Sparse autoencoders (SAEs) seek to decompose the dense activations of large language models (LLMs) into sparse, potentially interpretable features using a learned overcomplete dictionary. In this work, we ask whether it is possible to enhance the semantic selectivity of the learned dictionary atoms by linearly \emph{shaping} the dense activations prior to learning an SAE. We explore rank-reducing transformations based on the recently introduced J-space corresponding to the averaged Jacobian from the current layer to the LLM output, projecting activations onto the ``live'' subspace spanned by eigenmodes of the Jacobian with larger singular values (which we expect to be more semantically relevant). We also introduce a new approach for training SAEs in which (a)~an affine shaping transformation is learned along with the dictionary atoms, (b)~the atom selection mechanism is based on the TopK projections along the one-dimensional subspace spanned by the atoms, and (c)~reconstruction is carried out using the same atoms. We call this approach a Matched Filter SAE (MF-SAE) based on analogies with communication theory. The promise of shaping is demonstrated via evaluations against dense activations and state-of-the-art SAEs on benchmarks including SAEBench and SAE Probes.

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