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The Underlying Topology of Data

Jose Perea

Mathematics

Computer Sciences

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Königsberg, 1700s

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“This question is so banal, but seemed to me worthy of attention in that [neither] geometry, nor algebra, nor even the art of counting was sufficient to solve it”

Hopkins, Brian, and Robin Wilson. "The Truth about Königsberg." College Mathematics Journal (2004), 35, 198-207.

Leonhard Euler, 1707 - 1783

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Leonhard Euler, 1707 - 1783

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Leonhard Euler, 1707 - 1783

3

5

degree

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Euler, L. Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae, 1741

Leonhard Euler, 1707 - 1783

Has an Eulerian

path

# of nodes with odd degree

=

THEN

Theorem

3

5

3

Königsberg

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In Topology

Objects are equal up to continuous deformations:

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Topological Data Analysis

Computer Vision

Computational Biology

Dimensionality Reduction

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Betti Numbers:

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Betti Numbers:

Components

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Betti Numbers:

Components

Holes

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Betti Numbers:

Components

Holes

Voids

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barcode

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The Persistent Homology of Data:

Betti Numbers

Barcodes

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Data

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The Persistent Homology of Data:

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Detecting Recurrence in Time Series Data

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Global control of cell-cycle transcription by coupled SDK and network oscillators, D. Orlando et. al., Nature, 2008

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What is recurrence, and how do we quantify it?

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Sliding Windows

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Sliding Windows

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Sliding window embedding

window

Sliding window point-cloud

Sliding Windows and Persistence: An application of topology to signal analysis, J. Perea and J. Harer, FOCM 2015

Step/delay

Dimension

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SW1PerS: Sliding Windows and 1-Persistence Scoring

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1D- Persistence Barcode

Recurrence score

Sliding Windows and Persistence: An application of topology to signal analysis, J. Perea and J. Harer, FOCM 2015

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Biological Clocks

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SW1PerS: Sliding Windows and 1-Persistence Scoring, J. Perea et. al., 2015

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SW1PerS: Sliding Windows and 1-Persistence Scoring, J. Perea et. al., 2015

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SW1PerS: Sliding Windows and 1-Persistence Scoring, J. Perea et. al., 2015

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Yeast Metabolic Cycle Data

SW1PerS: Sliding Windows and 1-Persistence Scoring, J. Perea et. al., 2015

Rankings of genes in the top 10% (out of 9,330 ) according to SW, and not in the top 10% for any other algorithm

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Persistence Diagram

Barcode

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arXiv:2103.04540

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Time Series

Commensurate

Non- Commensurate

Sliding Window Point Cloud

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Time Series

Persistent Homology

Sliding Window Point Cloud

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Recurrence in video data (SW1PerS-video)

Sliding window embedding

C. Tralie and J. Perea, (Quasi)Periodicity Quantification in Video Data, Using Topology, 2018.

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Experiment: Mechanical Turk

C. Tralie and J. Perea, (Quasi)Periodicity Quantification in Video Data, Using Topology, 2018.

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Results: Humans (Amazon turk) vs Computers

Kendall’s Tau	SW	CutlerDavis Freq	CutlerDavis Lattice	Humans
SW	1	-0.315	0.221	0.663
CutlerDavis Freq		1	-0.0842	-0.294
CutlerDavis Lattice			1	0.347
Humans				1

Correlation of rakings

(from most periodic to least periodic)

across 20 videos

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Recurrence in video data (SW1PerS-video)

C. Tralie and J. Perea, (Quasi)Periodicity Quantification in Video Data, Using Topology, 2018.

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Recurrence in video data (SW1PerS-video)

normal

C. Tralie and J. Perea, (Quasi)Periodicity Quantification in Video Data, Using Topology, 2018.

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Recurrence in video data (SW1PerS-video)

Clinical asymmetry

C. Tralie and J. Perea, (Quasi)Periodicity Quantification in Video Data, Using Topology, 2018.

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Laryngeal video-endoscopy

normal

Mucus irregular

AP-Biphonation

Clinical asymmetry

C. Tralie and J. Perea, (Quasi)Periodicity Quantification in Video Data, Using Topology, 2018.

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Continuous and compactly supported

Templates

Remark:

Is continuous.

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Theorem

Let be compact and let be continuous.

Then, given and nonzero, there exist

and a polynomial so that

for every .

J. A. Perea, E. Munch and F. Khasawneh, Approximating Continuous Functions on Persistence Diagrams, FoCM 2022

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Theorem

Let be compact and let be continuous.

Then, given and nonzero, there exist

and a polynomial so that

for every .

J. A. Perea, E. Munch and F. Khasawneh, Approximating Continuous Functions on Persistence Diagrams, FoCM 2022

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Learning with Templates:

Training data:

Model parameters: , ,

Loss function: For , let

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A protein classification benchmark collection for machine learning, P. Sonego et. al., Nucleic acids research, 2007.

Protein Classification Benchmark Collection

(PCB00019) – SCOP40mini

1,357 proteins	# atoms ~ 1K	55 classification tasks

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L. Polanco and J. A. Perea, Adaptive template systems: Data-driven feature selection for learning with persistence, ICMLA 2019

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template

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Theorem [Elchesen, Hartsock, P. , Rask]

Let be compact and let be continuous.

Then, given , there exist functions

and a polynomial so that

for every .

A. Elchesen, I. Harstock, J. A. Perea and T. Rask, Learning on Persistence Diagrams as Radon Measures, arXiv 2022

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Experiment: Shape classification

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Experiment: Texture classification

Sublevel set persistence

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Experiment: Texture classification

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Experiment: Texture classification

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Experiment: Satellite cloud pattern classification

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Thanks!

http://www.joperea.com