new dpt. of computational mathematics, science & engineering … · 2017. 3. 20. · thanks!!...
TRANSCRIPT
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The shape of data
Jose PereaDpt. of Computational Mathematics, Science & Engineering (CMSE)
Dpt. of Mathematics
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Model
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Model Error
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Model Error
Least-squaresproblem
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Let’s (try to) do it again
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• 7 × 7 grey-scale images depicting a single straight line
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• 7 × 7 grey-scale images depicting a single straight line
• intensity-centered
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• 7 × 7 grey-scale images depicting a single straight line
• intensity-centered
• subset of , via column concatenation
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The shape of data
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In Topology
Objects are equal up to continuous deformations:
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The circle The torus
The projectiveplane
The Klein bottle
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Homology of simplicial complexes
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Homology of simplicial complexes
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Fix a field ( e.g., , , , )
Homology of simplicial complexes
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Fix a field ( e.g., , , , )
Let and a simplicial complex
Homology of simplicial complexes
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Fix a field ( e.g., , , , )
Let and a simplicial complex
Homology of simplicial complexes
- th homology group of with coefficients in
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Fix a field ( e.g., , , , )
Let and a simplicial complex
Homology of simplicial complexes
- th homology group of with coefficients in
Vector space over
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Fix a field ( e.g., , , , )
Let and a simplicial complex
Homology of simplicial complexes
- th homology group of with coefficients in
Vector space over
Number of -dimensional -holes
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Homology of simplicial complexes
- th homology group of with coefficients in
Vector space over Number of -dimensional -holes
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Homology of simplicial complexes
- th homology group of with coefficients in
Vector space over Number of -dimensional -holes
# of connected components
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Homology of simplicial complexes
- th homology group of with coefficients in
Vector space over Number of -dimensional -holes
# of connected components
# of 1-dimensional holes
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The torus
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The torus
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The torus
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The torusNumber of -dimensional -holes
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The torusNumber of -dimensional -holes
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The torusNumber of -dimensional -holes
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The torusNumber of -dimensional -holes
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The Klein bottle
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The Klein bottle
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The Klein bottle
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The Klein bottleNumber of -dimensional -holes
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The Klein bottleNumber of -dimensional -holes
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The Klein bottleNumber of -dimensional -holes
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The Klein bottleNumber of -dimensional -holes
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The projective plane
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Number of -dimensional -holes The projective plane
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Number of -dimensional -holes The projective plane
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Number of -dimensional -holes The projective plane
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The Persistent Homology of Data
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Barcodes: The Persistent Topology of Data, R. Ghrist, 2008
The Persistent Homology of Data:
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Barcodes: The Persistent Topology of Data, R. Ghrist, 2008
The Persistent Homology of Data:
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Barcodes: The Persistent Topology of Data, R. Ghrist, 2008
The Persistent Homology of Data:
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Barcodes: The Persistent Topology of Data, R. Ghrist, 2008
The Persistent Homology of Data:
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The Persistent Homology of Data:
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Back to
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The Projective Plane
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Main idea
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Main idea
First cohomology
group
Homotopyclasses of maps
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Main idea
Lines in
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Main idea
Lines in
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Main idea
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Main idea
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Main idea
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Main idea
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Main idea
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How we do this in practice
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The Nerve Complex
• Open cover for
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The Nerve Complex
• Open cover for
•
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The Nerve Complex
• Open cover for
•
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The Nerve Complex
• Open cover for
•
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Theorem (P.)
Let be a metric space and let
be so that
If and then
is well defined and classifies the - vector bundle with transition functions
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Theorem (P.)
Let be a metric space and let
be so that
If and then
is well defined and classifies the - vector bundle with transition functions
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Theorem (P.)
Let be a metric space and let
be so that
If and then
is well defined and classifies the - vector bundle with transition functions
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Theorem (P.)
