Mark Crowley | Exciting new paper accepted to ACML 2021!

Our new collaborative paper (Godaz et al., 2021) on an exciting, if a bit theoretical, method for improving a common optimziation method has been accepted to ACML 2021.

RLBFGS is a common quasi-Newton method for optimization using Riemannian manifolds, but the method is very computationally expensive. In this paper we present how altering two mappings (vector transport and adjoint) in the algorithm can lead to a significant computational performance gain with very little loss in accuracy.

ACML is the Asian Conference on Machine Learning, for the 2021 edition there were 378 submissions with an acceptance rate of around 30% via double-blind reviewing.

References:

VTFR-LBFGS-21

Vector Transport Free Riemannian LBFGS for Optimization on Symmetric Positive Definite Matrix Manifolds

Reza Godaz, Benyamin Ghojogh, Reshad Hosseini, Reza Monsefi, Fakhri Karray, and Mark Crowley

In Asian Conference on Machine Learning (ACML). Virtual. Nov, 2021.

Abs arXiv PDF Supp Poster URL

This work concentrates on optimization on Riemannian manifolds. The Limited-memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) algorithm is a commonly used quasi-Newton method for numerical optimization in Euclidean spaces. Riemannian LBFGS (RLBFGS) is an extension of this method to Riemannian manifolds. RLBFGS involves computationally expensive vector transports as well as unfolding recursions using adjoint vector transports. In this article, we propose two mappings in the tangent space using the inverse second root and Cholesky decomposition. These mappings make both vector transport and adjoint vector transport identity and therefore isometric. Identity vector transport makes RLBFGS less computationally expensive and its isometry is also very useful in convergence analysis of RLBFGS. Moreover, under the proposed mappings, the Riemannian metric reduces to Euclidean inner product, which is much less computationally expensive. We focus on the Symmetric Positive Definite (SPD) manifolds which are beneficial in various fields such as data science and statistics. This work opens a research opportunity for extension of the proposed mappings to other well-known manifolds.