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.

翻译：超球面原型学习（HPL）是一种监督式表示学习方法，其在单位超球面上设计类别原型。这些原型在尺度不变且已知的几何结构中，使表示偏向于类别分离。以往的HPL方法存在以下不足之一：(i) 采用缺乏理论依据的优化过程；或 (ii) 虽理论严谨，但仅限于单一可能的潜在维度。本文同时解决了这两个问题。针对(i)，我们提出了一种具有理论依据的优化过程，并证明其解是最优的。针对(ii)，我们利用线性分组码在广泛的维度范围内构建了充分分离的原型。此外，我们通过可达界与逆界完整刻画了最优原型放置的特性，表明所提方法接近最优。