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Register a new userUniversal low-rank matrix recovery from Pauli measurements
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Yi-Kai Liu
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We study the problem of reconstructing an unknown matrix M of rank r and dimension d using O(rd poly log d) Pauli measurements. This has applications in quantum state tomography, and is a non-commutative analogue of a well-known problem in compressed sensing: recovering a sparse vector from a few of its Fourier coefficients. We show that almost all sets of O(rd log^6 d) Pauli measurements satisfy the rank-r restricted isometry property (RIP). This implies that M can be recovered from a fixed (universal) set of Pauli measurements, using nuclear-norm minimization (e.g., the matrix Lasso), with nearly-optimal bounds on the error. A similar result holds for any class of measurements that use an orthonormal operator basis whose elements have small operator norm. Our proof uses Dudleys inequality for Gaussian processes, together with bounds on covering numbers obtained via entropy duality.
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Tensors play a central role in many modern machine learning and signal processing applications. In such applications, the target tensor is usually of low rank, i.e., can be expressed as a sum of a small number of rank one tensors. This motivates us to consider the problem of low rank tensor recovery from a class of linear measurements called separable measurements. As specific examples, we focus on two distinct types of separable measurement mechanisms (a) Random projections, where each measurement corresponds to an inner product of the tensor with a suitable random tensor, and (b) the completion problem where measurements constitute revelation of a random set of entries. We present a computationally efficient algorithm, with rigorous and order-optimal sample complexity results (upto logarithmic factors) for tensor recovery. Our method is based on reduction to matrix completion sub-problems and adaptation of Leurgans method for tensor decomposition. We extend the methodology and sample complexity results to higher order tensors, and experimentally validate our theoretical results.
In this paper, we propose a new global analysis framework for a class of low-rank matrix recovery problems on the Riemannian manifold. We analyze the global behavior for the Riemannian optimization with random initialization. We use the Riemannian gradient descent algorithm to minimize a least squares loss function, and study the asymptotic behavior as well as the exact convergence rate. We reveal a previously unknown geometric property of the low-rank matrix manifold, which is the existence of spurious critical points for the simple least squares function on the manifold. We show that under some assumptions, the Riemannian gradient descent starting from a random initialization with high probability avoids these spurious critical points and only converges to the ground truth in nearly linear convergence rate, i.e. $mathcal{O}(text{log}(frac{1}{epsilon})+ text{log}(n))$ iterations to reach an $epsilon$-accurate solution. We use two applications as examples for our global analysis. The first one is a rank-1 matrix recovery problem. The second one is a generalization of the Gaussian phase retrieval problem. It only satisfies the weak isometry property, but has behavior similar to that of the first one except for an extra saddle set. Our convergence guarantee is nearly optimal and almost dimension-free, which fully explains the numerical observations. The global analysis can be potentially extended to other data problems with random measurement structures and empirical least squares loss functions.
Low rank matrix recovery is the focus of many applications, but it is a NP-hard problem. A popular way to deal with this problem is to solve its convex relaxation, the nuclear norm regularized minimization problem (NRM), which includes LASSO as a special case. There are some regularization parameter selection results for LASSO in vector case, such as screening rules, which improve the efficiency of the algorithms. However, there are no corresponding parameter selection results for NRM in matrix case. In this paper, we build up a novel rule to choose the regularization parameter for NRM under the help of duality theory. This rule claims that the regularization parameter can be easily chosen by feasible points of NRM and its dual problem, when the rank of the desired solution is no more than a given constant. In particular, we apply this idea to NRM with least square and Huber functions, and establish the easily calculated formula of regularization parameters. Finally, we report numerical results on some signal shapes, which state that our proposed rule shrinks the interval of the regularization parameter efficiently.
Estimating the rank of a corrupted data matrix is an important task in data science, most notably for choosing the number of components in principal component analysis. Significant progress on this task has been made using random matrix theory by characterizing the spectral properties of large noise matrices. However, utilizing such tools is not straightforward when the data matrix consists of count random variables, such as Poisson or binomial, in which case the noise can be heteroskedastic with an unknown variance in each entry. In this work, focusing on a Poisson random matrix with independent entries, we propose a simple procedure termed textit{biwhitening} that makes it possible to estimate the rank of the underlying data matrix (i.e., the Poisson parameter matrix) without any prior knowledge on its structure. Our approach is based on the key observation that one can scale the rows and columns of the data matrix simultaneously so that the spectrum of the corresponding noise agrees with the standard Marchenko-Pastur (MP) law, justifying the use of the MP upper edge as a threshold for rank selection. Importantly, the required scaling factors can be estimated directly from the observations by solving a matrix scaling problem via the Sinkhorn-Knopp algorithm. Aside from the Poisson distribution, we extend our biwhitening approach to other discrete distributions, such as the generalized Poisson, binomial, multinomial, and negative binomial. We conduct numerical experiments that corroborate our theoretical findings, and demonstrate our approach on real single-cell RNA sequencing (scRNA-seq) data, where we show that our results agree with a slightly overdispersed generalized Poisson model.
This paper develops a new class of nonconvex regularizers for low-rank matrix recovery. Many regularizers are motivated as convex relaxations of the matrix rank function. Our new factor group-sparse regularizers are motivated as a relaxation of the number of nonzero columns in a factorization of the matrix. These nonconvex regularizers are sharper than the nuclear norm; indeed, we show they are related to Schatten-$p$ norms with arbitrarily small $0 < p leq 1$. Moreover, these factor group-sparse regularizers can be written in a factored form that enables efficient and effective nonconvex optimization; notably, the method does not use singular value decomposition. We provide generalization error bounds for low-rank matrix completion which show improved upper bounds for Schatten-$p$ norm reglarization as $p$ decreases. Compared to the max norm and the factored formulation of the nuclear norm, factor group-sparse regularizers are more efficient, accurate, and robust to the initial guess of rank. Experiments show promising performance of factor group-sparse regularization for low-rank matrix completion and robust principal component analysis.
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