Low-rank approximation ≈ Low-rank approximation
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Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview Open
Substantial progress has been made recently on developing provably accurate\nand efficient algorithms for low-rank matrix factorization via nonconvex\noptimization. While conventional wisdom often takes a dim view of nonconvex\noptimizatio…
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Low-Rank Matrix Recovery via Efficient Schatten p-Norm Minimization Open
As an emerging machine learning and information retrieval technique, the matrix completion has been successfully applied to solve many scientific applications, such as collaborative prediction in information retrieval, video completion in …
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No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis Open
In this paper we develop a new framework that captures the common landscape underlying the common non-convex low-rank matrix problems including matrix sensing, matrix completion and robust PCA. In particular, we show for all above problems…
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Low rank alternating direction method of multipliers reconstruction for MR fingerprinting Open
Purpose The proposed reconstruction framework addresses the reconstruction accuracy, noise propagation and computation time for magnetic resonance fingerprinting. Methods Based on a singular value decomposition of the signal evolution, mag…
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Quantum Recommendation Systems Open
A recommendation system uses the past purchases or ratings of n products by a group of m users, in order to provide personalized recommendations to individual users. The information is modeled as an m \times n preference matrix which is as…
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Why Are Big Data Matrices Approximately Low Rank? Open
Matrices of (approximate) low rank are pervasive in data science, appearing in movie preferences, text documents, survey data, medical records, and genomics. While there is a vast literature on how to exploit low rank structure in these da…
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Light Field Inpainting Propagation via Low Rank Matrix Completion Open
Building up on the advances in low rank matrix completion, this article presents a novel method for propagating the inpainting of the central view of a light field to all the other views. After generating a set of warped versions of the in…
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Global optimality of local search for low rank matrix recovery Open
We show that there are no spurious local minima in the non-convex factorized parametrization of low-rank matrix recovery from incoherent linear measurements. With noisy measurements we show all local minima are very close to a global optim…
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Enhanced Low-Rank Matrix Approximation Open
This letter proposes to estimate low-rank matrices by formulating a convex\noptimization problem with non-convex regularization. We employ parameterized\nnon-convex penalty functions to estimate the non-zero singular values more\naccuratel…
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Randomized Matrix Decompositions Using <i>R</i> Open
Matrix decompositions are fundamental tools in the area of applied\nmathematics, statistical computing, and machine learning. In particular,\nlow-rank matrix decompositions are vital, and widely used for data analysis,\ndimensionality redu…
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Sparse and Low-Rank Decomposition of a Hankel Structured Matrix for Impulse Noise Removal Open
Recently, the annihilating filter-based low-rank Hankel matrix (ALOHA) approach was proposed as a powerful image inpainting method. Based on the observation that smoothness or textures within an image patch correspond to sparse spectral co…
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Discretized Dynamical Low-Rank Approximation in the Presence of Small Singular Values Open
Low-rank approximations to large time-dependent matrices and tensors are the subject of this paper. These matrices and tensors either are given explicitly or are the unknown solutions of matrix and tensor differential equations. Based on s…
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Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization Open
This paper studies noisy low-rank matrix completion: given partial and noisy entries of a large low-rank matrix, the goal is to estimate the underlying matrix faithfully and efficiently. Arguably one of the most popular paradigms to tackle…
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$\ell_0$ -Motivated Low-Rank Sparse Subspace Clustering Open
In many applications, high-dimensional data points can be well represented by low-dimensional subspaces. To identify the subspaces, it is important to capture a global and local structure of the data which is achieved by imposing low-rank …
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Riemannian Optimization for High-Dimensional Tensor Completion Open
Tensor completion aims to reconstruct a high-dimensional data set where the vast majority of entries is missing. The assumption of low-rank structure in the underlying original data allows us to cast the completion problem into an optimiza…
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Matrix completion via max-norm constrained optimization Open
