Equation 4 · What a Circuit Explains: The State and Limits of Mechanistic Interpretability
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol L
L is part of the quantity the equation computes from the expression on the right.
Symbol hatx
hatx is one of the signed contributions combined to compute the quantity on the left.
Symbol λ
λ is one of the signed contributions combined to compute the quantity on the left.
Symbol f
f is one of the signed contributions combined to compute the quantity on the left.
Symbol W_d
is one of the signed contributions combined to compute the quantity on the left.
Symbol b_d
is one of the signed contributions combined to compute the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subtraction
Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.
subscript
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.
superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
See an illustrated explanation →How to interpret it
Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
Now the careful part. Consider what the training objective actually asks for. Writing x for an activation vector, f(x) for the sparse code and for the reconstruction, the objective has the form . Every term refers to the activation vector. No term refers to what the model does with that activation afterwards. The objective rewards a code that reconstructs the activation sparsely; it is indifferent to whether the dictionary elements correspond to anything the network’s downstream layers treat as a unit. Low reconstruction error at high sparsity is therefore evidence that the activation distribution is sparsely decomposable in the trained basis. It is not evidence…
Read the full surrounding passage
Now the careful part. Consider what the training objective actually asks for. Writing x for an activation vector, f(x) for the sparse code and for the reconstruction, the objective has the form . Every term refers to the activation vector. No term refers to what the model does with that activation afterwards. The objective rewards a code that reconstructs the activation sparsely; it is indifferent to whether the dictionary elements correspond to anything the network’s downstream layers treat as a unit. Low reconstruction error at high sparsity is therefore evidence that the activation distribution is sparsely decomposable in the trained basis. It is not evidence that the basis is the model’s own.
Sources cited in the article section
- [7] Sparse Autoencoders Find Highly Interpretable Features in Language Models ↗
- [8] Scaling and evaluating sparse autoencoders ↗
- [9] Scaling Monosemanticity: Extracting Interpretable Features from Claude 3 Sonnet ↗
- [10] Automated Interpretability Metrics Do Not Distinguish Trained and Random Transformers ↗
- [11] Are Sparse Autoencoders Useful? A Case Study in Sparse Probing ↗
These citations give research context. Read each source to check which claims it supports.
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