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Topological Dimensionality Reduction Jose Perea Dpt. of Computational Mathematics, Science & Engineering (CMSE) Dpt. of Mathematics

Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

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Page 1: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

TopologicalDimensionalityReduction

JosePereaDpt.ofComputationalMathematics,Science&Engineering(CMSE)

Dpt.ofMathematics

Page 2: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete
Page 3: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete
Page 4: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete
Page 5: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete
Page 6: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete
Page 7: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Partitionofunity:

Page 8: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Partitionof1

Page 9: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Example:

Partitionof1

Page 10: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Example:

Partitionof1

Page 11: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Topological Dimensionality Reduction

Page 12: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Topologicalgroup

Page 13: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Topologicalgroup

Page 14: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Topologicalgroup

Page 15: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Topologicalgroup

Page 16: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Topologicalgroup

Page 17: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Topologicalgroup

Fiberbundle

Page 18: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Topologicalgroup

Fiberbundle fiberwisefreeandtransitive

Page 19: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Topologicalgroup

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

Page 20: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Topologicalgroup

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

Let be the sheaf (over B) of continuous G-valued

functions; , , thenTheorem:

Page 21: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Topologicalgroup

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

Let be the sheaf (over B) of continuous G-valued

functions; , , thenTheorem:

Page 22: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

UniversalBundle:

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

Page 23: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

UniversalBundle:

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

discrete

Page 24: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

UniversalBundle:

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

discrete

Page 25: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

UniversalBundle:

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

Page 26: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

UniversalBundle:

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

Page 27: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

UniversalBundle:

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

Page 28: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

UniversalBundle:

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

Page 29: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

UniversalBundle:

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

Page 30: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

UniversalBundle:

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

Page 31: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

UniversalBundle:

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

Page 32: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

UniversalBundle:

IsomorphismclassesofPrincipalG-bundles

Fiberbundle fiberwisefreeandtransitive

Theorem:

Page 33: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Partitionof1

Page 34: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Partitionof1

Page 35: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Partitionof1

Page 36: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Partitionof1

Page 37: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Partitionof1

Page 38: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Partitionof1

Page 39: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Partitionof1

Page 40: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Partitionof1

Page 41: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

(Milnor)

Partitionof1

Page 42: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

(Milnor)

Partitionof1

Page 43: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

CircularCoordinates:

Partitionof1

Page 44: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

CircularCoordinates:

Partitionof1

Page 45: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

CircularCoordinates:

Partitionof1

Page 46: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

CircularCoordinates:HarmonicCocycle

Partitionof1

Page 47: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

CircularCoordinates:HarmonicCocycle

Partitionof1

Page 48: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

COIL-5 (Columbia Image Library)

72Rotations

(Unprocessed)

5Objects

Page 49: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

ProjectiveCoords():

Page 50: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

ProjectiveCoords():

Page 51: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

ProjectiveCoords():

Page 52: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

ProjectiveCoords():

SupportsaversionofPrincipalComponentAnalysis

(ProjectivePCA)

Page 53: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

ConformationspaceofCyclo-Octane

TopologyofCyclo-OctaneEnergyLandscape,S.Martinet.al.,JCP,2010

Page 54: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

ConformationspaceofCyclo-Octane

TopologyofCyclo-OctaneEnergyLandscape,S.Martinet.al.,JCP,2010

Page 55: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

ConformationspaceofCyclo-Octane

TopologyofCyclo-OctaneEnergyLandscape,S.Martinet.al.,JCP,2010

Page 56: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

ConformationspaceofCyclo-Octane

TopologyofCyclo-OctaneEnergyLandscape,S.Martinet.al.,JCP,2010

Page 57: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete
Page 58: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete
Page 59: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete
Page 60: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

The state of the art (BG-Coordinates):

• Sparse,stableandtransductivecircularcoordinates:

• RealProjectivecoordinates:

• ComplexProjectivecoordinates:

• LensSpaceCoordinates:

Page 61: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

The state of the art (BG-Coordinates):

• Sparse,stableandtransductivecircularcoordinates:

• RealProjectivecoordinates:

• ComplexProjectivecoordinates:

• LensSpaceCoordinates:

Page 62: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

The state of the art (BG-Coordinates):

• Sparse,stableandtransductivecircularcoordinates:

