The research deals with the application of manifold optimization techniques to large-scale convex optimization problems whose expected or desired solutions have low rank. We focus specifically on convex relaxations of large-scale rank-constrained problems encountered in machine learning, data reduction and bioinformatics.

As of now, I am trying to understand this comic strip.

1. qGeomMC: A Quotient Geometric approach to low-rank Matrix Completion
2. Trace norm (nuclear norm) minimization [TraceNorm code], [poster]
3. Low-rank distance matrix completion (low-rank EDMCP)

Manopt comes with a large library of manifolds and ready-to-use Riemannian optimization algorithms. It is well documented and includes diagnostics tools to help you get started quickly and make sure you make no mistakes along the way. It is designed to provide great flexibility in describing your cost function and incorporates an optional caching system for more efficiency.

This package contains a MATLAB implementation of algorithms for the low-rank matrix completion problem. The present version includes gradient descent and conjugate gradient algorithms based on the fixed-rank geometry proposed in the techreport [arXiv:1211.1550].