It has been observed in a variety of contexts that gradient descent methods have great success in solving low-rank matrix factorization problems, despite the relevant problem formulation being non-convex. We tackle a particular instance of this scenario, where we seek the @math -dimensional subspace spanned by a streaming data matrix. We apply the natural first order incremental gradient descent method, constraining the gradient method to the Grassmannian. In this paper, we propose an adaptive step size scheme that is greedy for the noiseless case, that maximizes the improvement of our metric of convergence at each data index @math , and yields an expected improvement for the noisy case. We show that, with noise-free data, this method converges from any random initialization to the global minimum of the problem. For noisy data, we provide the expected convergence rate of the proposed algorithm per iteration.
Turning to batch methods, @cite_23 @cite_30 provided the first theoretical guarantee for an alternating minimization algorithm for low-rank matrix recovery in the undersampled case. Under typical assumptions required for the matrix recovery problems @cite_20 , they established geometric convergence to the global optimal solution. Earlier work @cite_22 @cite_1 considered the same undersampled problem formulation and established convergence guarantees for a steepest descent method (and a preconditioned version) on the full gradient, performed on the Grassmannian. @cite_15 @cite_9 @cite_28 considered low rank semidefinite matrix estimation problems, where they reparamterized the underlying matrix as @math , and update @math via a first order gradient descent method. However, all these results require batch processing and a decent initialization that is close enough to the optimal point, resulting in a heavy computational burden and precluding problems with streaming data. We study random initialization, and our algorithm has fast, computationally efficient updates that can be performed in an online context.
@cite_30 @cite_22 @cite_28 @cite_9 @cite_1 @cite_23 @cite_15 @cite_20
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Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in statistics, machine learning, computer vision, as well as signal and image processing. In theory, this problem can be solved by the natural convex joint mixed relaxations (i.e., l 1 -norm and trace norm) under certain conditions. However, all current provable algorithms suffer from superlinear per-iteration cost, which severely limits their applicability to large-scale problems. In this paper, we propose a scalable, provable and structured robust bilinear factorization (RBF) method to recover low-rank and sparse matrices from missing and grossly corrupted data, i.e., robust matrix completion (RMC), or incomplete and grossly corrupted measurements, i.e., compressive principal component pursuit (CPCP). Specifically, we first present two small-scale matrix trace norm regularized bilinear factorization models for RMC and CPCP problems, in which repetitively calculating SVD of a large-scale matrix is replaced by updating two much smaller factor matrices. Then, we apply the alternating direction method of multipliers (ADMM) to efficiently solve the RMC problems. Finally, we provide the convergence analysis of our algorithm, and extend it to address general CPCP problems. Experimental results verified both the efficiency and effectiveness of our method compared with the state-of-the-art methods. In this paper we propose a global optimization-based approach to jointly matching a set of images. The estimated correspondences simultaneously maximize pairwise feature affinities and cycle consistency across multiple images. Unlike previous convex methods relying on semidefinite programming, we formulate the problem as a low-rank matrix recovery problem and show that the desired semidefiniteness of a solution can be spontaneously fulfilled. The low-rank formulation enables us to derive a fast alternating minimization algorithm in order to handle practical problems with thousands of features. Both simulation and real experiments demonstrate that the proposed algorithm can achieve a competitive performance with an order of magnitude speedup compared to the state-of-the-art algorithm. In the end, we demonstrate the applicability of the proposed method to match the images of different object instances and as a result the potential to reconstruct category-specific object models from those images. In this paper we propose a global optimization-based approach to jointly matching a set of images. The estimated correspondences simultaneously maximize pairwise feature affinities and cycle consistency across multiple images. Unlike previous convex methods relying on semidefinite programming, we formulate the problem as a low-rank matrix recovery problem and show that the desired semidefiniteness of a solution can be spontaneously fulfilled. The low-rank formulation enables us to derive a fast alternating minimization algorithm in order to handle practical problems with thousands of features. Both simulation and real experiments demonstrate that the proposed algorithm can achieve a competitive performance with an order of magnitude speedup compared to the state-of-the-art algorithm. In the end, we demonstrate the applicability of the proposed method to match the images of different object instances and as a result the potential to reconstruct category-specific object models from those images. The problem of minimizing the rank of a matrix subject to affine constraints has applications in several areas including machine learning, and is known to be NP-hard. A tractable relaxation for this problem is nuclear norm (or trace norm) minimization, which is guaranteed to find the minimum rank matrix under suitable assumptions. In this paper, we propose a family of Iterative Reweighted Least Squares algorithms IRLS-p (with 0 ≤ p ≤ 1), as a computationally efficient way to improve over the performance of nuclear norm minimization. The algorithms can be viewed as (locally) minimizing certain smooth approximations to the rank function. When p = 1, we give theoretical guarantees similar to those for nuclear norm minimization, that is, recovery of low-rank matrices under certain assumptions on the operator defining the constraints. For p < 1, IRLS-p shows better empirical performance in terms of recovering low-rank matrices than nuclear norm minimization. We provide an efficient implementation for IRLS-p, and also present a related family of algorithms, sIRLS-p. These algorithms exhibit competitive run times and improved recovery when compared to existing algorithms for random instances of the matrix completion problem, as well as on the MovieLens movie recommendation data set.