Optimization on the biorthogonal manifold

TL;DR

This paper introduces the biorthogonal manifold, a new Riemannian manifold for optimizing pairs of orthogonal matrices, with applications in shape correspondence, and provides foundational geometric tools for its analysis.

Contribution

The paper defines and studies the biorthogonal manifold, including tangent space projection, exponential map, and retraction, which were not previously explored.

Findings

The biorthogonal manifold is a valid Riemannian manifold.

Explicit formulas for tangent space projection, exponential map, and retraction are provided.

Numerical implementation strategies are discussed.

Abstract

In this paper, we consider optimization problems w.r.t. to pairs of orthogonal matrices $XY = I$. Problems of this form arise in several applications such as finding shape correspondence in computer graphics. We show that the space of such matrices is a Riemannian manifold, which we call the biorthogonal manifold. To our knowledge, this manifold has not been studied before. We give expressions of tangent space projection, exponential map, and retraction operators of the biorthogonal manifold, and discuss their numerical implementation.