Matrix completion has been well studied under the uniform sampling model and the trace-norm regularized methods perform well both theoretically and numerically in such a setting. However, the uniform sampling model is unrealistic for a ran…
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A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion Open
Low-rank matrix completion (LRMC) problems arise in a wide variety of\napplications. Previous theory mainly provides conditions for completion under\nmissing-at-random samplings. This paper studies deterministic conditions for\ncompletion.…
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TRP: Trained Rank Pruning for Efficient Deep Neural Networks Open
To enable DNNs on edge devices like mobile phones, low-rank approximation has been widely adopted because of its solid theoretical rationale and efficient implementations. Several previous works attempted to directly approximate a pre-trai…
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Motion‐robust reconstruction of multishot diffusion‐weighted images without phase estimation through locally low‐rank regularization Open
Purpose The goal of this work is to propose a motion robust reconstruction method for diffusion‐weighted MRI that resolves shot‐to‐shot phase mismatches without using phase estimation. Methods Assuming that shot‐to‐shot phase variations ar…
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Fast Low-Rank Bayesian Matrix Completion With Hierarchical Gaussian Prior Models Open
The problem of low rank matrix completion is considered in this paper. To\nexploit the underlying low-rank structure of the data matrix, we propose a\nhierarchical Gaussian prior model, where columns of the low-rank matrix are\nassumed to …
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Sketchy Decisions: Convex Low-Rank Matrix Optimization with Optimal Storage Open
This paper concerns a fundamental class of convex matrix optimization problems. It presents the first algorithm that uses optimal storage and provably computes a low-rank approximation of a solution. In particular, when all solutions have …
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Rapid Robust Principal Component Analysis: CUR Accelerated Inexact Low Rank Estimation Open
Robust principal component analysis (RPCA) is a widely used tool for\ndimension reduction. In this work, we propose a novel non-convex algorithm,\ncoined Iterated Robust CUR (IRCUR), for solving RPCA problems, which\ndramatically improves …
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Beyond Low Rank + Sparse: Multiscale Low Rank Matrix Decomposition Open
We present a natural generalization of the recent low rank + sparse matrix decomposition and consider the decomposition of matrices into components of multiple scales. Such decomposition is well motivated in practice as data matrices often…
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Low-tubal-rank tensor completion using alternating minimization Open
The low-tubal-rank tensor model has been recently proposed for real-world\nmultidimensional data. In this paper, we study the low-tubal-rank tensor\ncompletion problem, i.e., to recover a third-order tensor by observing a subset\nof its el…
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An asymptotic-preserving dynamical low-rank method for the multi-scale multi-dimensional linear transport equation Open
We propose a dynamical low-rank method to reduce the computational complexity for solving the multi-scale multi-dimensional linear transport equation. The method is based on a macro-micro decomposition of the equation; the low-rank approxi…
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MRI artifact correction using sparse + low‐rank decomposition of annihilating filter‐based hankel matrix Open
Purpose Magnetic resonance imaging (MRI) artifacts are originated from various sources including instability of an magnetic resonance (MR) system, patient motion, inhomogeneities of gradient fields, and so on. Such MRI artifacts are usuall…
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Subspace-Orbit Randomized Decomposition for Low-Rank Matrix Approximations Open
An efficient, accurate and reliable approximation of a matrix by one of lower rank is a fundamental task in numerical linear algebra and signal processing applications. In this paper, we introduce a new matrix decomposition approach termed…
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Sparse principal component analysis via regularized low rank matrix approximation Open
Principal component analysis (PCA) is a widely used tool for data analysis and dimension reduction in applications throughout science and engineering. However, the principal components (PCs) can sometimes be difficult to interpret, because…
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A Fast Low Rank Hankel Matrix Factorization Reconstruction Method for Non-Uniformly Sampled Magnetic Resonance Spectroscopy Open
Multidimensional magnetic resonance spectroscopy (MRS) serves as a valuable tool to analyze metabolites in medical imaging, complex chemical compounds in the chemistry, and protein structures in biology. The data acquisition time, however,…
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Joint Capped Norms Minimization for Robust Matrix Recovery Open
The low-rank matrix recovery is an important machine learning research topic with various scientific applications. Most existing low-rank matrix recovery methods relax the rank minimization problem via the trace norm minimization. However,…