Elina Robeva

Computer vision models, such as image and video classifiers, are increasingly prevalent in Internet-of-Things systems. Since the advent of the AlexNet neural network model in 2012, convolutional neural networks have been demonstrated to be very effective at performing many computer vision tasks. However, convolutional neural networks’ high computational and storage costs hinder the wider adoption of computer vision models in smaller Internet-of-Things devices such as mobile phones or embedded systems. As larger neural network models increase hardware costs, industry and academia have come together to tackle the problem of how to compress convolutional neural networks. Convolutional neural network compression via tensor decomposition has been shown to reduce the memory and storage requirements for devices to perform computer vision tasks successfully. In this work, we first review the preliminaries of tensor decomposition and define the four major types of tensor decompositions and their related decomposition algorithms in Chapter 1. Afterward, we introduce the building blocks of neural networks and describe convolutional neural networks in Chapter 2. Finally, we overview the different tensor decomposition approaches for convolutional neural network compression and display the results of two experiments using the PyTorch-TedNet CIFAR10 and CIFAR100 model benchmarks in Chapter 3.

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This thesis addresses the multivariate super-resolution problem commonly encountered in pixelized images. The study uses concepts from measure theory, linear algebra, and functional analysis. It builds on previous work conducted in the field of two-dimensional image analysis by Eftekhari, Bendory, and Tang [14]. Specifically, this research addresses an open problem posted by Schiebinger, Robeva, and Recht [21], with a particular emphasis on scenarios where the higher dimensional point-spread function can be decomposed.The first chapter of this thesis provides an introduction to the super-resolution image problem. It acquaints the reader with the topic by presenting an overview of previous works and research. Additionally, a mathematical problem is introduced that focuses on the case where the point-spread function of the imaging device can be decomposed component-wise.Chapter 2 presents a comprehensive overview of the subject matter, along with the foundational theoretical concepts that will be vital for understanding the subsequent chapters. Chapter 3 presents the findings of the study. The research demonstrates that accurate recovery of true images is possible under mild conditions of component-wise functions, even in the absence of noise. Furthermore, the study shows the construction of dual certificates using T-systems and T*-systems. Chapter 4 outlines future work on extending this research to include the general Gaussian case, defining the generalization of T-systems to higher dimensions, and contextualizing their relevance in this context. Overall, this research contributes to the field of image analysis by providing valuable insights into the complexities of the multivariate super-resolution problem.

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In this thesis, we study three problems, each of which concerns inferring certain pieces of information from some observed data. We apply tools arising from algebraic geometry, statistics, and combinatorics to approach these problems.In Chapter 2, we consider data that can be recorded in the form of a tensor admitting a special type of decomposition called an orthogonal tensor-train decomposition. Finding equations defining varieties of low-rank tensors is generally a hard problem, however, the set of orthogonally-decomposable tensors is defined by appealing quadratic equations. The tensors we consider are an extension of orthogonally-decomposable tensors. We show that they are defined by similar quadratic equations, as well as linear equations and a higher-degree equation.In Chapter 3, we study the problem of maximum likelihood estimation of log-concave densities that lie in the graphical model of a given undirected graph G, and factorize according to this graph with log-concave factors. We show that the maximum likelihood estimate (MLE) is the product of the exponentials of several tent functions, one for each maximal clique of G. While the family of densities in question is infinite-dimensional, our results imply the MLE can be found by solving a finite-dimensional convex optimization problem. We provide an implementation. Furthermore, when G is chordal, we prove that the MLE exists and is unique with probability 1 as long as the number of sample points is larger than the size of the largest clique of G. Finally, we discuss the conditions under which a log-concave density in the graphical model of G has a log-concave factorization according to G.In Chapter 4, we study the problem of inferring causality from an observed i.i.d. sample arising from a distribution faithful to a directed graph G which can possibly have directed cycles. In particular, our goal is to recover the Markov equivalence class of G. We propose an algorithm, and conjecture that it is consistent, i.e., if the set of conditional independence relations satisfied by the distribution is precisely inferred from the observed data, then the output of the algorithm is Markov equivalent to G.

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