Computing Kurdyka-\L{}ojasiewicz exponents via composition and symmetry

TL;DR

This paper introduces calculus rules for computing Kurdyka-ojasiewicz exponents applicable to composite and invariant functions, facilitating convergence analysis in matrix-related optimization problems without relying on smoothness.

Contribution

It presents a unified framework using the rank theorem and Lie group actions to compute Kurdyka-ojasiewicz exponents for a broad class of functions, especially nonisolated minima.

Findings

Applicable to nonisolated local minima

No reliance on gradient or Hessian computations

Enables linear convergence proofs for various algorithms

Abstract

We devise calculus rules for the Kurdyka-\L{}ojasiewicz exponent using the rank theorem and Lie group actions. They apply to a wide class of composite and invariant functions, and are particularly suitable for handling nonisolated local minima. Notably, smoothness plays no role, eschewing gradient and Hessian computations. This provides a unified framework for establishing linear convergence of various algorithms in matrix factorization, $\ell_1$-matrix factorization, matrix sensing, and linear neural networks.