NDLI: Information-theoretic limits of matrix completion

Content Provider	IEEE Xplore Digital Library
Author	Riegler, E. Stotz, D. Bolcskei, H.
Copyright Year	2015
Description	Author affiliation: Dept. IT & EE, ETH Zurich, Zurich, Switzerland (Riegler, E.; Stotz, D.; Bolcskei, H.)
Abstract	We propose an information-theoretic framework for matrix completion. The theory goes beyond the low-rank structure and applies to general matrices of “low description complexity”. Specifically, we consider random matrices X ∈ $ℝ^{m×n}$ of arbitrary distribution (continuous, discrete, discrete-continuous mixture, or even singular). With S ⊆ $ℝ^{m×n}$ an ε-support set of X, i.e., P[X ∈ S] ≥ 1 - ε, and equation denoting the lower Minkowski dimension of S, we show that equation measurements of the form $〈A_{i},X〉,$ with $A_{i}$ denoting the measurement matrices, suffice to recover X with probability of error at most ε. The result holds for Lebesgue a.a. $A_{i}$ and does not need incoherence between the $A_{i}$ and the unknown matrix X. We furthermore show that equation measurements also suffice to recover the unknown matrix X from measurements taken with rank-one $A_{i},$ again this applies to a.a. rank-one $A_{i}.$ Rank-one measurement matrices are attractive as they require less storage space than general measurement matrices and can be applied faster. Particularizing our results to the recovery of low-rank matrices, we find that k > (m+n-r)r measurements are sufficient to recover matrices of rank at most r. Finally, we construct a class of rank-r matrices that can be recovered with arbitrarily small probability of error from k <; (m + n - r)r measurements.
Starting Page	1836
Ending Page	1840
File Size	314319
Page Count	5
File Format	PDF
ISSN	21578117
e-ISBN	9781467377041
DOI	10.1109/ISIT.2015.7282773
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2015-06-14
Publisher Place	China
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Atmospheric measurements Particle measurements Measurement uncertainty Manganese Decoding Matrix decomposition Silicon
Content Type	Text
Resource Type	Article

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

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