Top Banner
Topological Dimensionality Reduction Jose Perea Dpt. of Computational Mathematics, Science & Engineering (CMSE) Dpt. of Mathematics
89

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

Apr 19, 2021

Download

Documents

dariahiddleston
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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


Related Documents
Teo perea individual1_los_movimientos_de_la_tierra_(estaciones)

Teo perea...

Category: Education
Discrete Morse Theory for Filtrations - Peoplepeople.maths.ox.ac.uk/nanda/source/Thesis.pdf · Discrete Morse Theory for Filtrations Vidit Nanda A Thesis submitted for the degree

Discrete Morse Theory for Filtrations -...

Category: Documents
Teo perea el_lince_ibérico

Teo perea el_lince_ibérico

Category: Education
Andrea Perea

Andrea Perea

Category: Documents
Zuta perea

Zuta perea

Category: Documents
Teo perea grupal1_sist_circulatorio

Teo perea grupal1_sist_circulatorio

Category: Education
Cech cohomology

Cech cohomology

Category: Documents
beta$-FILTRATION OF COHOMOLOGY OF GROUPS (Cohomology ... · Title $\beta$-FILTRATION OF COHOMOLOGY OF GROUPS (Cohomology Theory of Finite Groups and Related Topics) Author(s) YAGITA,

beta$-FILTRATION OF COHOMOLOGY OF GROUPS (Cohomology ... ·...

Category: Documents
arantxa lara perea

arantxa lara perea

Category: Documents
Group Cohomology and Bounded Cohomology · – Cohomology of finite groups and periodic cohomology – The Hochschild-Serre spectral sequence In the second part of the semester we

Group Cohomology and Bounded Cohomology · – Cohomology.....

Category: Documents
Benjamin perea

Benjamin perea

Category: Art & Photos
Future Filtrations by LiqTech International

Future Filtrations by LiqTech International

Category: Documents