dpt. of computational mathematics, science & engineering … · 2018. 6. 15. · j. a. perea,...
TRANSCRIPT
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TopologicalDimensionalityReduction
JosePereaDpt.ofComputationalMathematics,Science&Engineering(CMSE)
Dpt.ofMathematics
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Partitionofunity:
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Partitionof1
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Example:
Partitionof1
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Example:
Partitionof1
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Topological Dimensionality Reduction
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Topologicalgroup
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Topologicalgroup
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Topologicalgroup
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Topologicalgroup
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Topologicalgroup
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Topologicalgroup
Fiberbundle
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Topologicalgroup
Fiberbundle fiberwisefreeandtransitive
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Topologicalgroup
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
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Topologicalgroup
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
Let be the sheaf (over B) of continuous G-valued
functions; , , thenTheorem:
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Topologicalgroup
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
Let be the sheaf (over B) of continuous G-valued
functions; , , thenTheorem:
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UniversalBundle:
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
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UniversalBundle:
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
discrete
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UniversalBundle:
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
discrete
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UniversalBundle:
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
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UniversalBundle:
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
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UniversalBundle:
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
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UniversalBundle:
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
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UniversalBundle:
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
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UniversalBundle:
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
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UniversalBundle:
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
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UniversalBundle:
IsomorphismclassesofPrincipalG-bundles
Fiberbundle fiberwisefreeandtransitive
Theorem:
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Partitionof1
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Partitionof1
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Partitionof1
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Partitionof1
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Partitionof1
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Partitionof1
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Partitionof1
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Partitionof1
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(Milnor)
Partitionof1
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(Milnor)
Partitionof1
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CircularCoordinates:
Partitionof1
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CircularCoordinates:
Partitionof1
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CircularCoordinates:
Partitionof1
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CircularCoordinates:HarmonicCocycle
Partitionof1
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CircularCoordinates:HarmonicCocycle
Partitionof1
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COIL-5 (Columbia Image Library)
72Rotations
(Unprocessed)
5Objects
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ProjectiveCoords():
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ProjectiveCoords():
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ProjectiveCoords():
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ProjectiveCoords():
SupportsaversionofPrincipalComponentAnalysis
(ProjectivePCA)
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ConformationspaceofCyclo-Octane
TopologyofCyclo-OctaneEnergyLandscape,S.Martinet.al.,JCP,2010
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ConformationspaceofCyclo-Octane
TopologyofCyclo-OctaneEnergyLandscape,S.Martinet.al.,JCP,2010
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ConformationspaceofCyclo-Octane
TopologyofCyclo-OctaneEnergyLandscape,S.Martinet.al.,JCP,2010
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ConformationspaceofCyclo-Octane
TopologyofCyclo-OctaneEnergyLandscape,S.Martinet.al.,JCP,2010
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The state of the art (BG-Coordinates):
• Sparse,stableandtransductivecircularcoordinates:
• RealProjectivecoordinates:
• ComplexProjectivecoordinates:
• LensSpaceCoordinates:
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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](https://reader035.vdocuments.mx/reader035/viewer/2022071507/61286d71bf040d091369ddf4/html5/thumbnails/62.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022071507/61286d71bf040d091369ddf4/html5/thumbnails/63.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022071507/61286d71bf040d091369ddf4/html5/thumbnails/64.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022071507/61286d71bf040d091369ddf4/html5/thumbnails/65.jpg)
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](https://reader035.vdocuments.mx/reader035/viewer/2022071507/61286d71bf040d091369ddf4/html5/thumbnails/66.jpg)
The state of the art (BG-Coordinates):
• Sparse,stableandtransductivecircularcoordinates:
• RealProjectivecoordinates:
• ComplexProjectivecoordinates:
• LensSpaceCoordinates:
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Reconstructing Dynamics (Sliding Windows + Persistence + BG-Coordinates)
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Slidingwindowembedding
SlidingWindowsandPersistence:Anapplicationoftopologytosignalanalysis,J.PereaandJ.Harer,FOCM2015
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Slidingwindowembedding
SlidingWindowsandPersistence:Anapplicationoftopologytosignalanalysis,J.PereaandJ.Harer,FOCM2015
Step/delay
Windowsize
Dimension
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Slidingwindowembeddingwindow
SlidingWindowsandPersistence:Anapplicationoftopologytosignalanalysis,J.PereaandJ.Harer,FOCM2015
Step/delay
Windowsize
Dimension
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Slidingwindowembeddingwindow
SlidingWindowsandPersistence:Anapplicationoftopologytosignalanalysis,J.PereaandJ.Harer,FOCM2015
Step/delay
Windowsize
Dimension
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Slidingwindowembeddingwindow
Slidingwindowpoint-cloud
SlidingWindowsandPersistence:Anapplicationoftopologytosignalanalysis,J.PereaandJ.Harer,FOCM2015
Step/delay
Windowsize
Dimension
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TimeSeries PersistentHomologySlidingWindowPointCloud
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TimeSeries PersistentHomologySlidingWindowPointCloud
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TimeSeries PersistentHomologySlidingWindowPointCloud
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TimeSeries PersistentHomologySlidingWindowPointCloud
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PersistentHomologyofSlidingWindowPointCloud
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CircularCoordinatesofSlidingWindowPointCloud
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OriginalDynamics RecoveredDynamics
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OriginalDynamics
RecoveredDynamics
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OriginalDynamics
RecoveredDynamics
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OriginalDynamics
RecoveredDynamics
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OriginalDynamics
RecoveredDynamics
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ProjectiveCoordinatesofKlein-BottleTimeSeries
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Code
• Today’sdemos:MATLABwrapperofRIPSER+Cocyclerepresentatives
• PythonLibrary(w/ChrisTralie):
DREiMac:DimensionReductionwithEilenberg-MacLane Coordinates
https://github.com/ctralie/DREiMac
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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