Non-Convex Rank Minimization via an Empirical Bayesian Approach

In many applications that require matrix solutions of minimal rank, the underlying cost function is non-convex leading to an intractable, NP-hard optimization problem. Consequently, the convex nuclear norm is frequently used as a surrogate penalty term for matrix rank. The problem is that in many practical scenarios there is no longer any guarantee that we can correctly estimate generative low-rank matrices of interest, theoretical special cases notwithstanding. Consequently, this paper proposes an alternative empirical Bayesian procedure build upon a variational approximation that, unlike the nuclear norm, retains the same globally minimizing point estimate as the rank function under many useful constraints. However, locally minimizing solutions are largely smoothed away via marginalization, allowing the algorithm to succeed when standard convex relaxations completely fail. While the proposed methodology is generally applicable to a wide range of low-rank applications, we focus our attention on the robust principal component analysis problem (RPCA), which involves estimating an unknown low-rank matrix with unknown sparse corruptions. Theoretical and empirical evidence are presented to show that our method is potentially superior to related MAP-based approaches, for which the convex principle component pursuit (PCP) algorithm (Candes et al., 2011) can be viewed as a special case.Comment: 10 pages, 6 figures, UAI 2012 pape

Exploring Algorithmic Limits of Matrix Rank Minimization under Affine Constraints

Many applications require recovering a matrix of minimal rank within an affine constraint set, with matrix completion a notable special case. Because the problem is NP-hard in general, it is common to replace the matrix rank with the nuclear norm, which acts as a convenient convex surrogate. While elegant theoretical conditions elucidate when this replacement is likely to be successful, they are highly restrictive and convex algorithms fail when the ambient rank is too high or when the constraint set is poorly structured. Non-convex alternatives fare somewhat better when carefully tuned; however, convergence to locally optimal solutions remains a continuing source of failure. Against this backdrop we derive a deceptively simple and parameter-free probabilistic PCA-like algorithm that is capable, over a wide battery of empirical tests, of successful recovery even at the theoretical limit where the number of measurements equal the degrees of freedom in the unknown low-rank matrix. Somewhat surprisingly, this is possible even when the affine constraint set is highly ill-conditioned. While proving general recovery guarantees remains evasive for non-convex algorithms, Bayesian-inspired or otherwise, we nonetheless show conditions whereby the underlying cost function has a unique stationary point located at the global optimum; no existing cost function we are aware of satisfies this same property. We conclude with a simple computer vision application involving image rectification and a standard collaborative filtering benchmark

Asymptotic Analysis of MAP Estimation via the Replica Method and Applications to Compressed Sensing

The replica method is a non-rigorous but well-known technique from statistical physics used in the asymptotic analysis of large, random, nonlinear problems. This paper applies the replica method, under the assumption of replica symmetry, to study estimators that are maximum a posteriori (MAP) under a postulated prior distribution. It is shown that with random linear measurements and Gaussian noise, the replica-symmetric prediction of the asymptotic behavior of the postulated MAP estimate of an n-dimensional vector "decouples" as n scalar postulated MAP estimators. The result is based on applying a hardening argument to the replica analysis of postulated posterior mean estimators of Tanaka and of Guo and Verdu. The replica-symmetric postulated MAP analysis can be readily applied to many estimators used in compressed sensing, including basis pursuit, lasso, linear estimation with thresholding, and zero norm-regularized estimation. In the case of lasso estimation the scalar estimator reduces to a soft-thresholding operator, and for zero norm-regularized estimation it reduces to a hard-threshold. Among other benefits, the replica method provides a computationally-tractable method for precisely predicting various performance metrics including mean-squared error and sparsity pattern recovery probability.Comment: 22 pages; added details on the replica symmetry assumptio

Off-The-Grid Spectral Compressed Sensing With Prior Information

Recent research in off-the-grid compressed sensing (CS) has demonstrated that, under certain conditions, one can successfully recover a spectrally sparse signal from a few time-domain samples even though the dictionary is continuous. In this paper, we extend off-the-grid CS to applications where some prior information about spectrally sparse signal is known. We specifically consider cases where a few contributing frequencies or poles, but not their amplitudes or phases, are known a priori. Our results show that equipping off-the-grid CS with the known-poles algorithm can increase the probability of recovering all the frequency components.Comment: 5 pages, 4 figure

Super-resolution Line Spectrum Estimation with Block Priors

We address the problem of super-resolution line spectrum estimation of an undersampled signal with block prior information. The component frequencies of the signal are assumed to take arbitrary continuous values in known frequency blocks. We formulate a general semidefinite program to recover these continuous-valued frequencies using theories of positive trigonometric polynomials. The proposed semidefinite program achieves super-resolution frequency recovery by taking advantage of known structures of frequency blocks. Numerical experiments show great performance enhancements using our method.Comment: 7 pages, double colum

Non-parametric statistical thresholding for sparse magnetoencephalography source reconstructions.

Uncovering brain activity from magnetoencephalography (MEG) data requires solving an ill-posed inverse problem, greatly confounded by noise, interference, and correlated sources. Sparse reconstruction algorithms, such as Champagne, show great promise in that they provide focal brain activations robust to these confounds. In this paper, we address the technical considerations of statistically thresholding brain images obtained from sparse reconstruction algorithms. The source power distribution of sparse algorithms makes this class of algorithms ill-suited to "conventional" techniques. We propose two non-parametric resampling methods hypothesized to be compatible with sparse algorithms. The first adapts the maximal statistic procedure to sparse reconstruction results and the second departs from the maximal statistic, putting forth a less stringent procedure that protects against spurious peaks. Simulated MEG data and three real data sets are utilized to demonstrate the efficacy of the proposed methods. Two sparse algorithms, Champagne and generalized minimum-current estimation (G-MCE), are compared to two non-sparse algorithms, a variant of minimum-norm estimation, sLORETA, and an adaptive beamformer. The results, in general, demonstrate that the already sparse images obtained from Champagne and G-MCE are further thresholded by both proposed statistical thresholding procedures. While non-sparse algorithms are thresholded by the maximal statistic procedure, they are not made sparse. The work presented here is one of the first attempts to address the problem of statistically thresholding sparse reconstructions, and aims to improve upon this already advantageous and powerful class of algorithm

A closed-form solution to estimate uncertainty in non-rigid structure from motion

Semi-Definite Programming (SDP) with low-rank prior has been widely applied in Non-Rigid Structure from Motion (NRSfM). Based on a low-rank constraint, it avoids the inherent ambiguity of basis number selection in conventional base-shape or base-trajectory methods. Despite the efficiency in deformable shape reconstruction, it remains unclear how to assess the uncertainty of the recovered shape from the SDP process. In this paper, we present a statistical inference on the element-wise uncertainty quantification of the estimated deforming 3D shape points in the case of the exact low-rank SDP problem. A closed-form uncertainty quantification method is proposed and tested. Moreover, we extend the exact low-rank uncertainty quantification to the approximate low-rank scenario with a numerical optimal rank selection method, which enables solving practical application in SDP based NRSfM scenario. The proposed method provides an independent module to the SDP method and only requires the statistic information of the input 2D tracked points. Extensive experiments prove that the output 3D points have identical normal distribution to the 2D trackings, the proposed method and quantify the uncertainty accurately, and supports that it has desirable effects on routinely SDP low-rank based NRSfM solver.Comment: 9 pages, 2 figure