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The formal objective has evolved since the earliest versions, and the direction of that evolution is itself informative. A dictionary decoder reconstructs the activation x from a sparse code f(x) : $\hat x = W_d f(x) + b_d, \qquad f(x) = \mathrm{JumpReLU}_\theta\big(W_e x + b_e\big), \qquad \mathcal L(x) = \lVert x - \hat x \rVert_2^2 + \lambda \lVert f(x) \rVert_0$. Earlier versions of this objective penalised the code’s $\ell_1$ norm as a differentiable stand-in for sparsity, but an $\ell_1$ penalty also shrinks the magnitude of every active feature, distorting reconstruction in a way that has nothing to do with how many features are active. Rajamanoharan and colleagues introduced JumpReLU, a thresholded activation function with a learned per-feature cutoff $\theta$ trained through a straight-through gradient estimator, which lets the objective penalise the true count of active features, $\lVert$ f(x) $\rVert_0$ , directly rather than through the $\ell_1$ proxy, and reported state-of-the-art reconstruction fidelity at matched sparsity against both the earlier $\ell_1$ formulation and a competing gated variant [ 7 ] . That the field kept revising the sparsity penalty is itself evidence of how much the objective’s exact shape affects what gets recovered.