Testing link embed with bugfix on context importer
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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
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:
- This produces a sparse vector where only out of latents are non-zero
- The encoder computation itself is dense (full matrix multiply)
Decoder:
- Uses DenseSparseMatmul: (dense, ) multiplied by (sparse, only non-zeros)
- Only need to compute weighted sum of columns of instead of all columns
Backward Pass
Let's denote as the gradient w.r.t. reconstruction.
1. Decoder gradient:
- Uses DenseSparseMatmul: outer product of dense with sparse
- Only updates columns of the weight gradient
2. Latent gradient:
- Uses MatmulAtSparseIndices: only compute gradients for the active latents
- No need to compute gradients for zero latents (which would be zeroed anyway by TopK)
3. Encoder gradient:
- Uses DenseSparseMatmul: only the non-zero rows of contribute
- Results in updates to only 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 latents are active, allowing all downstream computations to exploit this sparsity for up to 6x speedup in the limit.
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?
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 :
- Normally this would produce an -dimensional gradient vector (one entry per latent)
- But we know from the forward pass that only 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 entries we actually need:
- Instead of computing all dot products in
- We only compute the dot products corresponding to the indices where was non-zero
- This turns an operation into
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.