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Convergence of a data-driven time–frequency analysis method

Hou, Thomas Y. and Shi, Zuoqiang and Tavallali, Peyman (2014) Convergence of a data-driven time–frequency analysis method. Applied and Computational Harmonic Analysis, 37 (2). pp. 235-270. ISSN 1063-5203. http://resolver.caltech.edu/CaltechAUTHORS:20140731-103641810

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Abstract

In a recent paper [11], Hou and Shi introduced a new adaptive data analysis method to analyze nonlinear and non-stationary data. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t)cos(θ(t))}, where a∈V(θ),V(θ) consists of the functions that are less oscillatory than cos(θ(t)) and θ′⩾0. This problem was formulated as a nonlinear L^0 optimization problem and an iterative nonlinear matching pursuit method was proposed to solve this nonlinear optimization problem. In this paper, we prove the convergence of this nonlinear matching pursuit method under some scale separation assumptions on the signal. We consider both well-resolved and poorly sampled signals, as well as signals with noise. In the case without noise, we prove that our method gives exact recovery of the original signal.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/j.acha.2013.12.004DOIArticle
http://www.sciencedirect.com/science/article/pii/S1063520313001127PublisherArticle
http://arxiv.org/abs/1303.7048arXivDiscussion Paper
Additional Information:© 2014 Elsevier Inc. Received 28 March 2013, Revised 1 August 2013, Accepted 29 December 2013, Available online 2 January 2014. Communicated by Stephane G. Mallat. This work was in part supported by the AFOSR MURI grant FA9550-09-1-0613, a DOE grant DE-FG02-06ER25727, and NSF Grants DMS-1159138 and DMS-1318377. The research of Dr. Z. Shi was in part supported by a NSFC Grant 11201257.
Funders:
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI)FA9550-09-1-0613
Department of Energy (DOE)DE-FG02-06ER25727
NSFDMS-1159138
NSFDMS-1318377
NSFC (China)11201257
Subject Keywords:Sparse representation; Data-driven; Time–frequency analysis; Matching pursuit
Record Number:CaltechAUTHORS:20140731-103641810
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20140731-103641810
Official Citation:Thomas Y. Hou, Zuoqiang Shi, Peyman Tavallali, Convergence of a data-driven time–frequency analysis method, Applied and Computational Harmonic Analysis, Volume 37, Issue 2, September 2014, Pages 235-270, ISSN 1063-5203, http://dx.doi.org/10.1016/j.acha.2013.12.004. (http://www.sciencedirect.com/science/article/pii/S1063520313001127)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:47705
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:31 Jul 2014 18:02
Last Modified:31 Jul 2014 18:29

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