High-Dimensional Markov Chains and Applications

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LADDHA-DISSERTATION-2023.pdf (1021.99 KB)
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Laddha, Aditi
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School of Computer Science
School established in 2007
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https://hdl.handle.net/1853/72703
Abstract
A Markov chain is a stochastic process satisfying the Markovian property, meaning that future events are independent of the past states given the present state. Markov chains form the basis for numerous randomized algorithms used in optimization, integration, linear programming, approximate counting, and more. They also play a crucial role in understanding the geometry of subsets in high-dimensional Euclidean space. In this study, we use high-dimensional Markov chains to develop efficient algorithms for two problems: sampling uniformly from convex bodies and bounding the discrepancy of set systems. We investigate two Markov chain-based algorithms for uniformly sampling convex bodies and introduce novel isoperimetric inequalities to analyze the mixing time of these chains. We propose a unified framework for constructive discrepancy minimization by discretizing a Brownian motion-based stochastic process.
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2023-07-24
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