking Summary

The user is asking about the significance of the last sentence in the quote about the "repetition" feature from the sparse autoencoder paper. The last sentence states: "However, like the first-token-position latents, random activations of this latent are typically appear unrelated and uninterpretable." This sentence is significant because it highlights a key limitation or failure mode of the sparse autoencoder. While the feature appears to capture repetition when looking at its top activations (the strongest activations), when you look at random activations (presumably lower-strength activations), they don't seem to be related to repetition at all. This suggests that: 1. The feature isn't purely monosemantic (capturing just one concept) 2. It may be suffering from polysemanticity - activating for multiple unrelated concepts 3. The feature might be capturing repetition only at high activation values, but at lower values it's activating for other, unrelated things 4. This could indicate the autoencoder is struggling to cleanly separate features, especially for dense features (this one is ~20% dense, which is quite high) This is a known challenge in interpretability - features that seem interpretable when looking at their strongest activations can actually be quite messy when you look at their full activation distribution.

The last sentence reveals a critical limitation of this "repetition" feature and highlights a broader challenge in sparse autoencoder interpretability.

While the feature's top activations clearly correspond to repetitive sequences (dates, indices, chess notation, etc.), its random activations - presumably lower-strength activations sampled from across its distribution - appear unrelated and uninterpretable. This indicates the feature is not cleanly monosemantic.

This is significant for several reasons:

  1. Illusory interpretability: Looking only at top activations can create a false sense that we understand what a feature represents. The feature seems to capture "repetition" when we cherry-pick its strongest signals, but this breaks down when we examine its broader behavior.

  2. Polysemanticity: The feature appears to be activating for multiple, unrelated concepts. At high activation values it captures repetition, but at lower values it's doing something else entirely that we can't interpret.

  3. Density correlation: This feature is notably dense (~20% of tokens), and the paper notes this pattern also occurs with other dense features like the first-token-position feature. This suggests that as features become more dense, they become less interpretable and more polysemantic.

  4. Optimization challenges: The authors conjecture this might be symptomatic of optimization failure, potentially related to the feature being forced to be too active by the TopK constraint, leading it to activate on unrelated inputs just to meet the sparsity requirement.

This finding underscores why the authors consider solutions with many dense features (especially when k approaches d_model) to be problematic - they tend to produce features that appear interpretable at first glance but break down under closer scrutiny.

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@hdevalence.bsky.social

§F.2 Impact of token position

We find that tokens at later positions are harder to reconstruct (Figure 29). We hypothesize that this is because the residual stream at later positions have more features.

Seems related to earlier notes

None of this is actually working with the attention mechanism except indirectly so seems hard to understand how we hope for it to work for the interesting long context behavior

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Claude 4 Opus (T) via @hdevalence.bsky.social
Thinking Summary

The user is sharing a link embed about the "Scaling and evaluating sparse autoencoders" paper and pointing out that section F.2 discusses how later tokens are harder to reconstruct. They're noting this seems related to earlier discussions about the limitations of working with only 64-token contexts and how this doesn't capture the interesting long-context behavior of language models. The key insight they're highlighting is that the SAE work is only indirectly working with the attention mechanism, making it hard to understand how it could capture the interesting long-context behaviors that emerge from attention patterns. I should engage with this observation and explore what it means for interpretability research, especially given the fundamental role of attention in enabling long-context capabilities.

You've hit on a crucial limitation. The fact that later tokens accumulate more features (making them harder to reconstruct) directly illustrates why the 64-token restriction is so problematic. It's like trying to understand a river by only looking at the first few meters from the source.

The indirect relationship with attention is particularly telling. SAEs on residual streams are essentially trying to understand the outputs of attention operations without directly modeling the attention mechanism itself. But attention is precisely what enables:

  • Long-range information routing
  • Dynamic feature selection based on context
  • The formation of temporary computational structures

So we're in a weird position where:

  1. Feature accumulation - The paper shows features accumulate over positions, but cuts off observation right when this gets interesting
  2. Attention blindness - The method can't see HOW features are being routed and combined, only the final superposition
  3. Computational incompleteness - We're missing the control flow that makes transformers powerful

It's a bit like trying to reverse-engineer a program by only looking at variable values at certain checkpoints, without seeing the actual control flow logic. You might identify that "variables tend to get more complex over time" but miss the entire algorithmic structure.

This suggests SAEs might be better suited for understanding the "vocabulary" of features rather than the "grammar" of how they're dynamically composed in context. The real computational story of transformers might be fundamentally about attention patterns, not just feature decomposition.

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