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Visualizing Topological Importance:

A Class-Driven Approach

Yu (Demi) Qin, Tulane University

Brittany Fasy, Montana State University

Carola Wenk, Tulane University

Presented by Brian Summa, Tulane University

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Motivation

  • Topological data analysis (TDA) extracts meaningful shape from complex data.
  • These features are represented compactly in persistence diagrams.
  • Not all topological features are equally important for analysis tasks!
  • Typically, persistence (lifetime) is used to (implicitly or explicitly) weigh feature importance.
  • But, persistence may not be the optimal weight for all datasets or tasks!

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Low Persistence can be More Important

Example Datasets

References

[1] C. Hofer, NeurIPS 2017

[2] H Riihimaki, ATDA 2019

[3] V Patrangenaru, Sankhya A 2019

[4] Q Zhao, NeurIPS 2019

[5] P Bubenik, Inverse Problems 2020

…

More!

Protein Molecules

Histological Images

Leaf Images

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Which topological feature is more important?

Persistence Diagram

Original Data

H1

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Wait, if we have two classes….

Class A

Persistence Diagram

Original Data

H1

Persistence Diagram

Original Data

H1

Class B

Not all topological features are equally important for analysis tasks!

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Can we learn the right weights?

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Approach Overview

  • Use deep metric learning on vectorized topological features.
  • Train CNN with attention module and metric loss.

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Results: Topological Classification

Dataset

W1

PWPI

WKPI

BC

PWGK

SWK

Ours

3D Shape

0.92

0.89

1.0

0.91

0.9

0.88

1.0

COLLAB

0.76

0.73

0.77

0.76

0.71

0.78

0.84

PROTEINS

0.77

0.76

0.79

0.76

0.72

0.76

0.87

Prostate

0.82

0.85

0.88

0.86

0.83

0.84

0.95

Colorectal

0.77

0.78

0.81

0.77

0.78

0.75

0.85

Typical

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Importance Field Visualization

  • Create a 2D heatmap over a persistence diagram - Importance Field.
  • Importance field shows the importance of region of the diagram for classification.
  • Based on the gradient of class score in our model.

Our Model

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In-Image Visualization

  • Maps importance field back to original image data.
  • Uses persistence diagram to lookup pixel locations of 0D feature critical pairs.
  • Renders interlevel set between critical pairs based on learned importance.

Importance Field

In-Image

VIS

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Different Classes Different Importance Features

Gleason 3

Gleason 5

Original

Image

Importance

Field

In-Image

VIS

Prostate Cancer

Gleason 4

Similar Diagram,

Different Importance

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Different Classes Different Importance Features

Gleason 3

Gleason 5

Original

Image

Importance

Field

In-Image

VIS

NORM

MUS

Prostate Cancer

Colorectal Cancer

Gleason 4

STR

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Same Class Similar Important Features

MUC

TUM

Colorectal Cancer

Different Diagram,

Similar Importance

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Same Class Similar Important Features

Gleason 3

MUC

Gleason 4

Original

Image

Importance

Field

In-Image

VIS

TUM

Prostate Cancer

Colorectal Cancer

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Conclusions

  • The first approach is to visually present topological feature importance.
  • State-of-the-art topological classification accuracy with learned weights.
  • Novel in-image visualization revealing topological structures.

ML

VIS

TDA

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Thank you to our funders

  • DOE ASCR DE-SC0022873
  • NIH R01GM143789
  • NSF-IIS 2136744
  • NSF-CCF 2046730
  • NSF-DMS 1664858

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Thank you!

Q&A

Demi will be graduating soon!

Email: yqin2@tulane.edu

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EXTRA SLIDES

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Graph Dataset

  • Extended Filtration to capture H0 and H1 in one diagram
  • Our model can indicate H0 vs H1, which is more important

Extended Filtration

Protein Graph

Collab Graph

Ours Persistence

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3D Shapes

  • Our model can learn the weight on high persistence as well

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Prostate Cancer Images

Original Image

Importance Field VIS

In-Image

VIS

High Importance

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Colorectal Cancer Images

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Pipeline - Prostate Cancer

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Ours vs Persistence - Colorectal Cancer

  • Our Learned Importance vs Persistence Importance

Ours

Persistence