A Coding-Theoretic Analysis of Hyperspherical Prototypical Learning Geometry

TL;DR

This paper provides a rigorous analysis and improved optimization method for hyperspherical prototypical learning, enabling well-separated class prototypes across various dimensions with near-optimal placement.

Contribution

It introduces a principled optimization procedure and constructs well-separated prototypes in multiple dimensions using linear block codes, addressing previous limitations.

Findings

Proposed an optimal, principled optimization method.

Constructed well-separated prototypes in many dimensions.

Achieved near-optimal prototype placement bounds.

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.