The Exploration of Neural Collapse under Imbalanced Data

TL;DR

This paper investigates neural collapse phenomena in imbalanced data scenarios, providing a theoretical analysis of the global minimizer and revealing geometric structures and feature behaviors during training.

Contribution

It offers a novel theoretical analysis of neural collapse under imbalanced data, including geometric structures and feature convergence properties, supported by numerical experiments.

Findings

Features within the same class converge to their class mean.

The geometric structure involves orthonormal transformations of classifiers and class means.

Some rows of the transformation matrices collapse to zeros or are orthogonal.

Abstract

Neural collapse, a newly identified characteristic, describes a property of solutions during model training. In this paper, we explore neural collapse in the context of imbalanced data. We consider the $L$-extended unconstrained feature model with a bias term and provide a theoretical analysis of global minimizer. Our findings include: (1) Features within the same class converge to their class mean, similar to both the balanced case and the imbalanced case without bias. (2) The geometric structure is mainly on the left orthonormal transformation of the product of $L$ linear classifiers and the right transformation of the class-mean matrix. (3) Some rows of the left orthonormal transformation of the product of $L$ linear classifiers collapse to zeros and others are orthogonal, which relies on the singular values of $\hat Y=(I_K-1/N\mathbf{n}1^\top_K)D$, where $K$ is class size, $\mathbf{n}$ is the vector of sample size for each class, $D$ is the diagonal matrix whose diagonal entries are given by $\sqrt{\mathbf{n}}$. Similar results are for the columns of the right orthonormal transformation of the product of class-mean matrix and $D$. (4) The $i$-th row of the left orthonormal transformation of the product of $L$ linear classifiers aligns with the $i$-th column of the right orthonormal transformation of the product of class-mean matrix and $D$. (5) We provide the estimation of singular values about $\hat Y$. Our numerical experiments support these theoretical findings.