• RealProjectivecoordinates:

• ComplexProjectivecoordinates:

• LensSpaceCoordinates:

Page 63: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

The state of the art (BG-Coordinates):

• Sparse,stableandtransductivecircularcoordinates:

• RealProjectivecoordinates:

• ComplexProjectivecoordinates:

• LensSpaceCoordinates:

Page 64: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

The state of the art (BG-Coordinates):

• Sparse,stableandtransductivecircularcoordinates:

• RealProjectivecoordinates:

• ComplexProjectivecoordinates:

• LensSpaceCoordinates:

Page 65: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

The state of the art (BG-Coordinates):

• Sparse,stableandtransductivecircularcoordinates:

• RealProjectivecoordinates:

• ComplexProjectivecoordinates:

• LensSpaceCoordinates:

Page 66: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

The state of the art (BG-Coordinates):

• Sparse,stableandtransductivecircularcoordinates:

• RealProjectivecoordinates:

• ComplexProjectivecoordinates:

• LensSpaceCoordinates:

Page 67: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Reconstructing Dynamics (Sliding Windows + Persistence + BG-Coordinates)

Page 68: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Slidingwindowembedding

SlidingWindowsandPersistence:Anapplicationoftopologytosignalanalysis,J.PereaandJ.Harer,FOCM2015

Page 69: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Slidingwindowembedding

SlidingWindowsandPersistence:Anapplicationoftopologytosignalanalysis,J.PereaandJ.Harer,FOCM2015

Step/delay

Windowsize

Dimension

Page 70: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Slidingwindowembeddingwindow

SlidingWindowsandPersistence:Anapplicationoftopologytosignalanalysis,J.PereaandJ.Harer,FOCM2015

Step/delay

Windowsize

Dimension

Page 71: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Slidingwindowembeddingwindow

SlidingWindowsandPersistence:Anapplicationoftopologytosignalanalysis,J.PereaandJ.Harer,FOCM2015

Step/delay

Windowsize

Dimension

Page 72: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Slidingwindowembeddingwindow

Slidingwindowpoint-cloud

SlidingWindowsandPersistence:Anapplicationoftopologytosignalanalysis,J.PereaandJ.Harer,FOCM2015

Step/delay

Windowsize

Dimension

Page 73: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

TimeSeries PersistentHomologySlidingWindowPointCloud

Page 74: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

TimeSeries PersistentHomologySlidingWindowPointCloud

Page 75: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

TimeSeries PersistentHomologySlidingWindowPointCloud

Page 76: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

TimeSeries PersistentHomologySlidingWindowPointCloud

Page 77: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete
Page 78: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

PersistentHomologyofSlidingWindowPointCloud

Page 79: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

CircularCoordinatesofSlidingWindowPointCloud

Page 80: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

OriginalDynamics RecoveredDynamics

Page 81: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

OriginalDynamics

RecoveredDynamics

Page 82: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

OriginalDynamics

RecoveredDynamics

Page 83: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

OriginalDynamics

RecoveredDynamics

Page 84: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

OriginalDynamics

RecoveredDynamics

Page 85: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete
Page 86: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete
Page 87: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

ProjectiveCoordinatesofKlein-BottleTimeSeries

Page 88: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

Code

• Today’sdemos:MATLABwrapperofRIPSER+Cocyclerepresentatives

• PythonLibrary(w/ChrisTralie):

DREiMac:DimensionReductionwithEilenberg-MacLane Coordinates

https://github.com/ctralie/DREiMac

Page 89: Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea, Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations, Discrete

J.A.Perea,MultiscaleProjectiveCoordinatesviaPersistentCohomologyofSparseFiltrations,Discrete&ComputationalGeometry,2018

Thanks!

J.A.Perea,PersistentHomologyofToroidalSlidingWindowEmbeddings,IEEE-ICASSP,2016

J.A.PereaandJ.Harer,SlidingWindowsandPersistence:AnApplicationofTopologicalMethodstoSignalAnalysis,FoundationsofComputationalMathematics,2015

J.A.Perea,TowardsSparse,StableandTransductiveCircularCoordinates,Preprint,2018

B.Xu,C.J.Tralie,A.Antia,M.LinandJ.A.Perea,TwistyTakens:AGeometricCharacterizationofGoodObservationsonDenseTrajectories,Preprint,2018


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