Incoherence is Sufficient for Statistical RIP of Unit Norm Tight Frames: Constructions and Properties

Sasmal, P and Murthy, CR (2021) Incoherence is Sufficient for Statistical RIP of Unit Norm Tight Frames: Constructions and Properties. In: IEEE Transactions on Signal Processing, 69 . pp. 2343-2355.

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Abstract

In this work, we show that the incoherence alone is sufficient to establish the statistical restricted isometry property (StRIP) and statistical incoherence property (SInCoP) for unit norm tight frames (UNTFs). Further, we derive three simple properties that binary matrices need to satisfy, in order to produce incoherent UNTFs (IUNTFs) with high redundancy (ratio of the number of columns to the number of rows) via an existing embedding operation. We show that biadjacency matrices corresponding to biregular graphs satisfy the required properties. Thereby, we provide a connection between graph theory and the construction of IUNTFs. We also provide a bouquet of constructions of IUNTFs from finite fields and combinatorial designs. Another important aspect of our construction is that the sparse recovery guarantees for the embedded IUNTFs can in fact be translated to the constituent binary matrix. We show that if the constituent mtimes M binary matrix has constant row and column weight, it can support sparse recovery through ell_{1}-minimization for all but an epsilon-fraction of t-sparse signals chosen from a random signal model, provided m=O(t(log(frac{M}{epsilon})){3}), which is a significant improvement over the existing m=O(t{2}) bound, where m denotes the number of measurements. Also, the StRIP and SInCoP based approach results in matrices whose column size is exponential in the fourth root of the row size. To the best of our knowledge, this is the first construction of deterministic matrices satisfying StRIP and SInCoP with such high redundancy.

Item Type:	Journal Article
Publication:	IEEE Transactions on Signal Processing
Publisher:	Institute of Electrical and Electronics Engineers Inc.
Additional Information:	The copyright for this article belongs to Institute of Electrical and Electronics Engineers Inc.
Keywords:	Graph theory; Recovery; Redundancy, Combinatorial design; Computationally efficient; First constructions; High redundancy; L1 minimizations; Restricted isometry properties (RIP); Sparse recovery; Theoretical guarantees, Matrix algebra
Department/Centre:	Division of Electrical Sciences > Electrical Communication Engineering > Electrical Communication Engineering - Technical Reports
Date Deposited:	18 Apr 2023 07:01
Last Modified:	18 Apr 2023 07:01
URI:	https://eprints.iisc.ac.in/id/eprint/80640

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