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Optimization for high-dimensional analysis and estimation in signal processing and machine learning
Li, Shuang
Li, Shuang
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2020
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Abstract
High-dimensional signal analysis and estimation appears in many signal processing and machine learning applications, including modal analysis, airborne radar system demixing, parameter estimation in spectrally sparse signals, and simultaneous blind deconvolution and phase retrieval. The underlying low-dimensional structure in these high-dimensional signals inspires us to develop optimality guarantees as well as some optimization-based techniques for the fundamental problems in signal processing and machine learning.In many applications, high-dimensional signals often have a certain concise representation, which is a linear combination of a small number of atoms in a dictionary with elements drawn from the signal space. In compressive sensing,L1-minimization is a widely used framework to find the sparse representations of a signal. It has recently been shown that atomic norm minimization (ANM), which is a generalization of L1-minimization, is an efficient and powerful way for exactly recovering unobserved time-domain samples and identifying unknown frequencies in signals having sparse frequency spectra, namely, finding a concise representation for spectrally sparse signals. This new technique works on a continuous dictionary and can completely avoid the effects of basis mismatch, which can plague conventional grid-based compressive sensing techniques.Almost every problem in the fields of signal processing and machine learning can be formulated as either a convex or non-convex optimization problem. With convex formulations, we can get guaranteed global optimality but we often encounter problems with large size. Though non-convex optimization often lacks global optimality, it can have much lower computational and storage complexity. The objective of this dissertation is to (i) analyze and estimate high-dimensional signals or parameters with convex optimization-based techniques; (ii) analyze and estimate high-dimensional signals or parameters with non-convex optimization-based techniques; (iii) generalize the ideas and techniques used in optimization to differentiable games, which are games with continuous decision variables and differentiable cost functions, and have been gradually adapted to model many signal processing, communication, and networking problems in the last two decades.Our main contributions include a new (i) method to estimate the modal parameters in modal analysis, non-asymptotic bound on the sample complexity of modal analysis with random temporal compression, and non-asymptotic bound on the recovery error of an atomic norm denoising problem in the multiple measurement vector setting; (ii) analysis for the airborne radar system demixing problem, where the received signal consists of contributions from targets, jammers, and clutter; (iii) optimization-based perspective on the classical MUSIC algorithm that could lead to future developments and understanding, and non-asymptotic theoretical guarantees for the proposed algorithms; (iv) non-asymptotic theoretical bound on the mean square error of atomic norm denoising for complex exponentials with unknown waveform modulations; (v) strategy to address the problem of simultaneous blind deconvolution and phase retrieval; (vi) analysis for landscape correspondence between the critical points of non-convex empirical risk and its population counterpart without using the strongly Morse assumption required in some existing literature; (vii) analysis for landscape correspondence between empirical and population games; (viii) cubic regularization based methods for differential games.
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