More Like this

1. Tensor Methods for Nonlinear Matrix Completion

   Ongie, Greg ; Pimentel-Alarcon, Daniel ; Nowak, Robert ; Willett, Rebecca ; Balzano, Laura ( September 2020 , SIAM journal on mathematics of data science)

   null (Ed.)

   In the low-rank matrix completion (LRMC) problem, the low-rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. This paper extends this thinking to cases where the columns are points on a low-dimensional nonlinear algebraic variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC). Matrices whose columns belong to a union of subspaces are an important special case. We propose a LADMC algorithm that leverages existing LRMC methods on a tensorized representation of the data. For example, a second-order tensorized representation is formed by taking the Kronecker product of each column with itself, and we consider higher order tensorizations as well. This approach will succeed in many cases where traditional LRMC is guaranteed to fail because the data are low-rank in the tensorized representation but not in the original representation. We also provide a formal mathematical justication for the success of our method. In particular, we give bounds of the rank of these data in the tensorized representation, and we prove sampling requirements to guarantee uniqueness of the solution. We also provide experimental results showing that the new approach outperforms existing state-of-the-art methods for matrix completion under a union of subspaces model. 

   more » « less
2. High-Rank Matrix Completion with Side Information

   Wang, Y. ; Elhamifar, E. ( January 2018 , AAAI Conference on Artificial Intelligence)

   We address the problem of high-rank matrix completion with side information. In contrast to existing work dealing with side information, which assume that the data matrix is low-rank, we consider the more general scenario where the columns of the data matrix are drawn from a union of low-dimensional subspaces, which can lead to a high rank matrix. Our goal is to complete the matrix while taking advantage of the side information. To do so, we use the self-expressive property of the data, searching for a sparse representation of each column of matrix as a combination of a few other columns. More specifically, we propose a factorization of the data matrix as the product of side information matrices with an unknown interaction matrix, under which each column of the data matrix can be reconstructed using a sparse combination of other columns. As our proposed optimization, searching for missing entries and sparse coefficients, is non-convex and NP-hard, we propose a lifting framework, where we couple sparse coefficients and missing values and define an equivalent optimization that is amenable to convex relaxation. We also propose a fast implementation of our convex framework using a Linearized Alternating Direction Method. By extensive experiments on both synthetic and real data, and, in particular, by studying the problem of multi-label learning, we demonstrate that our method outperforms existing techniques in both low-rank and high-rank data regimes 

   more » « less
3. High-Rank Matrix Completion with Side Information

   Wang, Y. ; Elhamifar, E. ( January 2018 , AAAI Conference on Artificial Intelligence)

   We address the problem of high-rank matrix completion with side information. In contrast to existing work dealing with side information, which assume that the data matrix is low-rank, we consider the more general scenario where the columns of the data matrix are drawn from a union of low-dimensional subspaces, which can lead to a high rank matrix. Our goal is to complete the matrix while taking advantage of the side information. To do so, we use the self-expressive property of the data, searching for a sparse representation of each column of matrix as a combination of a few other columns. More specifically, we propose a factorization of the data matrix as the product of side information matrices with an unknown interaction matrix, under which each column of the data matrix can be reconstructed using a sparse combination of other columns. As our proposed optimization, searching for missing entries and sparse coefficients, is non-convex and NP-hard, we propose a lifting framework, where we couple sparse coefficients and missing values and define an equivalent optimization that is amenable to convex relaxation. We also propose a fast implementation of our convex framework using a Linearized Alternating Direction Method. By extensive experiments on both synthetic and real data, and, in particular, by studying the problem of multi-label learning, we demonstrate that our method outperforms existing techniques in both low-rank and high-rank data regimes. 

   more » « less
4. Recovery analysis of damped spectrally sparse signals and its relation to MUSIC

   https://doi.org/10.1093/imaiai/iaaa024  

   Li, Shuang ; Mansour, Hassan ; Wakin, Michael B ( October 2020 , Information and Inference: A Journal of the IMA)

   null (Ed.)

   Abstract One of the classical approaches for estimating the frequencies and damping factors in a spectrally sparse signal is the MUltiple SIgnal Classification (MUSIC) algorithm, which exploits the low-rank structure of an autocorrelation matrix. Low-rank matrices have also received considerable attention recently in the context of optimization algorithms with partial observations, and nuclear norm minimization (NNM) has been widely used as a popular heuristic of rank minimization for low-rank matrix recovery problems. On the other hand, it has been shown that NNM can be viewed as a special case of atomic norm minimization (ANM), which has achieved great success in solving line spectrum estimation problems. However, as far as we know, the general ANM (not NNM) considered in many existing works can only handle frequency estimation in undamped sinusoids. In this work, we aim to fill this gap and deal with damped spectrally sparse signal recovery problems. In particular, inspired by the dual analysis used in ANM, we offer a novel optimization-based perspective on the classical MUSIC algorithm and propose an algorithm for spectral estimation that involves searching for the peaks of the dual polynomial corresponding to a certain NNM problem, and we show that this algorithm is in fact equivalent to MUSIC itself. Building on this connection, we also extend the classical MUSIC algorithm to the missing data case. We provide exact recovery guarantees for our proposed algorithms and quantify how the sample complexity depends on the true spectral parameters. In particular, we provide a parameter-specific recovery bound for low-rank matrix recovery of jointly sparse signals rather than use certain incoherence properties as in existing literature. Simulation results also indicate that the proposed algorithms significantly outperform some relevant existing methods (e.g., ANM) in frequency estimation of damped exponentials. 

   more » « less
5. Optimal estimation of bacterial growth rates based on a permuted monotone matrix

   https://doi.org/10.1093/biomet/asaa082  

   Ma, Rong ; Cai, T Tony ; Li, Hongzhe ( October 2020 , Biometrika)

   Summary Motivated by the problem of estimating bacterial growth rates for genome assemblies from shotgun metagenomic data, we consider the permuted monotone matrix model $Y=\Theta\Pi+Z$ where $Y\in \mathbb{R}^{n\times p}$ is observed, $\Theta\in \mathbb{R}^{n\times p}$ is an unknown approximately rank-one signal matrix with monotone rows, $\Pi \in \mathbb{R}^{p\times p}$ is an unknown permutation matrix, and $Z\in \mathbb{R}^{n\times p}$ is the noise matrix. In this article we study estimation of the extreme values associated with the signal matrix $\Theta$, including its first and last columns and their difference. Treating these estimation problems as compound decision problems, minimax rate-optimal estimators are constructed using the spectral column-sorting method. Numerical experiments on simulated and synthetic microbiome metagenomic data are conducted, demonstrating the superiority of the proposed methods over existing alternatives. The methods are illustrated by comparing the growth rates of gut bacteria in inflammatory bowel disease patients and control subjects. 

   more » « less