Georgia Institute of Technology College of Sciences

In this thesis, we investigate three related problems at the intersection of analytic number theory and discrete harmonic analysis. Our primary goal is to understand discrete averaging operators over arithmetic sets—discrete analogues of classical continuous operators—and analyze their behavior using tools from harmonic analysis and additive combinatorics. The results deepen our understanding of how analytic and combinatorial techniques interact in the study of primes and other arithmetic structures.

The Zoom link to the meeting: https://gatech.zoom.us/j/99340322307

After quickly recalling the established theory on the combinatorics of orthogonal matroids, we define and study basic properties of the extended rank function and the modular tuples in orthogonal matroids. We then prove a weak version of the path theorem concerning the connectivity of circuits. 

Next, we consider representations of orthogonal matroids over fields (and more generally, over tracts) by bases. We then give a few applications, purely using this basis approach, to the representation theory of orthogonal matroids. We also give a different way of representing orthogonal matroids by circuit functions, which is proved to be equivalent to the basis approach. This is based on joint work with Matthew Baker and joint work with Donggyu Kim. 

The final part of the thesis focuses on the rescaling classes of representations. We construct the foundation of an orthogonal matroid, which possesses the universal property that the set of rescaling classes of representations is in one-to-one correspondence with the set of morphisms from the foundation to the target field. We also give explicit generators and relations of the foundation and an algorithm for computations. 

Zoom link: https://gatech.zoom.us/my/tongjinmath?pwd=QzRDalp2ditGL2tVNUozWm1RK1UwUT09

The orthogonal group synchronization problem, which focuses on recovering orthogonal group elements from their corrupted pairwise measurements, encompasses examples such as high-dimensional Kuramoto model on general signed networks, $\mathbb{Z}_2$-synchronization, community detection under stochastic block models, and orthogonal Procrustes problem. The semidefinite relaxation (SDR) has proven its power in solving this problem; however, its expensive computational costs impede its widespread practical applications. We consider the Burer-Monteiro factorization approach to the orthogonal group synchronization, an effective and scalable low-rank factorization to solve large scale SDPs. Despite the significant empirical successes of this factorization approach, it is still a challenging task to understand when the nonconvex optimization landscape is benign, i.e., the optimization landscape possesses only one local minimizer, which is also global. In this work, we demonstrate that if the degree of freedom within the factorization exceeds the condition number of the ``Laplacian" (certificate matrix) at the global minimizer, the optimization landscape is absent of spurious local minima. Our main theorem is purely algebraic and versatile, and it seamlessly applies to all the aforementioned examples: the nonconvex landscape remains benign under almost identical condition that enables the success of the SDR. Finally, we will discuss the statistical sides of group synchronization by quantifying the uncertainty of both MLE and spectral estimators.

Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, natural language processing, and many other domains. The involved deep neural network architectures and computational issues have been well studied in machine learning. But there is much less theoretical understanding about the modelling, approximation or generalization abilities of deep learning models with network architectures. An important family of structured deep neural networks is deep convolutional neural networks (CNNs) induced by convolutions. The convolutional architecture gives essential differences between deep CNNs and fully-connected neural networks, and the classical approximation theory for fully-connected networks developed around 30 years ago does not apply.  This talk describes approximation and generalization analysis of deep CNNs and related structured deep neural networks.