subtraction

The passage around this formula

The $\ell_1$ penalty has a known cost: it does not directly control how many dictionary elements fire, only how much their combined magnitude is discouraged, and it systematically shrinks the elements that do fire toward zero, biasing the reconstruction. Two later refinements address this more directly. Gao and colleagues introduced k -sparse encoding, which drops the tunable penalty in favour of a fixed sparsity budget enforced structurally, $z = \mathrm{TopK}_k\!\left(W_{\mathrm{enc}}(a - b_{\mathrm{dec}})\right)$. keeping only the k largest pre-activations and zeroing the rest, and used it to train a sixteen-million-latent dictionary on GPT-4 activations over forty billion tokens, reporting metrics that improve consistently as dictionary size…

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