TOPOLOGICAL DATA ANALYSIS FOR MULTIPARAMETER DATA

Abigail Hickok

Columbia University

TOPOLOGICAL DATA ANALYSIS (TDA)

Persistent homology uses homology to captures the “shape” of a data set (e.g., a point cloud)

PERSISTENT HOMOLOGY (FORMALLY)

PERSISTENT HOMOLOGY (FORMALLY)

simplicial complexes

SUBLEVEL FILTRATIONS

MULTIPARAMETER TOPOLOGICAL DATA ANALYSIS

MULTIPARAMETER TDA

• Previous examples involved a single parameter (e.g., the “scale parameter” for a point cloud filtration)
• Often desirable/necessary to consider multiple parameters

1. Multiparameter persistent homology (G. Carlsson and A. Zomorodian, 2009)

MULTIPARAMETER TDA

• Previous examples involved a single parameter (e.g., the “scale parameter” for a point cloud filtration)
• Often desirable/necessary to consider multiple parameters

A vineyard is a 1-parameter continuously-varying set of PDs (along with “connecting information” )

2. Vineyards (Cohen-Steiner, Edelsbrunner, Morozov 2006)

MULTIPARAMETER TDA

• Previous examples involved a single parameter (e.g., the “scale parameter” for a point cloud filtration)
• Often desirable/necessary to consider multiple parameters

3. Persistence Diagram Bundles (AH, 2023+)

PERSISTENCE DIAGRAM BUNDLES

Specials Cases Of PDBs

Image (right): Turner, Mukherjee, Boyer. Persistent Homology Transform for Modeling Shapes and Surfaces (2014).

RELATIONSHIP TO FIBERED BARCODE OF MULTIPARAMETER PERSISTENCE MODULES

Fibered barcode of a bifiltration

MOTIVATING QUESTIONS

• Are there discrete ways to encode the information in a persistence diagram bundle (PDB)?
• What kinds of geometric structure do these objects have, and does the geometry of the PDB say anything about the geometry of the data?
• How can we calculate PDBs so that they might be used for applications?

MOTIVATING QUESTIONS

• Are there discrete ways to encode the information in a persistence diagram bundle (PDB)?
• What kinds of geometric structure do these objects have, and does the geometry of the PDB say anything about the geometry of the data?
• How can we calculate PDBs so that they might be used for applications?

BACKGROUND: BIRTH AND DEATH SIMPLICES OF SINGLE-PARAMETER FILTRATIONS

PERSISTENCE DIAGRAMS ARE DETERMINED BY BIRTH, DEATH SIMPLICES

Generic PDBs are Determined by Finitely Many Base Points

STRATIFICATION EXAMPLE

STRATIFICATION EXAMPLE

The PDB is determined by the stratification, the (birth, death) pairs in each stratum, and the filtration values of each simplex

CELLULAR SHEAVES

Image: Hansen and Ghrist, 2019

A compatible cellular sheaf for a PDB

CELLULAR SHEAF EXAMPLE

Morphisms in an associated cellular sheaf

MOTIVATING QUESTIONS

• Are there discrete ways to encode the information in a persistence diagram bundle (PDB)?
• What kinds of geometric structure do these objects have, and does the geometry of the PDB say anything about the geometry of the data?
• How can we calculate PDBs so that they might be used for applications?

SECTIONS

• In a vineyard, every point can be extended to a section
• But in a PDB, sections may not exist

Each “vine” (curve) is a section of the vineyard

PDB WITH NO (NONTRIVIAL) SECTIONS

PDB WITH NO (NONTRIVIAL) SECTIONS

“Monodromy” in the PDB

CASE STUDY: MONODROMY IN THE �PERSISTENT HOMOLOGY TRANSFORM

Image: Turner, Mukherjee, Boyer. Persistent Homology Transform for Modeling Shapes and Surfaces (2014).

Question:

What does monodromy in the PHT of a shape means for the geometry of the shape itself?

(Subset of ongoing work with S. Arya, B. Giunti, AH, L. Kanari, S. McGuire, K. Turner)

PHT EXAMPLE WITH MONODROMY

PHT EXAMPLE WITHOUT MONODROMY

WHAT’S DIFFERENT ABOUT THESE SHAPES?

PHT of the spiral has monodromy

PHT of the five-arm star does not have monodromy

A SHAPE’S GEOMETRY IS REFLECTED �IN ITS PHT’S GEOMETRY

A convex shape

A SHAPE’S GEOMETRY IS REFLECTED �IN ITS PHT’S GEOMETRY

Example of a star shape.

PHT EXAMPLE WITHOUT MONODROMY

A SHAPE’S GEOMETRY IS REFLECTED �IN ITS PHT’S GEOMETRY

Example of a star shape.

MOTIVATING QUESTIONS

• Are there discrete ways to encode the information in a persistence diagram bundle (PDB)?
• What kinds of geometric structure do these objects have, and does the geometry of the PDB say anything about the geometry of the data?
• How do we calculate PDBs so that they might be used for applications?

COMPUTING A PDB

• The existence of a stratification makes it possible to compute PDBs
• We restrict to piecewise-linear fibered filtrations

COMPUTING A PDB

• The existence of a stratification makes it possible to compute PDBs
• We restrict to piecewise-linear fibered filtrations

COMPUTING A PDB�STEP 1: LINE ARRANGEMENT

COMPUTING A PDB

• The existence of a stratification makes it possible to compute PDBs
• We restrict to piecewise-linear fibered filtrations

EXAMPLE: PARAMETERIZED SET OF POINT CLOUDS

Distance between cluster centers

“Width” of clusters

EXAMPLE: PARAMETERIZED SET OF POINT CLOUDS

TP = Total persistence

AN APPLICATION TO GRAPH DATA

Curvature filtrations: Filtering a graph by ORC encodes structural information (e.g., communities) that we can use for graph ML tasks (Southern et al, 2023)

GRAPH CURVATURE PDB

DISCUSSION AND CONCLUSIONS

STABILITY

PDB’s are “pointwise” stable, but the global structure isn’t guaranteed to be stable

Two vines intersect

A perturbation of the vineyard

A different perturbation

Question: Is the global structure “generically” stable?

ALGEBRAIC STRUCTURE

CONCLUSIONS

Summary

• In persistent homology, we often have multiple parameters to account for
• PDBs are useful for identifying topological features of a parameterized set of point clouds or a parameterized set of filtrations of any form
• For generic PDBs, the information can be encoded discretely
• Monodromy within the PDB can reflect something geometrically about the data

​

Open questions

• What can we say about the algebraic structure?
• Are (generic) PDBs stable?
• When is it possible to get a “vine decomposition”?

THANK YOU

Questions?