Efficient Rank Reduction of Correlation Matrices

TL;DR

This paper introduces geometric optimization algorithms for efficiently finding the nearest low-rank correlation matrix, improving calibration of multi-factor interest rate models with flexible weighted norms.

Contribution

It presents novel geometric methods for low-rank correlation matrix approximation, connecting with Lagrange multipliers and enabling flexible norm applications.

Findings

Methods outperform existing algorithms in numerical tests

Connection established with Lagrange multiplier approach

Flexible weighted norm application demonstrated

Abstract

Geometric optimisation algorithms are developed that efficiently find the nearest low-rank correlation matrix. We show, in numerical tests, that our methods compare favourably to the existing methods in the literature. The connection with the Lagrange multiplier method is established, along with an identification of whether a local minimum is a global minimum. An additional benefit of the geometric approach is that any weighted norm can be applied. The problem of finding the nearest low-rank correlation matrix occurs as part of the calibration of multi-factor interest rate market models to correlation.