# Optimization on the biorthogonal manifold

arXiv: https://arxiv.org/abs/1609.04161 · HTML: https://mlsanity.com/paper/1609.04161/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Sep 2016.

## Built on by (most cited first, 1 of 1)

- [Geometric Dictionary Learning of Dynamical Systems with Optimal Transport](https://mlsanity.com/paper/2605.18276/) (2026)

## Abstract (arXiv)

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.