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Register a new userOptimal Weighted Low-rank Matrix Recovery with Subspace Prior Information
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Added by
Sajad Daei Omshi
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Matrix sensing is the problem of reconstructing a low-rank matrix from a few linear measurements. In many applications such as collaborative filtering, the famous Netflix prize problem, and seismic data interpolation, there exists some prior information about the column and row spaces of the ground-truth low-rank matrix. In this paper, we exploit this prior information by proposing a weighted optimization problem where its objective function promotes both rank and prior subspace information. Using the recent results in conic integral geometry, we obtain the unique optimal weights that minimize the required number of measurements. As simulation results confirm, the proposed convex program with optimal weights requires substantially fewer measurements than the regular nuclear norm minimization.
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This paper considers the problem of recovering a structured signal from a relatively small number of noisy measurements with the aid of a similar signal which is known beforehand. We propose a new approach to integrate prior information into the standard recovery procedure by maximizing the correlation between the prior knowledge and the desired signal. We then establish performance guarantees (in terms of the number of measurements) for the proposed method under sub-Gaussian measurements. Specific structured signals including sparse vectors, block-sparse vectors, and low-rank matrices are also analyzed. Furthermore, we present an interesting geometrical interpretation for the proposed procedure. Our results demonstrate that if prior information is good enough, then the proposed approach can (remarkably) outperform the standard recovery procedure. Simulations are provided to verify our results.
The orthogonal matching pursuit (OMP) algorithm is a commonly used algorithm for recovering $K$-sparse signals $xin mathbb{R}^{n}$ from linear model $y=Ax$, where $Ain mathbb{R}^{mtimes n}$ is a sensing matrix. A fundamental question in the performance analysis of OMP is the characterization of the probability that it can exactly recover $x$ for random matrix $A$. Although in many practical applications, in addition to the sparsity, $x$ usually also has some additional property (for example, the nonzero entries of $x$ independently and identically follow the Gaussian distribution), none of existing analysis uses these properties to answer the above question. In this paper, we first show that the prior distribution information of $x$ can be used to provide an upper bound on $|x|_1^2/|x|_2^2$, and then explore the bound to develop a better lower bound on the probability of exact recovery with OMP in $K$ iterations. Simulation tests are presented to illustrate the superiority of the new bound.
Suppose that a solution $widetilde{mathbf{x}}$ to an underdetermined linear system $mathbf{b} = mathbf{A} mathbf{x}$ is given. $widetilde{mathbf{x}}$ is approximately sparse meaning that it has a few large components compared to other small entries. However, the total number of nonzero components of $widetilde{mathbf{x}}$ is large enough to violate any condition for the uniqueness of the sparsest solution. On the other hand, if only the dominant components are considered, then it will satisfy the uniqueness conditions. One intuitively expects that $widetilde{mathbf{x}}$ should not be far from the true sparse solution $mathbf{x}_0$. We show that this intuition is the case by providing an upper bound on $| widetilde{mathbf{x}} - mathbf{x}_0|$ which is a function of the magnitudes of small components of $widetilde{mathbf{x}}$ but independent from $mathbf{x}_0$. This result is extended to the case that $mathbf{b}$ is perturbed by noise. Additionally, we generalize the upper bounds to the low-rank matrix recovery problem.
In matrix recovery from random linear measurements, one is interested in recovering an unknown $M$-by-$N$ matrix $X_0$ from $n<MN$ measurements $y_i=Tr(A_i^T X_0)$ where each $A_i$ is an $M$-by-$N$ measurement matrix with i.i.d random entries, $i=1,ldots,n$. We present a novel matrix recovery algorithm, based on approximate message passing, which iteratively applies an optimal singular value shrinker -- a nonconvex nonlinearity tailored specifically for matrix estimation. Our algorithm typically converges exponentially fast, offering a significant speedup over previously suggested matrix recovery algorithms, such as iterative solvers for Nuclear Norm Minimization (NNM). It is well known that there is a recovery tradeoff between the information content of the object $X_0$ to be recovered (specifically, its matrix rank $r$) and the number of linear measurements $n$ from which recovery is to be attempted. The precise tradeoff between $r$ and $n$, beyond which recovery by a given algorithm becomes possible, traces the so-called phase transition curve of that algorithm in the $(r,n)$ plane. The phase transition curve of our algorithm is noticeably better than that of NNM. Interestingly, it is close to the information-theoretic lower bound for the minimal number of measurements needed for matrix recovery, making it not only state-of-the-art in terms of convergence rate, but also near-optimal in terms of the matrices it successfully recovers.
In this work, we consider the problem of recovering analysis-sparse signals from under-sampled measurements when some prior information about the support is available. We incorporate such information in the recovery stage by suitably tuning the weights in a weighted $ell_1$ analysis optimization problem. Indeed, we try to set the weights such that the method succeeds with minimum number of measurements. For this purpose, we exploit the upper-bound on the statistical dimension of a certain cone to determine the weights. Our numerical simulations confirm that the introduced method with tuned weights outperforms the standard $ell_1$ analysis technique.
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