Optimizing Low-Dimensional Structure: Theory and Methods in Manifold Learning

High-dimensional data is ubiquitous in modern information systems, yet most of its
intrinsic structure lives in far fewer dimensions. This talk explores the theoretical
foundations of manifold learning and the optimization principles that underpin
dimensionality reduction techniques. We begin by formalizing the manifold hypothesis
and its implications for data representation, then survey key embedding methods
through a unified optimization lens. Special attention is given to probabilistic and
neighborhood-preserving approaches, examining how their objective functions balance
competing goals such as local fidelity, global structure, and computational tractability.
We conclude with practical insights on method selection, common pitfalls, and open
challenges in learning faithful low-dimensional representations from high-dimensional
data.