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.

The Importance of Phase in Neural Representations: An Internal Oppenheim-Lim Test of Image Classifiers

Oppenheim and Lim (1981) showed that natural images stay recognizable when reconstructed from their Fourier phase alone, while the magnitude carries little of their identity. We ask whether trained image classifiers reproduce this asymmetry inside their hidden layers, and we test it causally: given two images, we transplant the phase of one onto the magnitude of the other at a chosen layer and record which image the prediction follows. In PRISM2D, GFNet, and ViT-B/16 the prediction follows the phase or sign donor, and deleting all image-specific magnitude barely moves accuracy, so identity rides on phase while image-specific magnitude is largely dispensable to the readout. ResNet-50 at first seems to break the pattern, because transplanting sign after its ReLUs does nothing; a fair intervention before the ReLU reveals a strong latent sign code in the late blocks, and a DC-only control shows the readout consumes a channel-wise spatial average. Controls rule out the trivial case in which magnitude simply stops depending on the image. The architectures therefore share a phase/sign identity code but expose it in different bases, set by rectification and readout geometry, which gives a mechanistic account of the texture--shape gap between CNNs and attention models.

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.

A Cookbook of 3D Vision: Data, Learning Paradigms, and Application

3D vision has rapidly evolved, driven by increasingly diverse data representations, learning paradigms, and modeling strategies. Yet the field remains fragmented across representations and benchmarks, making it difficult to develop unified perspectives on efficiency, fidelity, and scalability. This work provides a data-centric taxonomy of 3D vision that connects geometric representations, datasets, learning frameworks, and applications within a single conceptual map. We begin by analysing the principal structural representations of 3D data--point clouds, meshes, voxels, and 3D Gaussians--along with their acquisition pipelines. We then examine how dataset design, benchmark construction, and supervision regimes shape recent advances, spanning 2D-supervised 3D learning, implicit neural representations, and 4D world modeling. Through this integrative lens, we clarify the relationships among representations, learning paradigms, and downstream tasks in reconstruction, generation, and video modeling, offering a consolidated view of emerging trends toward balancing efficiency and fidelity and toward multimodal geometric grounding.