NDLI: Fast and RIP-Optimal Transforms

Content Provider	Springer Nature Link
Author	Ailon, Nir Rauhut, Holger
Copyright Year	2014
Abstract	We study constructions of $$k \times n$$ matrices $$A$$ that both (1) satisfy the restricted isometry property (RIP) at sparsity $$s$$ with optimal parameters, and (2) are efficient in the sense that only $$O(n\log n)$$ operations are required to compute $$Ax$$ given a vector $$x$$ . Our construction is based on repeated application of independent transformations of the form $$DH$$ , where $$H$$ is a Hadamard or Fourier transform and $$D$$ is a diagonal matrix with random $$\{+1,-1\}$$ elements on the diagonal, followed by any $$k \times n$$ matrix of orthonormal rows (e.g. selection of $$k$$ coordinates). We provide guarantees (1) and (2) for a regime of parameters that is comparable with previous constructions, but using a construction that uses Fourier transforms and diagonal matrices only. Our main result can be interpreted as a rate of convergence to a random matrix of a random walk in the orthogonal group, in which each step is obtained by a Fourier transform $$H$$ followed by a random sign change matrix $$D$$ . After a few number of steps, the resulting matrix is random enough in the sense that any arbitrary selection of rows gives rise to an RIP matrix for, sparsity as high as slightly below $$s=\sqrt{n}$$ , with high probability. The proof uses a bootstrapping technique that, roughly speaking, says that if a matrix $$A$$ has some suboptimal RIP parameters, then the action of two steps in this random walk on this matrix has improved parameters. This idea is interesting in its own right, and may be used to strengthen other constructions.
Starting Page	780
Ending Page	798
Page Count	19
File Format	PDF
ISSN	01795376
Journal	Discrete & Computational Geometry
Volume Number	52
Issue Number	4
e-ISSN	14320444
Language	English
Publisher	Springer US
Publisher Date	2014-09-04
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Restricted isometry Johnson–Lindenstrauss transformations Compressive sensing Combinatorics Computational Mathematics and Numerical Analysis
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Theoretical Computer Science Computational Theory and Mathematics Geometry and Topology

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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
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