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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 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…

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