輔仁大學數學系學術演講

Title: Topological Alignment in Machine Learning:
Structure and Direction

Speaker：Professor Jae-Hun Jung
(Pohang University of Science and Technology in South Korea)

日 期：民國 115 年 10 月 5日（星期一）

演 講：下午1：40～2:30

地 點： SEB 140 (白恆光修女紀念講堂)

Abstract
Machine learning often asks how two representations of the same data
relate: an image encoder and a text encoder, a teacher and a student, or
the observed signals of two coupled dynamical systems. Persistent homology
offers a multiscale summary of such representations. The usual practice,
however, is to match persistence diagrams with a symmetric distance, which
leaves two questions open: what should align, and in which direction?
The first part of the talk asks what should align. The death edges on
zero-dimensional persistent homology are exactly the edges of a minimum
spanning tree, so topology records which samples are linked, not only when
components merge. Using this identity, our recent work revisits the Platonic
Representation Hypothesis across 204 vision-language model pairs. After
calibration, the relation structure of representations converges at both
local and global scales. Agreement on exact distances, by contrast, weakens
quickly and loses its dependence on model capacity.
The second part asks which direction alignment runs. We introduce
persistent cross entropy (PCE) that extends persistent entropy to pairs of
diagrams. One diagram induces a probability on the points of the other, and
an explicit event collects what it fails to explain. The excess over
persistent entropy is a Kullback-Leibler divergence and is stable in
Wasserstein distance. We present several numerical examples that shows that
PCE is directional and it recovers causal direction in dynamical systems
without a joint attractor, and in a first distillation experiment it serves
as a teacher-to-student topology loss.

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數學系 敬邀
民國115 年9 月29 日