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 Title: Polynomial methods in statistical inference: Theory and practice Author(s): Yang, Pengkun Director of Research: Wu, Yihong Doctoral Committee Chair(s): Raginsky, Maxim Doctoral Committee Member(s): Hajek, Bruce; Srikant, Rayadurgam; Oh, Sewoong Department / Program: Electrical & Computer Eng Discipline: Electrical & Computer Engr Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): polynomial methods statistical inference minimax estimation polynomial approximation method of moments efficient algorithm fundamental limits Abstract: Recent advances in genetics, computer vision, and text mining are accompanied by analyzing data coming from a large domain, where the domain size is comparable or larger than the number of samples. In this dissertation, we apply the polynomial methods to several statistical questions with rich history and wide applications. The goal is to understand the fundamental limits of the problems in the large domain regime, and to design sample optimal and time efficient algorithms with provable guarantees. The first part investigates the problem of property estimation. Consider the problem of estimating the Shannon entropy of a distribution over $k$ elements from $n$ independent samples. We obtain the minimax mean-square error within universal multiplicative constant factors if $n$ exceeds a constant factor of $k/\log(k)$; otherwise there exists no consistent estimator. This refines the recent result on the minimal sample size for consistent entropy estimation. The apparatus of best polynomial approximation plays a key role in both the construction of optimal estimators and, via a duality argument, the minimax lower bound. We also consider the problem of estimating the support size of a discrete distribution whose minimum non-zero mass is at least $\frac{1}{k}$. Under the independent sampling model, we show that the sample complexity, i.e., the minimal sample size to achieve an additive error of $\epsilon k$ with probability at least 0.1 is within universal constant factors of $\frac{k}{\log k}\log^2\frac{1}{\epsilon}$, which improves the state-of-the-art result of $\frac{k}{\epsilon^2 \log k}$. Similar characterization of the minimax risk is also obtained. Our procedure is a linear estimator based on the Chebyshev polynomial and its approximation-theoretic properties, which can be evaluated in $O(n+\log^2 k)$ time and attains the sample complexity within constant factors. The superiority of the proposed estimator in terms of accuracy, computational efficiency and scalability is demonstrated in a variety of synthetic and real datasets. When the distribution is supported on a discrete set, estimating the support size is also known as the distinct elements problem, where the goal is to estimate the number of distinct colors in an urn containing $k$ balls based on $n$ samples drawn with replacements. Based on discrete polynomial approximation and interpolation, we propose an estimator with additive error guarantee that achieves the optimal sample complexity within $O(\log\log k)$ factors, and in fact within constant factors for most cases. The estimator can be computed in $O(n)$ time for an accurate estimation. The result also applies to sampling without replacement provided the sample size is a vanishing fraction of the urn size. One of the key auxiliary results is a sharp bound on the minimum singular values of a real rectangular Vandermonde matrix, which might be of independent interest. The second part studies the problem of learning Gaussian mixtures. The method of moments is one of the most widely used methods in statistics for parameter estimation, by means of solving the system of equations that match the population and estimated moments. However, in practice and especially for the important case of mixture models, one frequently needs to contend with the difficulties of non-existence or non-uniqueness of statistically meaningful solutions, as well as the high computational cost of solving large polynomial systems. Moreover, theoretical analysis of the method of moments are mainly confined to asymptotic normality style of results established under strong assumptions. We consider estimating a $k$-component Gaussian location mixture with a common (possibly unknown) variance parameter. To overcome the aforementioned theoretic and algorithmic hurdles, a crucial step is to denoise the moment estimates by projecting to the truncated moment space (via semidefinite programming) before solving the method of moments equations. Not only does this regularization ensures existence and uniqueness of solutions, it also yields fast solvers by means of Gauss quadrature. Furthermore, by proving new moment comparison theorems in the Wasserstein distance via polynomial interpolation and majorization techniques, we establish the statistical guarantees and adaptive optimality of the proposed procedure, as well as oracle inequality in misspecified models. These results can also be viewed as provable algorithms for generalized method of moments which involves non-convex optimization and lacks theoretical guarantees. Issue Date: 2018-09-04 Type: Text URI: http://hdl.handle.net/2142/102774 Rights Information: Copyright 2018 Pengkun Yang Date Available in IDEALS: 2019-02-07 Date Deposited: 2018-12
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