An Alternating Direction Algorithm for Matrix Completion with Nonnegative Factors

This paper introduces an algorithm for the nonnegative matrix factorization-and-completion problem, which aims to find nonnegative low-rank matrices X and Y so that the product XY approximates a nonnegative data matrix M whose elements are partially known (to a certain accuracy). This problem aggregates two existing problems: (i) nonnegative matrix factorization where all entries of M are given, and (ii) low-rank matrix completion where nonnegativity is not required. By taking the advantages of both nonnegativity and low-rankness, one can generally obtain superior results than those of just using one of the two properties. We propose to solve the non-convex constrained least-squares problem using an algorithm based on the classic alternating direction augmented Lagrangian method. Preliminary convergence properties of the algorithm and numerical simulation results are presented. Compared to a recent algorithm for nonnegative matrix factorization, the proposed algorithm produces factorizations of similar quality using only about half of the matrix entries. On tasks of recovering incomplete grayscale and hyperspectral images, the proposed algorithm yields overall better qualities than those produced by two recent matrix-completion algorithms that do not exploit nonnegativity

Partial Sum Minimization of Singular Values in Robust PCA: Algorithm and Applications

Robust Principal Component Analysis (RPCA) via rank minimization is a powerful tool for recovering underlying low-rank structure of clean data corrupted with sparse noise/outliers. In many low-level vision problems, not only it is known that the underlying structure of clean data is low-rank, but the exact rank of clean data is also known. Yet, when applying conventional rank minimization for those problems, the objective function is formulated in a way that does not fully utilize a priori target rank information about the problems. This observation motivates us to investigate whether there is a better alternative solution when using rank minimization. In this paper, instead of minimizing the nuclear norm, we propose to minimize the partial sum of singular values, which implicitly encourages the target rank constraint. Our experimental analyses show that, when the number of samples is deficient, our approach leads to a higher success rate than conventional rank minimization, while the solutions obtained by the two approaches are almost identical when the number of samples is more than sufficient. We apply our approach to various low-level vision problems, e.g. high dynamic range imaging, motion edge detection, photometric stereo, image alignment and recovery, and show that our results outperform those obtained by the conventional nuclear norm rank minimization method.Comment: Accepted in Transactions on Pattern Analysis and Machine Intelligence (TPAMI). To appea

Non-Rigid Structure from Motion for Complex Motion

Recovering deformable 3D motion from temporal 2D point tracks in a monocular video is an open problem with many everyday applications throughout science and industry, or the new augmented reality. Recently, several techniques have been proposed to deal the problem called Non-Rigid Structure from Motion (NRSfM), however, they can exhibit poor reconstruction performance on complex motion. In this project, we will analyze these situations for primitive human actions such as walk, run, sit, jump, etc. on different scenarios, reviewing first the current techniques to finally present our novel method. This approach is able to model complex motion into a union of subspaces, rather than the summation occurring in standard low-rank shape methods, allowing better reconstruction accuracy. Experiments in a wide range of sequences and types of motion illustrate the benefits of this new approac