北京雁栖湖应用数学研究院 北京雁栖湖应用数学研究院

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关于我们
院长致辞
理事会
协作机构
参观来访
人员
管理层
科研人员
博士后
来访学者
行政团队
学术研究
研究团队
公开课
讨论班
招生招聘
教研人员
博士后
学生
会议
学术会议
工作坊
论坛
学院生活
住宿
交通
配套设施
周边旅游
新闻
新闻动态
通知公告
资料下载
清华大学 "求真书院"
清华大学丘成桐数学科学中心
清华三亚国际数学论坛
上海数学与交叉学科研究院
BIMSA > BIMSA Topology Seminar Cross-Barcodes and their applications in data analysis
Cross-Barcodes and their applications in data analysis
组织者
马修·伯菲特 , 李京艳 , 吴杰 , 杨南君 , 周嘉伟
演讲者
Serguei Barannikov
时间
2025年01月09日 13:30 至 14:30
地点
A3-2a-302
线上
Zoom 482 240 1589 (BIMSA)
摘要
In this talk, we describe a topological framework for comparing data representations, called R-Cross-Barcodes [1], and discuss its use in data analysis. R-Cross-Barcodes are a tool that measures multi-scale discrepancies in the topological structures of two point clouds with a one-to-one correspondence between points. The R-Cross-Barcodes track the discrepancies of topological features taking into account their localization, allowing comparison of data embeddings even when they lie in distinct ambient spaces. Based on R-Cross-Barcode, we define the Representation Topology Divergence (RTD), a scalar quantifying the topological differences in two data representations. We review the construction and principal properties of R-Cross-Barcodes, including the exact sequence:

\[
\cdots \;\to\;
H_i\bigl(\mathrm{VR}_\alpha(\mathcal{G}^w)\bigr)
\;\to\;
H_i\bigl(\mathrm{VR}_\alpha(\mathcal{G}^{\min(w,\tilde{w}}))\bigr)
\;\to\;
H_i\bigl(\mathrm{VR}_\alpha(\hat{\mathcal{G}}^{w,\tilde{w}})\bigr)
\;\to\;
H_{i-1}\bigl(\mathrm{VR}_\alpha(\mathcal{G}^w)\bigr)
\;\to\;
\cdots
\]

relating the R-Cross-Barcodes to localization-aware discrepancies in standard Bar-codes features.

We then incorporate RTD as a loss in deep auto-encoders to obtain topology-preserving data embeddings. Minimizing RTD aligns topological features between the original dataset and the embedding. We explain the stability, the continuity and the differentiability properties of the RTD loss. Our experiments with neural network representation analysis demonstrate that R-Cross-Barcodes and RTD capture and preserve topological structure, providing a robust method for analyzing, comparing and optimizing complex data representations
演讲者介绍
Prof. Serguei Barannikov earned his Ph.D. from UC Berkeley and has made contributions to algebraic topology, algebraic geometry, mathematical physics, and machine learning. His work, prior to his Ph.D., introduced canonical forms of filtered complexes, now known as persistence barcodes, which have become fundamental in topological data analysis. More recently, he has applied topological methods to machine learning, particularly in the study of large language models, with results published in leading ML conferences such as NeurIPS, ICML, and ICLR, effectively bridging pure mathematics and advanced AI research.
北京雁栖湖应用数学研究院
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北京雁栖湖应用数学研究院 101408

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