Geometric Analysis of Unconstrained Feature Models with $d=K$

TL;DR

This paper analyzes unconstrained feature models in deep learning, proving that all critical points are either global minima or strict saddles when feature dimension equals number of classes, confirming prior conjectures.

Contribution

It rigorously proves that two popular unconstrained feature models are strict saddle functions with no suboptimal local minima, confirming previous conjectures.

Findings

All critical points are either global minima or strict saddles.

The models are strict saddle functions with negative curvature at saddle points.

Confirms conjecture on the landscape of unconstrained feature models.

Abstract

Recently, interesting empirical phenomena known as Neural Collapse have been observed during the final phase of training deep neural networks for classification tasks. We examine this issue when the feature dimension d is equal to the number of classes K. We demonstrate that two popular unconstrained feature models are strict saddle functions, with every critical point being either a global minimum or a strict saddle point that can be exited using negative curvatures. The primary findings conclusively confirm the conjecture on the unconstrained feature models in previous articles.