Learning, Inference, and Unmixing of Weak, Structured Signals in Noise
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In this thesis, we study two methods that can be used to learn, infer, and unmix weak, structured signals in noise: the Dynamic Mode Decomposition algorithm and the sparse Principal Component Analysis problem. Both problems take as input samples of a multivariate signal that is corrupted by noise, and produce a set of structured signals. We present performance guarantees for each algorithm and validate our findings with numerical simulations.
First, we study the Dynamic Mode Decomposition (DMD) algorithm. We demonstrate that DMD can be used to solve the source separation problem. That is, we apply DMD to a data matrix whose rows are linearly independent, additive mixtures of latent time series. We show that when the latent time series are uncorrelated at a lag of one time-step then the recovered dynamic modes will approximate the columns of the mixing matrix. That is, DMD unmixes linearly mixed sources that have a particular correlation structure.
We next broaden our analysis beyond the noise-free, fully observed data setting. We study the DMD algorithm with a truncated-SVD denoising step, and present recovery guarantees for both the noisy data and missing data settings. We also present some preliminary characterizations of DMD performed directly on noisy data. We end with some complementary perspectives on DMD, including an optimization-based formulation.
Second, we study the sparse Principal Component Analysis (PCA) problem. We demonstrate that the sparse inference problem can be viewed in a variable selection framework and analyze the performance of various decision statistics. A major contribution of this work is the introduction of False Discovery Rate (FDR) control for the principal component estimation problem, made possible by the sparse structure. We derive lower bounds on the size of detectable coordinates of the principal component vectors, and utilize these lower bounds to derive lower bounds on the worst-case risk.
Chair: Professor Raj Rao Nadakuditi