Manifold optimization and recent applications

Optimization over smooth manifolds or manifold optimization involves minimizing an objective function over a smooth constrained set. Many such sets have usually a manifold structure. Some particularly useful manifolds include the set of orthogonal matrices, the set of symmetric positive definite matrices, the set of subspaces, the set of fixed-rank matrices/tensors, and the set of doubly stochastic matrices (optimal transport plans), to name a few [1]. Consequently, there has been a development of a number of manifold optimization toolboxes [2].

In this project, we make use of these wonderful tools to solve a few machine learning problems with manifold optimization. The aim would be to get a hands-on experience of manifold optimization.

[1] Boumal, N., 2020. An introduction to optimization on smooth manifolds. Web: http://sma.epfl.ch/~nboumal/book/index.html.

[2] Manopt, pymanopt, Manopt.jl, McTorch, Geomstats, ROPTLIB, and so on. The links to many of these toolboxes are available on https://www.manopt.org/about.html.

Project timezone: B