Let be a metric space and let
be so that
If and then
is well defined and classifies the - vector bundle with transition functions
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Projective coordinates
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
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Projective coordinates
Steps:
1. Compute salient cohomologicalfeature of the data
2. Use bundle theory (and ….) to construct classifying map
3. Do “PCA” in projective space
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
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Projective coordinates
Steps:
1. Compute salient cohomologicalfeature of the data
2. Use bundle theory (and ….) to construct classifying map
3. Do “PCA” in projective space
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
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Projective coordinates
Steps:
1. Compute salient cohomologicalfeature of the data
2. Use bundle theory (and ….) to construct classifying map
3. Do “PCA” in projective space
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
![Page 80: New Dpt. of Computational Mathematics, Science & Engineering … · 2017. 3. 20. · Thanks!! J. Perea and J. Harer, Sliding Windows and Persistence: An Application of Topological](https://reader035.vdocument.in/reader035/viewer/2022071507/61286a021f5a4710835592b4/html5/thumbnails/80.jpg)
Projective coordinates
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
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Projective coordinates
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
Data
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Suggested Model and map
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
Data
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More examples
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The torus
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
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The torus
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
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The torus
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The torus
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The torus
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The torus
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The Klein bottle
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
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This page is intentionally left blank
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The Klein bottle
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
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The Klein bottle
Projective coordinates for the analysis of data, J. Perea, Preprint, 2016
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Quantifying recurrence in time-evolving systems(no equations allowed)
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Recording
Dynamic networkVideo dataEvolving solution (RB conv.)
4 Data modalities – 1 Theme
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Problem:
Given time series data
• Determine
• Quantify
• Leverage
the global geometric/topological structure of attractors
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Problem:
Given time series data
• Determine
• Quantify
• Leverage
the global geometric/topological structure of attractors
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Problem:
Given time series data
• Determine
• Quantify
• Leverage
the global geometric/topological structure of attractors
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Problem:
Given time series data
• Determine – generative models
• Quantify
• Leverage
the global geometric/topological structure of attractors
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Problem:
Given time series data
• Determine – generative models
• Quantify – dynamic regimes
• Leverage
the global geometric/topological structure of attractors
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Problem:
Given time series data
• Determine – generative models
• Quantify – dynamic regimes
• Leverage – for prediction and control
the global geometric/topological structure of attractors
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Topological Time Series Analysis
Set up:
Metric space Time series
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Topological Time Series Analysis
Set up:
Metric space Time series
• Real/complex numbers • Images• Functions• Graphs
Examples ( ):
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Topological Time Series Analysis
Set up:
Metric space Time series
• Real/complex numbers • Images• Functions• Graphs
Examples ( ):
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Topological Time Series Analysis
Set up:
Metric space Time series
Sliding window embedding:
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Topological Time Series Analysis
Set up:
Metric space Time series
Sliding window embedding:
parameters
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Topological Time Series Analysis
Set up:
Metric space Time series
Sliding window embedding:
parameters
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Topological Time Series Analysis
Set up:
Metric space Time series
Sliding window embedding:
parameters
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Topological Time Series Analysis
Set up:
Metric space Time series
Sliding window embedding:
parameters
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Time Series
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Time Series
Commensurate
Non-Commensurate
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Sliding Window EmbeddingTime Series
Commensurate
Non-Commensurate
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Sliding Window EmbeddingTime Series
Commensurate
Non-Commensurate
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(joint with Chris Tralie)
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(joint with Chris Tralie)
PCA of sliding window embedding
Eigenvalues
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Applications
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Results: Humans (Amazon turk) vs Computers
Kendall’s Tau SWCutlerDavis
FreqCutlerDavis
LatticeHumans
SW 1 -0.315 0.221 0.663
CutlerDavisFreq
1 -0.0842 -0.294
CutlerDavisLattice
1 0.347
Humans 1
Correlation of rakings (from most periodic to least periodic)
across 20 videos
(joint with Chris Tralie)
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Physiology of speech Disorders
normal Mucus irregular AP-BiphonationClinical asymmetry
(joint with Chris Tralie)
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Video Name Periodicity Score Quasiperiodicity Score
HerbstPeriodic 0.794 0.005
NormalPeriodic 0.561 0.010
APBiphonation 0.296 0.298
APBiphonation2 0.486 0.133
ClinicalAsymmetry 0.072 0.423
MucusPerturbedPeriodic 0.019 0.005
HerbstIrregular 0.019 0.037
(joint with Chris Tralie)
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Thanks!!J. Perea and J. Harer, Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis, Foundations of Computational Mathematics, 2014.
J. Perea, A. Deckard, S. Haase and J. Harer, SW1PerS: Sliding Windows and 1-Persistence Socring; Discovering Periodicity in Time Series Data, BMC Bioinformatics, 2015.
J. Perea, Multi-Scale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, arXiv, 2016.
J. Perea, Persistent Homology of Toroidal Sliding Window Embeddings, ICASSP, 2016.