Low-Rank Latent Factor Models for Multiview Analysis, Supervised Learning, and Density Estimation 23 views

This dissertation explores a sequence of problems within the framework of low rank latent factor models. It begins with extensions of Canonical Correlation Analysis (CCA) and progresses to density estimation and function approximation, with particular emphasis on the variational aspects that affect interpolation and learning in high-dimensional settings. Specifically, we consider two novel extensions of classical CCA: Deep Generalized CCA (DGCCA) and Subspace Clustering of Subspaces (SCoS), a unifying framework that bridges Generalized CCA and Subspace Clustering. We point out certain drawbacks in the current formulations of DGCCA, and for both DGCCA and SCoS we establish state of the art identifiability conditions and novel estimation algorithms. Our identifiability analysis of DGCCA relies on conditional independence assumptions, which naturally lead to latent factor models for multivariate density estimation. This brings upon us the curse of dimensionality and the emergence of energy concentration phenomena in high-dimensional spaces. Both can exacerbate sample complexity and hinder generalization. To address these challenges, we consider tensor product splines and propose a variational framework grounded in the use of Dirichlet energy. These tools allow us to control smoothness and regularity, while also promoting generalization and robustness in learning. Three concrete applications are presented: graph alignment using CCA, pixel clustering in hyperspectral imaging, and supervised learning for tabular data in the high-dimensional, scarce-sample regime.

PHD (Doctor of Philosophy)

English

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2025-12-15