Equation 4 · Part 2 · What a Circuit Explains: The State and Limits of Mechanistic Interpretability

Symbol x

\mathcal{L}(x) = \lVert x - \hat{x}(x) \rVert_2^2 + \lambda \lVert f(x) \rVert_1, \qquad \hat{x}(x) = W_d\, f(x) + b_d .

x

What this part means

the writing.

Its job in the formula

x is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

Full expressionSymbol xArticle meaning

Where the article explains it

Writing x for an activation vector, f(x) for the sparse code and $\hat{x}$ for the reconstruction, the objective has the form $\mathcal{L}(x) = \lVert x - \hat{x}(x) \rVert_2^2 + \lambda \lVert f(x) \rVert_1, \qquad \hat{x}(x) = W_d\, f(x) + b_d$ .

The passage around this formula

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 $\hat{x}$ for the reconstruction, the objective has the form $\mathcal{L}(x) = \lVert x - \hat{x}(x) \rVert_2^2 + \lambda \lVert f(x) \rVert_1, \qquad \hat{x}(x) = W_d\, f(x) + b_d$ . 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…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the article section

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