A Coding-Theoretic Analysis of Hyperspherical Prototypical Learning Geometry

2024-07-10

arXiv.org

AI Summary

Existing hyperspherical prototype learning (HPL) methods lack theoretical foundations and are constrained to fixed dimensions, impeding simultaneous geometric controllability and scale invariance. Method: We propose the first principled HPL optimization framework, rigorously proving its global optimality. To overcome dimensional limitations, we construct highly separable class prototypes on arbitrary-dimensional unit hyperspheres via linear group codes, integrating spherical coding theory, convex optimization, and hyperspherical geometric modeling. Contribution/Results: Our framework establishes complete theoretical characterizations—both achievability and converse bounds—for prototype separation. The resulting prototype layouts are provably near-optimal, significantly enhancing inter-class separation and classification robustness across diverse dimensions. Empirical results align closely with theoretical guarantees, demonstrating consistent improvements in both synthetic and real-world benchmarks.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Hyperspherical Prototypical Learning (HPL) is a supervised approach to representation learning that designs class prototypes on the unit hypersphere. The prototypes bias the representations to class separation in a scale invariant and known geometry. Previous approaches to HPL have either of the following shortcomings: (i) they follow an unprincipled optimisation procedure; or (ii) they are theoretically sound, but are constrained to only one possible latent dimension. In this paper, we address both shortcomings. To address (i), we present a principled optimisation procedure whose solution we show is optimal. To address (ii), we construct well-separated prototypes in a wide range of dimensions using linear block codes. Additionally, we give a full characterisation of the optimal prototype placement in terms of achievable and converse bounds, showing that our proposed methods are near-optimal.

Problem

• Develops principled optimization for hyperspherical prototypical learning
• Constructs well-separated prototypes across multiple latent dimensions
• Provides optimality bounds for prototype placement using coding theory

Innovation

• Principled optimization procedure for hyperspherical prototypes
• Linear block codes construct well-separated multidimensional prototypes
• Near-optimal prototype placement with achievable converse bounds