An Exact Linear Solver for Content-Loss Neural Code Conversion

Functional alignment maps neural activity between individuals so that decoding models can be transferred across subjects. Content-loss neural code conversion targets this goal by training an inter-subject mapping to minimize the mismatch between stimulus content and the content that fixed target-subject decoders read from converted activity. The published method defines a high-dimensional loss in a deep neural network feature space and fits the converter by finite-budget iterative optimization. The fitted converter therefore depends on the optimization route and stopping budget as well as on the objective, leaving open what the complete objective alone determines. We show that, with fixed linear decoders, the complete squared content loss for any converter reduces exactly to a constant and two matrices expressed in target-response coordinates. For an $\ell_2$-regularized linear converter, the contracted problem has a unique spectral solution requiring no iterative optimization or stopping rule. On public fMRI data from five subjects who viewed natural images, linear converters fitted under the published optimization budget remained above the exact minimum of the regularized objective in all 20 source-target directions. Images reconstructed from converted responses nevertheless appeared similar for the exact linear converter and for the matched linear control fitted under the published budget. Pairwise-identification scores likewise differed only slightly across nine image feature spaces, with identification near ceiling in most of them. The exact linear solution thereby provides a cross-subject converter determined solely by the complete objective and independent of optimization choices.