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New Applications of Random Matrices in Spin Glass and Machine Learning
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| 자료유형 | 학위논문 |
|---|---|
| 서명/저자사항 | New Applications of Random Matrices in Spin Glass and Machine Learning. |
| 개인저자 | Wu, Hao. |
| 단체저자명 | University of Michigan. Applied and Interdisciplinary Mathematics. |
| 발행사항 | [S.l.]: University of Michigan., 2019. |
| 발행사항 | Ann Arbor: ProQuest Dissertations & Theses, 2019. |
| 형태사항 | 238 p. |
| 기본자료 저록 | Dissertations Abstracts International 81-06B. Dissertation Abstract International |
| ISBN | 9781687999146 |
| 학위논문주기 | Thesis (Ph.D.)--University of Michigan, 2019. |
| 일반주기 |
Source: Dissertations Abstracts International, Volume: 81-06, Section: B.
Advisor: Baik, Jinho |
| 이용제한사항 | This item must not be sold to any third party vendors.This item must not be added to any third party search indexes. |
| 요약 | Recent advancement in random matrix theory is beneficial to challenging problems in many disciplines of science and engineering. In another direction, these applications motivate a lot of new questions in random matrix theory. In this thesis, we present two applications of random matrix theory to statistical physics and machine learning.The first part of this thesis is about the spherical Sherrington-Kirkpatrick (SSK) model in statistical physics. The SSK model is defined by a random probability measure on a high dimensional sphere. The probability measure involves the temperature and a random Hamiltonian. We consider the simplest non-trivial case where the Hamiltonian is a random symmetric quadratic form perturbed by a specific symmetric polynomial of degree one or two. It is interesting to consider the interaction between the quadratic form and the perturbations. In particular, using the obvious connection between random quadratic forms and random matrices, we study the free energies and obtain the limiting law of their fluctuations as the dimension becomes large.The second part is devoted to an application of the random matrix theory in machine learning. We develope Free component analysis (FCA) for unmixing signals in the matrix form from their linear mixtures with little prior knowledge. The matrix signals are modeled as samples of random matrices, which are further regarded as non-commutative random variables. The counterpart of scalar probability for non-commutative random variables is the free probability. Our principle of separation is to maximize free independence between the unmixed signals. This is achieved in a manner analogous to the independent component analysis (ICA) based method for unmixing independent random variables from their additive mixtures. We describe the theory, the various algorithms, and compare FCA to ICA. We show that FCA performs comparably to, and often better than, ICA in every application, such as image and speech unmixing, where ICA has been known to succeed. |
| 일반주제명 | Physics. Mathematics. Computer science. |
| 언어 | 영어 |
| 바로가기 | 
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