On the Convex Interpolation for Linear Operators

TL;DR

This paper develops new interpolation conditions for linear operators based on their spectral properties, enabling improved worst-case performance analysis of optimization algorithms like Gradient and Chambolle-Pock methods.

Contribution

It introduces novel spectral interpolation conditions for linear operators, extending previous characterizations to unions of spectral subsets, and applies these to analyze optimization algorithms.

Findings

New spectral interpolation conditions for linear operators.

Enhanced worst-case guarantees for Gradient and Chambolle-Pock methods.

Characterization of linear operators with specific eigenvalue and singular value constraints.

Abstract

The worst-case performance of an optimization method on a problem class can be analyzed using a finite description of the problem class, known as interpolation conditions. In this work, we study interpolation conditions for linear operators given scalar products between discrete inputs and outputs. First, we show that if only convex constraints on the scalar products of inputs and outputs are allowed,it is only possible to characterize classes of linear operators or symmetric linear operator whose all singular values or eigenvalues belong to some subset of R. Then, we propose new interpolation conditions for linear operators with minimal and maximal singular values and linear operators whose eigenvalues or singular values belong to unions of subsets. Finally, we illustrate the new interpolation conditions through the analysis of the Gradient and Chambolle-Pock methods. It allows to obtain new numerical worst-case guarantees on these methods.