However, the proposed method and results do not have big impact in practical perspective because the convex regularized matrix factorization itself is very naive, and non-convex regularized low-rank matrix recovery is now widely studied and there are many related works such as truncated nuclear-norm[ex1], weighted nuclear-norm[ex2], capped-l1[ex3], LSP[ex4], SCAD[ex5], and MCP[ex6]. Also in another perspective, greedy rank-increment approach [ex7,ex8,ex9] for low-rank matrix recovery should be referred for discussion. This does not need to estimate initial d unlike regularized matrix factorization methods, and it is usually memory efficient.

The user ratings in recommendation systems are usually in the form of ordinal discrete values. To give more accurate prediction of such rating data, maximum margin matrix factorization (M3F) was proposed. Existing M3F algorithms, however, either have massive computational cost or require expensive model selection procedures to determine the number of latent factors (i.e. the rank of the matrix to be recovered), making them less practical for large scale data sets. To address these two challenges, in this paper, we formulate M3F with a known number of latent factors as the Riemannian optimization problem on a fixed-rank matrix manifold and present a block-wise nonlinear Riemannian conjugate gradient method to solve it efficiently. We then apply a simple and efficient active subspace search scheme to automatically detect the number of latent factors. Empirical studies on both synthetic data sets and large real-world data sets demonstrate the superior efficiency and effectiveness of the proposed method.