Sparse Prototype Code Underlies Classification and Prediction Across Modalities

Neural representations have become a central tool for studying the internal mechanisms of modern AI models, yet their complex high-dimensional structure makes them difficult to interpret. We show that classification tasks give rise to a universal representational geometry, shared across state-of-the-art models in vision, audio, and language processing. The key structure is that within-class variability is not random in representation space. Instead, its classifier-relevant component has strong and structured correlations with the class's own centroid and with the centroids of its competing classes. Building on this observation, we derive an analytical mean-field theory governed mainly by the variability along true-class and rival-class centroid coordinates, together with a global renormalization of the class radius that compensates for the non-Gaussian statistics of real representations. The theory accurately predicts classification accuracy across architectures and modalities. The relevant geometric quantities improve systematically with model scale, mirroring the observed gains in accuracy. A striking feature of the theory is its sparsity: accurate prediction requires only a small set of centroid coordinates associated with the true class and its strongest rivals - connecting our framework to sparse-feature extraction approaches such as sparse autoencoders. Together, these results provide a parsimonious predictive theory of neural representations and suggest that classification in deep networks is governed by a sparse, centroid-aligned structure embedded within the full high-dimensional representation space.

A Geometric Measure of Linear Separability for Neural Representations

Modern neural classifiers commonly rely on linear readouts, yet predictive metrics alone do not characterize the class-wise geometry of the representations on which such readouts operate. We introduce the directional linear separability measure (LSM), a finite-sample diagnostic for one-sided affine separability. For a target class A and a competing set B, LSM searches over affine halfspaces that contain all samples in A and measures the smallest competing-sample intrusion that must remain on the target side, normalized by |A|. The resulting quantity is asymmetric, class-wise, target-normalized, and applicable to finite representations extracted from neural networks. We establish its supporting-hyperplane characterization, relate it to optimal affine classification accuracy, and prove invariance under full-rank linear embeddings. These results separate changes caused by linear reparameterization from those caused by information loss or nonlinear geometric transformations. We also give a penalty-based affine search for estimating class-wise LSM in high-dimensional features, with reported values computed from the original discrete preservation and violation criterion. Finally, we analyze coordinatewise gated nonlinearities as finite-sample geometric operators and empirically use LSM to diagnose class-wise intrusion across common deep-learning components and architectures.