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Haoyu's FinalThesis.pdf (1.96 MB)

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High-Dimensional Statistical Inference from Coarse and Nonlinear Data: Algorithms and Guarantees

Fu, Haoyu
http://rave.ohiolink.edu/etdc/view?acc_num=osu1574381893567471

Abstract Details

2019, Doctor of Philosophy, Ohio State University, Electrical and Computer Engineering.
Learning a postulated parametric model from the acquired data to extract useful information is of great importance in modern signal processing, machine learning and statistics. The linear model, where the observed data are assumed to depend linearly on the input data, has been studied extensively and applied successfully to many applications. However, the linear assumption is quite restricted, creating a major roadblock for its accuracy and universality, since the dependency of the data is nonlinear in general, and cannot be approximated by a linear model. The challenges of learning these nonlinear models include high computational cost and susceptibility to local minima in their associated optimization problems. Through case studies, we highlight the statistical and computational issues when learning from high-dimensional coarse and nonlinear data, with the hope of shedding light on resolving these challenges. In this thesis, we consider data from typical signal processing and machine learning applications. In the context of signal processing, we study the problem of estimating spectrally-sparse signals from their quantized noisy complex-valued random linear measurements, a problem arising naturally from analog-to-digital conversion in sub-Nyquist spectrum sampling. We first study the effects of quantization on estimating the spectrum by characterizing the Cram\'er-Rao bound under the additive white Gaussian noise. We use the calculated bound to highlight the trade-off between the sample complexity and the bit depth under different signal-to-noise ratios for a fixed budget of bits. Secondly, we formulate a convex optimization approach based on atomic norm soft thresholding to estimate the spectrum of the signal, which is computationally more efficient than the maximum-likelihood estimator. Moving to the context of machine learning, we study several one-hidden-layer neural network models for nonlinear regression using both cross-entropy and least-squares loss functions. The neural-network-based models have attracted a significant amount of research interest due to the success of deep learning in practical domains such as computer vision and natural language processing. Learning such neural-network-based models often requires solving a non-convex optimization problem. We propose different strategies to characterize the optimization landscape of the non-convex loss functions and provide guarantees on the statistical and computational efficiency of optimizing these loss functions via gradient descent.
Liang Yingbin (Advisor)
Chi Yuejie (Advisor)
Lee Kiryung (Committee Member)
Schniter Philip (Committee Member)
156 p.
Electrical Engineering
quantization; atomic norms; line spectrum estimation; neural-network-based model; nonconvex optimization

Recommended Citations

Citations

  • Fu, H. (2019). High-Dimensional Statistical Inference from Coarse and Nonlinear Data: Algorithms and Guarantees [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1574381893567471

    APA Style (7th edition)

  • Fu, Haoyu. High-Dimensional Statistical Inference from Coarse and Nonlinear Data: Algorithms and Guarantees. 2019. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1574381893567471.

    MLA Style (8th edition)

  • Fu, Haoyu. "High-Dimensional Statistical Inference from Coarse and Nonlinear Data: Algorithms and Guarantees." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1574381893567471

    Chicago Manual of Style (17th edition)

osu1574381893567471
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This open access ETD is published by The Ohio State University and OhioLINK.