RTRMC : A Riemannian Trust-Region method for low-rank Matrix Completion

RTRMC is an algorithm developed by Nicolas Boumal (contact person) and Pierre-Antoine Absil at UCLouvain to solve low-rank matrix completion problems.

The associated paper is available here:

N. Boumal and P.-A. Absil, RTRMC : A Riemannian trust-region method for low-rank matrix completion, NIPS 2011,

with a corresponding BibTex entry.

An extended version of the paper, with more detailed mathematical developments and numerical experiments, is available too:

N. Boumal and P.-A. Absil, .

Since then, we developed a preconditioner for RTRMC and added Riemannian conjugate gradients support, thanks to the Manopt toolbox.
As a consequence, the code changed significantly: please see below to download version 3.0 of RTRMC and RCGMC (released March 7, 2014).
This version of the algorithms is described in Nicolas Boumal's PhD thesis.

The abstract of the extended paper reads:

We consider large matrices of low rank. We address the mathematical problem of recovering such matrices when most of the entries are unknown. We follow an approach that exploits the geometry of the low-rank constraint to recast the problem as an unconstrained optimization problem on the Grassmann manifold. We then apply first- and second-order Riemannian trust-region methods to solve it. The cost of each iteration is linear in the number of known entries. The proposed methods, RTRMC 1 and 2, outperform state-of-the-art algorithms on a wide range of problem instances. In particular, RTRMC performs very well on rectangular matrices and we note that second-order methods such as RTRMC 2 are well suited to solve badly conditioned or nonuniformly sampled matrix completion tasks.

Here is a plot of one of the experiments, where different matrix completion algorithms compete on rectangular matrices of size 1000-by-30000:

RTRMC Scenario 2
Figure 2: Evolution of the Root Mean Square Error for six matrix completion methods under Scenario 2 of the RTRMC paper (m = 1000, n = 30000, rank = 5, sampling ratio = 2.6%). For rectangular matrices, RTRMC is especially efficient owing to the linear growth of the dimension of the search space in min(m,n), whereas for most methods the growth is linear in m+n.

Here is the Matlab code for RTRMC v3.0 under BSD licence.
This version of the code was put online on March 7, 2014. It changed significantly since version 1.1 and now uses Manopt.
Changes since version 2: now bundled with Manopt 1.0.5 instead of 1.0.4, and the buildproblem function now allows the data I, J, X, C to be given in any order.

To install and use the software:

If this procedure fails (quite probably somewhere in the installrtrmc.m script), then either there is trouble with the C compiler Matlab tries to use, which you can check by typing 'mex -setup' at the Matlab prompt, or the installation script is incompatible with your OS (it was written for Windows users, but has been found to work as is on a MacOS and a Linux computer). If you need help, please feel free to contact us. If you successfully ran installrtrmc.m on your computer, we'd love to know about any, if any, difficulties you may have resolved.

If you get this error:
??? Error using ==> spbuildmatrix
dpotrf (in spbuildmatrix): a leading minor is not positive definite.
Error in ==> lsqfit>buildmatrix at 132
    Achol = spbuildmatrix(problem.mask, U, lambda^2);
Error in ==> lsqfit at 65
        compumem.Achol = buildmatrix(problem, U);
Error in ==> rtrmcobjective at 66
       [W compumem] = lsqfit(problem, U, compumem);
then try to raise the value of lambda (the regularisation parameter). This error triggers when the least-squares problem (the computation of WU) does not have a unique solution, so that one of the diagonal blocks of the large matrix A does not have a proper Cholesky factorization.

Raghunandan Keshavan, co-author of OptSpace, maintains an interesting web page listing matrix completion-related papers and software.
The blog Nuit-Blanche featured a post about RTRMC on Sept. 12, 2011. The Matrix Factorization Jungle is another list of related software.

Last update: March 7, 2014