UCLA

In this thesis, we discuss three topics in geometric deep learning and learning on manifolds. First, we study the geometric scattering transform, which is a wavelet-based transform defined on non-Euclidean domains such as graphs and manifolds. We introduce a framework for constructing and analyzing geometric scattering transforms on general measure spaces, and we propose methods for implementing a manifold scattering transform given only point-cloud data. We provide convergence analysis of one such method, and we perform numerical experiments on real-world data to validate the methods' utility. Next, we apply our framework to directed graphs and define a new scattering transform. We propose using an autoencoder to compress the scattering coefficients for better performance on downstream tasks, such as inference of cellular signaling networks. We experimentally demonstrate the merits of our methods on several data sets. The second main topic of this thesis is the design and analysis of manifold neural networks. We study a class of manifold neural networks implemented solely on point-cloud data sampled uniformly at random from an unknown manifold and prove that the outputs of such networks converge to their continuum counterparts. We supply a quantitative rate of convergence, and we experimentally investigate the sharpness of our rate. Next, we propose a framework for designing broad classes of manifold neural networks, and we prove convergence rates for such networks under nonuniform random sampling assumptions. We experimentally evaluate several architectures on synthetic data and single-cell RNA sequencing data. The final topic of this thesis is low-rank approximation of manifold-valued data. We propose an adaptation of nonnegative matrix factorization to manifold-valued data. We derive alternating updates for computing this factorization and apply our method to diffusion tensor magnetic resonance imaging data. The work in this thesis demonstrates ways in which geometric structure in data domains may be incorporated in the design and study of new data analysis tools.