Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea,...

TopologicalDimensionalityReduction

JosePereaDpt.ofComputationalMathematics,Science&Engineering(CMSE)

Dpt.ofMathematics

Partitionofunity:

Partitionof1

Example:

Partitionof1

Example:

Partitionof1

Topological Dimensionality Reduction

Topologicalgroup

Fiberbundle

Topologicalgroup

Fiberbundle fiberwisefreeandtransitive

Topologicalgroup

IsomorphismclassesofPrincipalG-bundles

Topologicalgroup

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

functions; , , thenTheorem:

Topologicalgroup

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

functions; , , thenTheorem:

UniversalBundle:

discrete

UniversalBundle:

discrete

UniversalBundle:

Theorem:

Partitionof1

(Milnor)

Partitionof1

(Milnor)

Partitionof1

CircularCoordinates:

Partitionof1

CircularCoordinates:HarmonicCocycle

Partitionof1

CircularCoordinates:HarmonicCocycle

Partitionof1

COIL-5 (Columbia Image Library)

72Rotations

(Unprocessed)

5Objects

ProjectiveCoords():

SupportsaversionofPrincipalComponentAnalysis

(ProjectivePCA)

ConformationspaceofCyclo-Octane

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

The state of the art (BG-Coordinates):

• Sparse,stableandtransductivecircularcoordinates:

• RealProjectivecoordinates:

• ComplexProjectivecoordinates:

• LensSpaceCoordinates:

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

Slidingwindowembedding

SlidingWindowsandPersistence:Anapplicationoftopologytosignalanalysis,J.PereaandJ.Harer,FOCM2015

Slidingwindowembedding

Step/delay

Windowsize

Dimension

Slidingwindowembeddingwindow

Step/delay

Windowsize

Dimension

Step/delay

Windowsize

Dimension

Slidingwindowpoint-cloud

Step/delay

Windowsize

Dimension

TimeSeries PersistentHomologySlidingWindowPointCloud

PersistentHomologyofSlidingWindowPointCloud

CircularCoordinatesofSlidingWindowPointCloud

OriginalDynamics RecoveredDynamics

OriginalDynamics

RecoveredDynamics

OriginalDynamics

RecoveredDynamics

OriginalDynamics

RecoveredDynamics

OriginalDynamics

RecoveredDynamics

ProjectiveCoordinatesofKlein-BottleTimeSeries

• Today’sdemos:MATLABwrapperofRIPSER+Cocyclerepresentatives

• PythonLibrary(w/ChrisTralie):

DREiMac:DimensionReductionwithEilenberg-MacLane Coordinates

https://github.com/ctralie/DREiMac

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

Dpt. of Computational Mathematics, Science & Engineering … · 2018. 6. 15. · J. A. Perea,...

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