Topological Data Analysis and Representation Theory 2025

位相的データ解析と表現論2025

Concept

Persistent homology is one of the main methods in topological data analysis, and can analyze the shape of data at multiple scales. It has been applied in many fields, for example in materials science, and also has been used as a preprocessing of data for machine learning. The theory of persistent homology is supported by algebra. In particular, understanding the category of persistence modules is important for multiparameter persistence and exploring issues of stability with respect noise. For further developments, we decided to organize this workshop to explore various interactions between topological data analysis, representation theory, and machine learning.

パーシステントホモロジーは位相的データ解析における主たる手法のうちの一つであり、データのかたちをマルチスケールで解析できる。材料科学に直接適用されるだけでなく機械学習の前処理にも用いられるパーシステントホモロジーの、その理論は代数によって支えられている。その中でもノイズ安定性についてはパーシステントホモロジーのなす圏を理解することが重要である。時空間同時解析などパーシステントホモロジーの理論の高次元化が望まれているが、その圏はより複雑になって行くため、代数的により多様な理論を必要とすることは間違いない。位相的データ解析、表現論と機械学習のさまざまなインタラクションを模索するため、この研究集会を企画することとした。

Details

Dates: 12 & 13 February 2025 （2025年2月12日、13日）

Hybrid event

In-person (registration required, limited to 25 people):
Kobe University, Graduate School of Human Development and Environment Bldg A room 739
現地開催場所（要参加登録、人数制約25名）：
神戸大学大学院人間発達環境学研究科 A棟739 (https://www.h.kobe-u.ac.jp/ja/access)
Online (registration required):
Microsoft Teams. We will send the link to your registered email address.

Notes:

In-person participation is limited to 25 people. Please register.
現地参加については人数の制約(約25名)があり先着順になります（要参加登録）。
Deadline for registration: 10 February 2025
参加登録締切：2025年2月10日

参加登録フォーム（Register Here）

Speakers and Talks

（敬称略）

Speaker: Woojin Kim

Title: Many Facets of the Generalized Rank Invariant

Abstract: The Generalized Rank Invariant (GRI) is a natural invariant for multi-parameter persistence modules, extending the concept of the persistence diagram from one-parameter to the multi-parameter setting. In this talk, we will discuss various aspects of the GRI, including its Möbius invertibility, discriminating power, connections to other invariants, computational aspects, and estimation methods, as time permits.

Speaker: Enhao Liu

Title: Invariants by Intervals, Essential Covers, and Their Applications

Abstract: Defining meaningful and computationally effective invariants is crucial in multiparameter persistent homology. Interval modules provide a well-structured subset of representations, facilitating visualization and practical data analysis. Restricting attention to interval-based invariants induces a tractable framework, balancing computational feasibility and theoretical insight. We present explicit formulas for two invariants—the interval rank invariant and interval multiplicity—along with a novel technique called the essential cover. This technique allows us not only to compute the invariants from certain special filtrations of simplicial complexes, but also to translate theories from one persistence to another.

Speaker: Ken Nakashima

Title: Connected Persistence Diagrams – From Theoretical Development to Practical Applications –

Abstract: A connected persistence diagram (cPD) is proposed as an extension of the conventional persistence diagram used in topological data analysis. In this talk, I will begin by reviewing the conventional persistence diagram and then introduce the concept of the cPD, describing its fundamental ideas. I will also present case studies demonstrating how the cPD can be applied to real materials data. Through these examples, I will explain how to interpret cPD and discuss their potential future applications. Furthermore, I will introduce “RuCPD,” a software package for computing cPD co-developed with Ippei Obayashi at Okayama University. If time allows, I will also touch on some of the considerations made during the software’s development, including improvements to computational efficiency.

Speaker: Kaveh Mousavand

Title: Hom-orthogonal modules, bricks and representation varieties

Abstract: In representation theory of algebras, bricks (a.k.a Schur representations) form an important subfamily of indecomposable modules, and they are known to play decisive roles in many areas of research. Originally motivated by some open conjectures in representation theory, so-called brick-Brauer-Thrall Conjectures, in our recent work we found new connections between the pairwise Hom-orthogonal modules and bricks. In particular, we use some algebraic and geometric tools to find explicit necessary and sufficient conditions for the existence of an infinite family of Hom-orthogonal modules of the same dimension. Consequently, we prove some new results on the geometry of representation varieties of the algebras under consideration. This is based on a series of joint work with Charles Paquette.

Speaker: Naoki Nishikawa

Title: Machine Learning with Topological Loss

Abstract: In the application of persistent homology (PH) to machine learning (ML), one can topologically control input data or ML models by optimizing a topological loss, which incorporates PH as part of the loss function. For instance, when we want to compress a high-dimensional point cloud into a 2D plane for visualization, we can use a topological loss to obtain a 2D point cloud that preserves the original data’s topology. Similarly, when segmenting a point cloud representing an object’s surface into its constituent parts, through training with a topological loss, we can obtain a ML model that provides topologically accurate segmentation. While the range of applications for topological loss continues to grow, challenges such as computational inefficiency for large-scale, high-dimensional data still remain. In this talk, I will introduce some applications of topological loss in machine learning and recent research aimed at improving efficiency.

Speaker: Shunsuke Kano

Title: Causal discovery via magnitude homology

Abstract: Causal discovery aims to identify causal relationships among variables in observational data, determining which variables act as causes and which as effects. DirectLiNGAM is one such method, resulting in a weighted directed acyclic graph (DAG) where edge weights represent causal strengths. These graphs are referred to as "causal graphs."

On the other hand, the AI system "WideLearning," developed by Fujitsu, identifies critical "conditions" important for causal discovery. Each condition corresponds to a subset of the original dataset, resulting in multiple causal graphs, one for each condition.

In this talk, I propose a new approach to analyze these numerous causal graphs by treating them as generalization of metric spaces and leveraging magnitude homology, an invariant of generalized metric spaces.

This talk is based on collaborative research conducted at the Fujitsu x Tohoku University Discovery Intelligence Laboratory.

Program Schedule

（敬称略）

12 February 2025

10:00-10:30 Announcements and Introductions （会場のアナウンスや自己紹介）

10:30-11:30 Woojin Kim "Many Facets of the Generalized Rank Invariant"

---　Lunch break（2h 30 mins.）　---

14:00-15:00 Enhao Liu "Invariants by Intervals, Essential Covers, and Their Applications"

15:30-16:30 Ken Nakashima "Connected Persistence Diagrams – From Theoretical Development to Practical Applications –"

16:30-18:00 Free discussion （自由討論）

13 February 2025

09:30-10:30 Kaveh Mousavand "Hom-orthogonal modules, bricks and representation varieties"

11:00-12:00 Naoki Nishikawa "Machine Learning with Topological Loss"

---　Lunch break（ 2 h ）　---

14:00-15:00 Shunsuke Kano "Causal discovery via magnitude homology"

15:30-16:30 Takuma Imamura "Similarities and differences between topology and coarse geometry from a logical view point"