We study the problem of estimating a low-rank positive semidefinite (PSD) matrix from a set of rank-one measurements using sensing vectors composed of i.i.d. standard Gaussian entries, which are possibly corrupted by arbitrary outliers. This problem arises from applications, such as phase retrieval, covariance sketching, quantum space tomography, and power spectrum estimation. We first propose a convex optimization algorithm that seeks the PSD matrix with the minimum @math -norm of the observation residual. The advantage of our algorithm is that it is free of parameters, therefore, eliminating the need for tuning parameters and allowing easy implementations. We establish that with high probability, a low-rank PSD matrix can be exactly recovered as soon as the number of measurements is large enough, even when a fraction of the measurements are corrupted by outliers with arbitrary magnitudes. Moreover, the recovery is also stable against bounded noise. With the additional information of an upper bound of the rank of the PSD matrix, we propose another nonconvex algorithm based on subgradient descent that demonstrates excellent empirical performance in terms of computational efficiency and accuracy.
Broadly speaking, our problem is related to low-rank matrix recovery from an under-determined linear system @cite_27 @cite_17 @cite_19 , where the linear measurements are drawn from inner products with rank-one sensing matrices. It is due to this special structure of the sensing matrices that we can eliminate the trace minimization, and only consider the feasibility constraint for PSD matrices. Standard approaches for separating low-rank and sparse components @cite_32 @cite_2 @cite_20 @cite_33 @cite_4 via convex optimization are given as where @math is a regularization parameter that requires to be tuned properly. In contrast, the formulation is parameter-free.
@cite_4 @cite_33 @cite_32 @cite_19 @cite_27 @cite_2 @cite_20 @cite_17
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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. Low-rank matrix recovery from a corrupted observation has many applications in computer vision. Conventional methods address this problem by iterating between nuclear norm minimization and sparsity minimization. However, iterative nuclear norm minimization is computationally prohibitive for large-scale data (e.g., video) analysis. In this paper, we propose a Robust Orthogonal Subspace Learning (ROSL) method to achieve efficient low-rank recovery. Our intuition is a novel rank measure on the low-rank matrix that imposes the group sparsity of its coefficients under orthonormal subspace. We present an efficient sparse coding algorithm to minimize this rank measure and recover the low-rank matrix at quadratic complexity of the matrix size. We give theoretical proof to validate that this rank measure is lower bounded by nuclear norm and it has the same global minimum as the latter. To further accelerate ROSL to linear complexity, we also describe a faster version (ROSL+) empowered by random sampling. Our extensive experiments demonstrate that both ROSL and ROSL+ provide superior efficiency against the state-of-the-art methods at the same level of recovery accuracy. 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 improve existing results in the field of compressed sensing and matrix completion when sampled data may be grossly corrupted. We introduce three new theorems. (1) In compressed sensing, we show that if the m×n sensing matrix has independent Gaussian entries, then one can recover a sparse signal x exactly by tractable l 1 minimization even if a positive fraction of the measurements are arbitrarily corrupted, provided the number of nonzero entries in x is O(m (log(n m)+1)). (2) In the very general sensing model introduced in Candes and Plan (IEEE Trans. Inf. Theory 57(11):7235–7254, 2011) and assuming a positive fraction of corrupted measurements, exact recovery still holds if the signal now has O(m (log2 n)) nonzero entries. (3) Finally, we prove that one can recover an n×n low-rank matrix from m corrupted sampled entries by tractable optimization provided the rank is on the order of O(m (nlog2 n)); again, this holds when there is a positive fraction of corrupted samples.