Approximation of sequences of symmetric matrices with the symmetric rank-one algorithm and applications

TL;DR

This paper extends the symmetric rank-one update method to approximate any sequence of symmetric invertible matrices, demonstrating its broader applicability in optimization and shape analysis with supporting numerical simulations.

Contribution

It generalizes the symmetric rank-one algorithm to approximate arbitrary symmetric invertible matrices sequences, beyond Hessian approximation.

Findings

The method effectively approximates matrix sequences in numerical experiments.

Applications include constrained geodesics in shape analysis imaging.

Numerical simulations validate the theoretical extensions.

Abstract

The symmetric rank-one update method is well-known in optimization for its applications in the quasi-Newton algorithm. In particular, Conn, Gould, and Toint proved in 1991 that the matrix sequence resulting from this method approximates the Hessian of the minimized function. Expanding their idea, we prove that the symmetric rank-one update algorithm can be used to approximate any sequence of symmetric invertible matrices, thereby adding a variety of applications to more general problems, such as the computation of constrained geodesics in shape analysis imaging problems. We also provide numerical simulations for the method and some of these applications.