The abstract reads:
| We consider large matrices of low rank. We address the problem of recovering such matrices when most of the entries are unknown. Matrix completion finds applications in recommender systems. In this setting, the rows of the matrix may correspond to items and the columns may correspond to users. The known entries are the ratings given by users to some items. The aim is to predict the unobserved ratings. This problem is commonly stated in a constrained optimization framework. 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. Our methods, RTRMC 1 and 2, outperform state-of-the-art algorithms on a wide range of problem instances. |
| 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. |
| 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. |