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We give a new framework for solving the fundamental problem of low-rank matrix completion, i.e., approximating a rank-$r$ matrix $\mathbf{M} \in \mathbb{R}^{m \times n}$ (where $m \ge n$) from random observations. First, we provide an…

Machine Learning · Computer Science 2023-08-08 Jonathan A. Kelner , Jerry Li , Allen Liu , Aaron Sidford , Kevin Tian

The task of reconstructing a matrix given a sample of observedentries is known as the matrix completion problem. It arises ina wide range of problems, including recommender systems, collaborativefiltering, dimensionality reduction, image…

Statistics Theory · Mathematics 2014-12-20 Jean Lafond , Olga Klopp , Eric Moulines , Jospeh Salmon

An incoherent low-rank matrix can be efficiently reconstructed after observing a few of its entries at random, and then solving a convex program that minimizes the nuclear norm. In many applications, in addition to these entries,…

Information Theory · Computer Science 2018-03-14 Armin Eftekhari , Dehui Yang , Michael B. Wakin

The problem of completing high-dimensional matrices from a limited set of observations arises in many big data applications, especially, recommender systems. Existing matrix completion models generally follow either a memory- or a…

Machine Learning · Computer Science 2019-09-30 Duc Minh Nguyen , Robert Calderbank , Nikos Deligiannis

Completing a data matrix X has become an ubiquitous problem in modern data science, with applications in recommender systems, computer vision, and networks inference, to name a few. One typical assumption is that X is low-rank. A more…

Machine Learning · Computer Science 2018-08-03 Daniel L. Pimentel-Alarcón

We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends…

Statistics Theory · Mathematics 2019-09-24 Daniel Irving Bernstein , Grigoriy Blekherman , Rainer Sinn

In this work, we consider the matrix completion problem, where the objective is to reconstruct a low-rank matrix from a few observed entries. A commonly employed approach involves nuclear norm minimization. For this method to succeed, the…

Signal Processing · Electrical Eng. & Systems 2024-06-25 Hamideh. Sadat Fazael Ardakani , Sajad Daei , Arash Amini , Mikael Skoglund , Gabor Fodor

As a paradigm to recover unknown entries of a matrix from partial observations, low-rank matrix completion (LRMC) has generated a great deal of interest. Over the years, there have been lots of works on this topic but it might not be easy…

Data Structures and Algorithms · Computer Science 2019-07-30 Luong Trung Nguyen , Junhan Kim , Byonghyo Shim

The problem of completing a low-rank matrix from a subset of its entries is often encountered in the analysis of incomplete data sets exhibiting an underlying factor model with applications in collaborative filtering, computer vision and…

Machine Learning · Computer Science 2009-02-24 Amit Singer , Mihai Cucuringu

We consider the matrix completion problem of recovering a structured low rank matrix with partially observed entries with mixed data types. Vast majority of the solutions have proposed computationally feasible estimators with strong…

Machine Learning · Statistics 2020-05-27 Daqian Sun , Martin T. Wells

We consider a problem of significant practical importance, namely, the reconstruction of a low-rank data matrix from a small subset of its entries. This problem appears in many areas such as collaborative filtering, computer vision and…

Machine Learning · Computer Science 2009-11-04 Raghunandan H. Keshavan , Andrea Montanari , Sewoong Oh

Matrix completion is a classical problem in data science wherein one attempts to reconstruct a low-rank matrix while only observing some subset of the entries. Previous authors have phrased this problem as a nuclear norm minimization…

Machine Learning · Computer Science 2019-04-19 Christian Parkinson , Kevin Huynh , Deanna Needell

Consider a movie recommendation system where apart from the ratings information, side information such as user's age or movie's genre is also available. Unlike standard matrix completion, in this setting one should be able to predict…

Machine Learning · Computer Science 2013-06-05 Prateek Jain , Inderjit S. Dhillon

The task of estimating a matrix given a sample of observed entries is known as the \emph{matrix completion problem}. Most works on matrix completion have focused on recovering an unknown real-valued low-rank matrix from a random sample of…

Statistics Theory · Mathematics 2014-08-27 Olga Klopp , Jean Lafond , Eric Moulines , Joseph Salmon

Low-rank matrix completion concerns the problem of estimating unobserved entries in a matrix using a sparse set of observed entries. We consider the non-uniform setting where the observed entries are sampled with highly varying…

Machine Learning · Statistics 2024-03-04 Xumei Xi , Christina Lee Yu , Yudong Chen

We present a novel algebraic combinatorial view on low-rank matrix completion based on studying relations between a few entries with tools from algebraic geometry and matroid theory. The intrinsic locality of the approach allows for the…

Machine Learning · Computer Science 2014-08-20 Franz J. Király , Louis Theran , Ryota Tomioka

Despite the popularity of low-rank matrix completion, the majority of its theory has been developed under the assumption of random observation patterns, whereas very little is known about the practically relevant case of non-random…

Machine Learning · Computer Science 2023-02-24 Manolis C. Tsakiris

This paper studies the low-rank matrix completion problem from an information theoretic perspective. The completion problem is rephrased as a communication problem of an (uncoded) low-rank matrix source over an erasure channel. The paper…

Information Theory · Computer Science 2016-09-08 Sriram Vishwanath

Exact matrix completion and low rank matrix estimation problems has been studied in different underlying conditions. In this work we study exact low-rank completion under non-degenerate noise model. Non-degenerate random noise model has…

Machine Learning · Computer Science 2022-04-06 Jafar Jafarov

In many applications, e.g., recommender systems and traffic monitoring, the data comes in the form of a matrix that is only partially observed and low rank. A fundamental data-analysis task for these datasets is matrix completion, where the…

Machine Learning · Computer Science 2017-05-02 Natali Ruchansky , Mark Crovella , Evimaria Terzi