Mathematicshttp://hdl.handle.net/1903/22612017-06-23T08:43:49Z2017-06-23T08:43:49ZNonlinear Analysis of Phase Retrieval and Deep LearningZou, Dongmianhttp://hdl.handle.net/1903/194872017-06-22T06:24:31Z2017-01-01T00:00:00ZNonlinear Analysis of Phase Retrieval and Deep Learning
Zou, Dongmian
Nonlinearity causes information loss. The phase retrieval problem, or the phaseless reconstruction problem, seeks to reconstruct a signal from the magnitudes of linear measurements. With a more complicated design, convolutional neural networks use nonlinearity to extract useful features. We can model both problems in a frame-theoretic setting. With the existence of a noise, it is important to study the stability of the phaseless reconstruction and the feature extraction part of the convolutional neural networks. We prove the Lipschitz properties in both cases. In the phaseless reconstruction problem, we show that phase retrievability implies a bi-Lipschitz reconstruction map, which can be extended to the Euclidean space to accommodate noises while remaining to be stable. In the deep learning problem, we set up a general framework for the convolutional neural networks and provide an approach for computing the Lipschitz constants.
2017-01-01T00:00:00ZTwo Goodness-of-Fit Tests for the Density Ratio ModelYu, Luquanhttp://hdl.handle.net/1903/194842017-06-22T06:23:49Z2017-01-01T00:00:00ZTwo Goodness-of-Fit Tests for the Density Ratio Model
Yu, Luquan
Under consideration are goodness-of-fit tests for the \textit{density ratio model}. The model stipulates that the log-likelihood ratio of two unknown densities is of a known form which depends on finite dimensional parameters and a tilt function. We can derive the empirical distribution estimator $\tilde{G}$ from a reference sample, and the semiparametric distribution estimator $\hat{G}$ under the density ratio model. Furthermore we can derive kernel density estimators $\tilde{g}$ and $\hat{g}$ corresponding to $\tilde{G}$ and $\hat{G}$ by choosing a bandwidth parameter. Goodness-of-fit test statistics can be constructed via the discrepancy between $\tilde{g}$ and $\hat{g}$ using Hellinger distance and a modification thereof. We propose two new test statistics by modifying the goodness-of-fit test statistics suggested by Bondell (2007) and by Cheng and Chu (2004). Asymptotic results and limiting distributions are derived for both new test statistics, and the selections of the kernel and bandwidth are discussed. Monte-Carlo simulations show that the new test statistics improve the accuracy of the the goodness-of-fit test and that the limiting distributions of the new test statistics are more symmetric.
2017-01-01T00:00:00ZTransfer of Representations and Orbital Integrals of Inner Forms of GL_nCohen, Jonathanhttp://hdl.handle.net/1903/194502017-06-22T06:16:00Z2016-01-01T00:00:00ZTransfer of Representations and Orbital Integrals of Inner Forms of GL_n
Cohen, Jonathan
Let $F$ be a nonarchimedean local field and $D$ an $F$-central division algebra. We characterize the Local Langlands Correspondence (LLC) for inner forms of $GL_n$ over $F$ via the Jacquet-Langlands Correspondence and compatibility with the Langlands Classification. We show that LLC satisfies a natural compatibility with parabolic induction and characterize the LLC for inner forms as a unique family of bijections $\Pi(GL_r(D)) \rightarrow \Phi(GL_r(D))$ for each $r$, (for a fixed $D$) satisfying certain properties. We construct a surjective map of Bernstein centers $\mathfrak{Z}(GL_n(F))\to \mathfrak{Z}(GL_r(D))$ and show this produces pairs of matching distributions in the sense of \cite{SBC}. Finally, we construct explicit Iwahori-biinvariant matching functions for unit elements in the parahoric Hecke algebras of $GL_r(D)$, and thereby produce many explicit pairs of matching functions.
2016-01-01T00:00:00ZSpectral Analysis of Markov Jump Processes with Rare Transitions: A Graph-Algorithmic ApproachGan, Tingyuehttp://hdl.handle.net/1903/194232017-06-22T06:09:46Z2017-01-01T00:00:00ZSpectral Analysis of Markov Jump Processes with Rare Transitions: A Graph-Algorithmic Approach
Gan, Tingyue
Parameter-dependent Markov jump processes with exponentially small transition
rates arise in modeling complex systems in physics, chemistry, and biology.
Long-term dynamics of these processes are largely governed by the spectral properties
of their generators. We propose a constructive graph-algorithmic approach to
computing the asymptotic estimates of eigenvalues and eigenvectors of the generator
matrix. In particular, we introduce the concepts of the hierarchy of Typical Transition
Graphs (T-graphs) and the associated sequence of Characteristic Timescales.
The hierarchy of T-graphs can be viewed as a unication of Wentzell's hierarchy of
optimal W-graphs and Friedlin's hierarchy of Markov chains. T-graphs are capable
of describing typical escapes from metastable classes as well as cyclic behaviors
within metastable classes, for both reversible and irreversible processes, with or
without symmetry. Moreover, the hierarchy of T-graphs can be used to construct
asymptotic estimates of eigenvalues and eigenvectors simultaneously. We apply the
proposed approach to investigate the biased random walk of a molecular motor and
conduct zero-temperature asymptotic analysis of the LJ75 network.
2017-01-01T00:00:00Z