Classification and Geometry of General Perceptual Manifolds

By SueYeon Chung, Daniel D. Lee, Haim Sompolinsky. 2017

193 citations recorded by the source corpus.

Perceptual manifolds arise when a neural population responds to an ensemble of sensory signals associated with different physical features (e.g., orientation, pose, scale, location, and intensity) of the same perceptual object. Object recognition and discrimination requires classifying the manifolds in a manner that is insensitive to variability within a manifold. How neuronal systems give rise to invariant object classification and recognition is a fundamental problem in brain theory as well as in machine learning. Here we study the ability of a readout network to classify objects from their perceptual manifold representations. We develop a statistical mechanical theory for the linear classification of manifolds with arbitrary geometry revealing a remarkable relation to the mathematics of conic decomposition. Novel geometrical measures of manifold radius and manifold dimension are introduced which can explain the classification capacity for manifolds of various geometries. The general theory is demonstrated on a number of representative manifolds, including L2 ellipsoids prototypical of strictly convex manifolds, L1 balls representing polytopes consisting of finite sample points, and orientation manifolds which arise from neurons tuned to respond to a continuous angle variable, such as object orientation. The effects of label sparsity on the classification capacity of manifolds are elucidated, revealing a scaling relation between label sparsity and manifold radius. Theoretical predictions are corroborated by numerical simulations using recently developed algorithms to compute maximum margin solutions for manifold dichotomies. Our theory and its extensions provide a powerful and rich framework for applying statistical mechanics of linear classification to data arising from neuronal responses to object stimuli, as well as to artificial deep networks trained for object recognition tasks.

Subfields: Artificial Intelligence, Cognitive Neuroscience, Computer Vision and Pattern Recognition

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Citations in the corpus

Re-evaluating circuit mechanisms underlying pattern separation (2019, neutral): “Even within a high-dimensional coding subspace, linear separability depends on the geometry of the variability of neural representations (Chung et al., 2018).”
Statistical mechanics of deep learning (2024, unknown): “Reprinted ﬁgure with permission from [11], Copyright (2018) by the American Physical Society. where, is the D -dimensional Chi probability density function.”
Intrinsic dimension of data representations in deep neural networks (2019, result): “Our results are broadly consistent with recent theoretical studies linking the classification capacity of data manifolds by perceptrons to their geometrical properties [29, 30].”
Convolutional Neural Networks as a Model of the Visual System: Past, Present, and Future (2020, methodology): “This theory has been developed using biological data [85], however, recently, techniques for describing the geometry of these clusters (or manifolds) have been developed and used to understand the untangling process in deep neural networks [86, 87].”
Interpreting neural computations by examining intrinsic and embedding dimensionality of neural activity (2021, background): “A theory for the ensuing trade-offs has been recently developed for the case where the activity is organized in multiple disjoint manifolds corresponding to different categories [66] and applied to interpret representations in the visual system [67] and in deep networks [68].”
On the geometry of generalization and memorization in deep neural networks (2021, methodology): “We run the MFTMA on each set of manifolds following the procedures in (Chung et al., 2018; Cohen et al., 2019; Stephenson et al., 2019).”
Machine learning and the physical sciences (2019, methodology): “For this we need to model the input data no longer as iid vectors, but for instance as outputs from a generative neural network as in (Gabrié et al., 2018), or as perceptual manifolds as in (Chung et al., 2018).”
Sparse Prototype Code Underlies Classification and Prediction Across Modalities (2026, unknown): “Predictive theories of representational geometry —manifold capacity [Chung et al., 2018, Cohen et al., 2020, Stephenson et al., 2021, Wakhloo et al., 2023] and neural collapse [Papyan et al., 2020, Han et al., 2021, Galanti et al., 2021] — have made important progress in connecting geometry to…”