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Series of quasi-uniform scatterings with fast search, root systems and neural network classifications

Igor V. Netay

TL;DR

The paper addresses scalable classification with many classes by predefining a structured, extendable set of embedding centers on the boundaries of weight polyhedra derived from Lie group representations. It constructs quasi-uniform scatterings of centers through a boundary enumeration via semistandard Young tableaux and a two-power subdivision, ensuring fast nearest-center search. An algorithmic framework projects queries to faces of the weight polyhedron and rounds to lattice points to identify nearest centers, enabling class expansion without full retraining. The method is developed primarily for A_n-type root systems, with discussion of extensions to other types and practical dimension considerations for real-world neural networks.

Abstract

In this paper we describe an approach to construct large extendable collections of vectors in predefined spaces of given dimensions. These collections are useful for neural network latent space configuration and training. For classification problem with large or unknown number of classes this allows to construct classifiers without classification layer and extend the number of classes without retraining of network from the very beginning. The construction allows to create large well-spaced vector collections in spaces of minimal possible dimension. If the number of classes is known or approximately predictable, one can choose sufficient enough vector collection size. If one needs to significantly extend the number of classes, one can extend the collection in the same latent space, or to incorporate the collection into collection of higher dimensions with same spacing between vectors. Also, regular symmetric structure of constructed vector collections can significantly simplify problems of search for nearest cluster centers or embeddings in the latent space. Construction of vector collections is based on combinatorics and geometry of semi-simple Lie groups irreducible representations with highest weight.

Series of quasi-uniform scatterings with fast search, root systems and neural network classifications

TL;DR

The paper addresses scalable classification with many classes by predefining a structured, extendable set of embedding centers on the boundaries of weight polyhedra derived from Lie group representations. It constructs quasi-uniform scatterings of centers through a boundary enumeration via semistandard Young tableaux and a two-power subdivision, ensuring fast nearest-center search. An algorithmic framework projects queries to faces of the weight polyhedron and rounds to lattice points to identify nearest centers, enabling class expansion without full retraining. The method is developed primarily for A_n-type root systems, with discussion of extensions to other types and practical dimension considerations for real-world neural networks.

Abstract

In this paper we describe an approach to construct large extendable collections of vectors in predefined spaces of given dimensions. These collections are useful for neural network latent space configuration and training. For classification problem with large or unknown number of classes this allows to construct classifiers without classification layer and extend the number of classes without retraining of network from the very beginning. The construction allows to create large well-spaced vector collections in spaces of minimal possible dimension. If the number of classes is known or approximately predictable, one can choose sufficient enough vector collection size. If one needs to significantly extend the number of classes, one can extend the collection in the same latent space, or to incorporate the collection into collection of higher dimensions with same spacing between vectors. Also, regular symmetric structure of constructed vector collections can significantly simplify problems of search for nearest cluster centers or embeddings in the latent space. Construction of vector collections is based on combinatorics and geometry of semi-simple Lie groups irreducible representations with highest weight.

Paper Structure

This paper contains 9 sections, 3 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weight polyhedron and its boundary points construction algorithm
  4. Construction of $\partial P_{\lambda} \cap \mathfrak{X}$
  5. Construction of $\partial P_{\lambda} \cap (\mathfrak{X} / 2^k)$
  6. Search algorithm
  7. Search for the nearest point in a face $F$ of $P_\lambda$
  8. Conclusion
  9. Acknowledgmenets

Key Result

Theorem 3.6

The algorithm above runs over all points of $C \cap \partial P_\lambda \cap \mathfrak{X}$.

Theorems & Definitions (18)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Remark 3.1
  • Remark 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Theorem 3.6
  • ...and 8 more