Geometric Regularization in Deep Representation Learning for Biological Inference and Model Compression

Abstract

Deep learning models are increasingly deployed in settings characterized by limited supervision, high-dimensional inputs, and substantial noise. In such regimes, unconstrained optimization often yields unstable representations and brittle generalization. This thesis studies how explicit geometric constraints can regularize deep representation learning in both biological data analysis and neural network compression. The central premise of this work is that many practical learning problems possess latent geometric structure that can be incorporated directly into model training. Enforcing such structure improves robustness, interpretability, and efficiency. First, in spatial transcriptomics, gene expression profiles are high-dimensional and spatially correlated, while cell-type labels are often unavailable. We introduce BioLACE, a representation learning framework that integrates marker-gene similarity as an auxiliary modality through contrastive regularization. This formulation reshapes the latent geometry to align with biologically meaningful structure, enabling stable clustering and label-free cell-type identification in heterogeneous tissues. Second, for single-cell DNA sequencing, mutation profiles are sparse and subject to substantial missingness and sequencing noise. We develop scECLIPSE, which incorporates geometric constraints from robust mutation similarity and biologically structured latent organization. These constraints guide latent representations toward biologically consistent organization and improve robustness under severe noise and missing data. Third, we examine singular-value-based model compression from a geometric perspective. Rather than focusing solely on post hoc truncation, we propose Low-Rank Prehab, a preconditioning stage that encourages low-rank parameter geometry prior to singular value decomposition. By shaping the weight landscape before compression, Prehab improves accuracy retention in large-scale architectures including vision transformers and language models. Across these domains, the results demonstrate that explicitly modeling geometric structure---whether in data manifolds or parameter spaces---provides a principled mechanism for improving stability, robustness, and efficiency in modern deep learning systems.