Robust Rotation Synchronization via Low-rank and Sparse Matrix Decomposition

This paper deals with the rotation synchronization problem, which arises in global registration of 3D point-sets and in structure from motion. The problem is formulated in an unprecedented way as a "low-rank and sparse" matrix decomposition that handles both outliers and missing data. A minimization strategy, dubbed R-GoDec, is also proposed and evaluated experimentally against state-of-the-art algorithms on simulated and real data. The results show that R-GoDec is the fastest among the robust algorithms.Comment: The material contained in this paper is part of a manuscript submitted to CVI

Greedy bilateral sketch, completion & smoothing

Copyright 2013 by the authors. Recovering a large low-rank matrix from highly corrupted, incomplete or sparse outlier over- whelmed observations is the crux of various in- triguing statistical problems. We explore the power of "greedy bilateral (GreB)" paradigm in reducing both time and sample complexities for solving these problems. GreB models a low- rank variable as a bilateral factorization, and up- dates the left and right factors in a mutually adap- tive and greedy incremental manner. We de- tail how to model and solve low-rank approx- imation, matrix completion and robust PCA in GreB's paradigm. On their MATLAB implemen- tations, approximating a noisy 104 × 104 matrix of rank 500 with SVD accuracy takes 6s; Movie- Lens10M matrix of size 69878 × 10677 can be completed in 10s from 30% of 107 ratings with RMSE 0.86 on the rest 70%; the low-rank back- ground and sparse moving outliers in a 120×160 video of 500 frames are accurately separated in 1s. This brings 30 to 100 times acceleration in solving these popular statistical problems

Rank-One Matrix Completion with Automatic Rank Estimation via L1-Norm Regularization

Completing a matrix from a small subset of its entries, i.e., matrix completion is a challenging problem arising from many real-world applications, such as machine learning and computer vision. One popular approach to solve the matrix completion problem is based on low-rank decomposition/factorization. Low-rank matrix decomposition-based methods often require a prespecified rank, which is difficult to determine in practice. In this paper, we propose a novel low-rank decomposition-based matrix completion method with automatic rank estimation. Our method is based on rank-one approximation, where a matrix is represented as a weighted summation of a set of rank-one matrices. To automatically determine the rank of an incomplete matrix, we impose L1-norm regularization on the weight vector and simultaneously minimize the reconstruction error. After obtaining the rank, we further remove the L1-norm regularizer and refine recovery results. With a correctly estimated rank, we can obtain the optimal solution under certain conditions. Experimental results on both synthetic and real-world data demonstrate that the proposed method not only has good performance in rank estimation, but also achieves better recovery accuracy than competing methods

Super-Resolution in Phase Space

This work considers the problem of super-resolution. The goal is to resolve a Dirac distribution from knowledge of its discrete, low-pass, Fourier measurements. Classically, such problems have been dealt with parameter estimation methods. Recently, it has been shown that convex-optimization based formulations facilitate a continuous time solution to the super-resolution problem. Here we treat super-resolution from low-pass measurements in Phase Space. The Phase Space transformation parametrically generalizes a number of well known unitary mappings such as the Fractional Fourier, Fresnel, Laplace and Fourier transforms. Consequently, our work provides a general super- resolution strategy which is backward compatible with the usual Fourier domain result. We consider low-pass measurements of Dirac distributions in Phase Space and show that the super-resolution problem can be cast as Total Variation minimization. Remarkably, even though are setting is quite general, the bounds on the minimum separation distance of Dirac distributions is comparable to existing methods.Comment: 10 Pages, short paper in part accepted to ICASSP 201