Exploiting Manifold Geometry for Multi-Modal Generative Modeling

PhD Qualifying Examination


Title: "Exploiting Manifold Geometry for Multi-Modal Generative Modeling"

by

Mr. Hongxu CHEN


Abstract:

Deep generative and vision-language models have emerged as leading paradigms for
high-fidelity visual synthesis, cross-modal alignment, and unified multi-modal
understanding. However, in this survey, we argue that mainstream generative
pipelines are inherently constrained by incompatible flat Euclidean priors,
which conflict with the intrinsic non-Euclidean manifold geometry of multi-modal
data and result in suboptimal performance across the modeling lifecycle. We
systematically review the literature through the unifying lens of manifold
geometry, organizing existing paradigms and their overlooked geometric
bottlenecks into three core stages: 1) Representation space construction:
Expanding the latent dimension of variational autoencoders (VAEs) pushes the
reconstruction upper bound of latent diffusion models, yet unregularized
expansion causes codimension explosion and yields a rugged manifold (i.e., the
off-manifold problem), whereas relying on external foundation priors (e.g.,
DINOv2) compromises out-of-distribution (OOD) generalization; 2) Generative and
cross-modal training: Modeling continuous transport or few-shot adaptation on
flat Euclidean chords or ad-hoc hyperbolic spaces induces radial-angular
coupling, resulting in angular dynamics distortion, truncation errors, and the
neglect of radial modality confidence; and 3) Inference-time solving and
guidance: Standard extrapolation and interpolation ODE solvers face a dichotomy
between large truncation drift and sequential evaluation latency at low Neural
Function Evaluations (NFEs), while full-channel guidance extrapolation corrupts
high-frequency details. To bridge these gaps, we critically compare classical
approaches against emerging manifold-aware frontiers-spanning internal
dual-manifold topological regularization, decoupled direct-product geodesic
flows, and bi-anchor trajectory integration with subspace-selective guidance-and
outline open challenges for future research.


Date:                   Monday, 23 November 2026

Time:                   3:00pm - 5:00pm

Venue:                  Room 3494
                        Lifts 25/26

Committee Members:      Dr. Long Chen (Supervisor)
                        Dr. Hao Chen (Chairperson)
                        Dr. Xiaomin Ouyang