NDLI: An Almost Optimal Algorithm for Computing Nonnegative Rank

Please wait, while we are loading the content...

An Almost Optimal Algorithm for Computing Nonnegative Rank

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Moitra, Ankur
Copyright Year	2016
Abstract	Here, we give an algorithm for deciding if the nonnegative rank of a matrix $M$ of dimension $m \times n$ is at most $r$ which runs in time $(nm)^{O(r^2)}$. This is the first exact algorithm that runs in time singly exponential in $r$. This algorithm (and earlier algorithms) are built on methods for finding a solution to a system of polynomial inequalities (if one exists). Notably, the best algorithms for this task run in time exponential in the number of variables but polynomial in all of the other parameters (the number of inequalities and the maximum degree). Hence, these algorithms motivate natural algebraic questions whose solution have immediate algorithmic implications: How many variables do we need to represent the decision problem, and does $M$ have nonnegative rank at most $r$? A naive formulation uses $nr + mr$ variables and yields an algorithm that is exponential in $n$ and $m$ even for constant $r$. Arora et al. [Proceedings of STOC, 2012, pp. 145--162] recently reduced the number of variables to $2r^2 2^r$, and here we exponentially reduce the number of variables to $2r^2$ and this yields our main algorithm. In fact, the algorithm that we obtain is nearly optimal (under the exponential time hypothesis) since an algorithm that runs in time $(nm)^{o(r)}$ would yield a subexponential algorithm for $3$-SAT [Proceedings of STOC, 2012, pp. 145--162]. Our main result is based on establishing a normal form for nonnegative matrix factorization---which in turn allows us to exploit algebraic dependence among a large collection of linear transformations with variable entries. Additionally, we also demonstrate that nonnegative rank cannot be certified by even a very large submatrix of $M$, and this property also follows from the intuition gained from viewing nonnegative rank through the lens of systems of polynomial inequalities.
Starting Page	156
Ending Page	173
Page Count	18
File Format	PDF
ISSN	00975397
DOI	10.1137/140990139
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	1
Volume Number	45
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2016-02-04
Access Restriction	Subscribed
Subject Keyword	Analysis of algorithms and problem complexity systems of polynomial inequalities extension complexity nonnegative rank
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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

An Almost Optimal Rank Bound for Depth-3 Identities

Computing a Nonnegative Matrix Factorization---Provably

A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation

Common Information and Unique Disjointness

A Polynomial Time Algorithm for Computing Extinction Probabilities of Multitype Branching Processes

Computing Optimal Steiner Trees in Polynomial Space

Nonnegative Weighted #CSP: An Effective Complexity Dichotomy

An Almost Space-Optimal Streaming Algorithm for Coresets in Fixed Dimensions

An Optimal-Time Algorithm for Slope Selection

An Almost Optimal Algorithm for Computing Nonnegative Rank

Similar Documents

An Almost Optimal Rank Bound for Depth-3 Identities

Computing a Nonnegative Matrix Factorization---Provably

A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation

Common Information and Unique Disjointness

A Polynomial Time Algorithm for Computing Extinction Probabilities of Multitype Branching Processes

Computing Optimal Steiner Trees in Polynomial Space

Nonnegative Weighted #CSP: An Effective Complexity Dichotomy

An Almost Space-Optimal Streaming Algorithm for Coresets in Fixed Dimensions

An Optimal-Time Algorithm for Slope Selection

An Almost Optimal Algorithm for Computing Nonnegative Rank