topological data analysis and stochastic topology - … · topological data analysis and stochastic...
Transcript of topological data analysis and stochastic topology - … · topological data analysis and stochastic...
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topological data analysisand stochastic topology
yuliy baryshnikovwaikiki, march 2013
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Promise of topological data analysis: extract the structure from the data.
In: point clouds
Out: hidden structure
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Underlying structure: manifolds?stratified sets?
Two ways to look at them: Poincare's and Riemann:
Correspondingly, there are two approaches in dimensionality reduction:
• Poincare view - use ambient coordinates - PCA, Kernel PCA,...
• Riemann view - only intrinsic proximity makes sense - IsoMap, EigenMap,...
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Topological Data Analysis aims at recovery of the most stable invariants of the data.
In particular, complete scale invariance is sought
This is both a curse and a blessing:
● very susceptible to noise● only vague notion of proximity matters
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Invariants of topological spaces: two spaces are the same if they are
homeomorphic?
diffeomorphic?
homotopic?
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Invariants of topological spaces:
Dimension?
Homotopy groups?
Homology and cohomology
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Homology and cohomology
come in different flavors and colors
singular, de Rham, simplicial
Simplicial - easiest to deal with. Works well with meshes...
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Algebraic formalism includes chains, cycles and boundaries, and the idiosyncratic relation
In essence, linear algebra encoding combinatorics of adjacencies.
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Homology groups depend on the coefficient ring: what are the entries of the boundary matrix.
Sometimes there is torsion:
Working over a field – ranks of the homology groups are called Betti numbers: numeric invariants of a topological space
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Morse theory – an important tool to generate the cellular partition of the underlying manifold, and estimate the Betti numbers
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What happens when one adds a simplex of dimension d: either a cycle in dimension d-1 disappears, or a cycle of dimension d appears.
This implies existence of an aggregate invariant, the Euler characteristic.
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Back to data: given a point cloud, how to arrive at a topological space? Taking into account desirable scale?
Old idea: alpha-shapes (Robbins, Edelsbrunner...)
Place a ball around each point in the dataset and construct the Čech complex: glue in a simplex for any non-empty intersection of the balls.
Nerve Theorem: If all intersections are contractible, the union of balls is homotopy equivalent to the Čech complex.
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Noise is a problem, however: how to get rid of the small features?Answer: increase the scale.and againand againUntil the features disappear altogether
Persistence is a way to address the onslaught of irrelevant features
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Tidal traces to be ignored: only the islands survive.
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Persistence homology
Start with a filtration, and record when a cycle appears, and when it dies.
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Encoding: either barcodes, or persistence diagram
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Important result: stability theorem – small perturbations of the filtration change the persistence diagram little
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Important tool: Morse theory
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Some examples
Mumford dataset and the Klein bottle
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Krioukov et al. Hyperbolic mappingof complex networks
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Simple test: what is the dimension of the internet?
How we find the dimension knowing a sample from a manifold?
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Testing on a special (stratified) set
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Testing on a special (stratified) set
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Autonomous Systems (“Internet”)
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Stochastic Topology: dealing with the baseline case
Niyogi, Smale, Weinberger: Cech complex for small enough r and dense enough sample with has the homotopy type of the underlying manifold
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Big Picture: from dust to foam to monolith
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Big Picture: timescale
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Big Picture: foam phase
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Foam phase: oscillation of Euler characteristic
Euler characteristic in dim=7
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Foam phase: concentrations of Betti numbers
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Life in the ambient space: crackle phenomenon
If the noise is Gaussian, the balls form a solid ball. In exponential and subexponential cases, the homologies form in layers.
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Plenty of further models and questions in stochastic topology
Persistence diagrams for Brownian bridge
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Plenty of further models and questions in stochastic topology
Topology of the level sets of random functions
Random simplicial complexes
Random configuration spaces
...
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