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Equation 14 · Part 8 · What Interpretability Actually Costs to Do at Scale

subscript

CSAE  ≈  2 d (n+k) T.C_{\mathrm{SAE}} \;\approx\; 2\,d\,(n + k)\,T .
subscript

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

The passage around this formula

Work out what that architecture actually spends compute on per token, because the two halves of it behave differently. The encoding step needs a score for every one of the n candidate latents before it can select the top k , so it is an unavoidably dense matrix multiply: roughly 2dn floating-point operations. The decoding step only touches the k latents that survived, so it is sparse: roughly 2dk operations. Summed and multiplied across T training tokens, a first-order compute model for training the dictionary is CSAE  ≈  2 d (n+k) TC_{\mathrm{SAE}} \;\approx\; 2\,d\,(n + k)\,T . Because published TopK configurations keep k in the tens to low hundreds while n runs into the millions, n ≫\gg k and the encoding term dominates almost…

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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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Sources cited in the surrounding passage

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