Persistent homology algebraic foundationscv.mikael.johanssons.org/talks/2017-01-JMM.pdf · The...
Transcript of Persistent homology algebraic foundationscv.mikael.johanssons.org/talks/2017-01-JMM.pdf · The...
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Mikael Vejdemo-JohanssonCUNY College of Staten Island
Persistent homology&
algebraic foundations
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• Topological Data Analysis and Persistent Homology
• The two persistences
• Algebraic foundations: the more we know…
• Sheaves
Outline
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• Topological Data Analysis and Persistent Homology
• The two persistences
• Algebraic foundations: the more we know…
• Sheaves
Outline
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Fundamental idea: use topology to understand data.
Data has shape. Shape carries meaning.
Key idea: how to create meaningful topology from discrete observations.
Topological Data Analysis
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• Mapper / Reeb graphs / Reeb spacesCreate topological models of data by breaking up fibres of a map
• Persistent (co)homologySweep across parametrized family of topological spaces, summarize changes in their (co)homology Works through creating simplicial complexes that carry shapeVietoris-Rips; Čech; α-shapes; witness complexes…
Several approaches
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Čech complex
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Čech complex
1 loop
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Čech complex
3 loops
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Čech complex
1 loop
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• Topological Data Analysis and Persistent Homology
• The two persistences
• Algebraic foundations: the more we know…
• Sheaves
Outline
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Two approaches to persistent homology
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Persistent homology is fundamentally about…
Two approaches to persistent homology
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Persistent homology is fundamentally about…
• Sublevel sets of some function on some manifold
Two approaches to persistent homology
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Persistent homology is fundamentally about…
• Sublevel sets of some function on some manifold
• Filtered (parametrized) families of topologicalspaces
Two approaches to persistent homology
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Sublevel sets
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Sublevel sets
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Filtered spaces
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Vietoris-Rips and Čech capture sublevel sets of the distance-to-data function
Perspectives are (kinda) equivalent
Manifold can be discretized, produces filtered topological spaces.
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• Topological Data Analysis and Persistent Homology
• The two persistences
• Algebraic foundations: the more we know…
• Sheaves
Outline
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Topological persistence and simplification
Defines persistent homology and provides an algorithm for computing with coefficients in ℤ2
With each span of values for a function on a manifold is associated the group of homological features that exist in all sublevel sets
Edelsbrunner, Letscher and Zomorodian (2000; 2002)
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We know what to capture
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We know what to capture
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Computing Persistent Homology
Observes there is a functor in play in the constructions in Edelsbrunner, Letscher and Zomorodian (2000).
Suggests that the structure to be captured can be described by a module over ℤ2[t].
Hence, arbitrary field coefficients 𝕜 work using 𝕜[t].
Carlsson and Zomorodian (2005)
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Functoriality𝕏
H1𝕏 0 𝕜 𝕜3 𝕜
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𝕜[t] moduleH1𝕏 0 𝕜 𝕜3 𝕜
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𝕜[t] moduleH1𝕏 0 𝕜 𝕜3 𝕜
Take direct sum H1𝕏 = 𝕜 ⊕ 𝕜3 ⊕ 𝕜
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𝕜[t] moduleH1𝕏 0 𝕜 𝕜3 𝕜
Take direct sum H1𝕏 = 𝕜 ⊕ 𝕜3 ⊕ 𝕜
Put each summand in module degree corresponding to position in sequence
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𝕜[t] moduleH1𝕏 0 𝕜 𝕜3 𝕜
Take direct sum H1𝕏 = 𝕜 ⊕ 𝕜3 ⊕ 𝕜
Put each summand in module degree corresponding to position in sequence
Module structure by defining ·t to be the induced map from the functor action
·t ·t ·t
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The Theory of Multidimensional Persistence
Also: Carlsson, Singh and Zomorodian (2009) and Biasotti, Cerri, Frosini, Giorgi and Landi (2008)
Suggests that tracking multiple parameters corresponds to working over 𝕜[t1, t2, …, td]
We are still today searching for a good invariant to describe the resulting modules.
Carlsson and Zomorodian (2009)
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Zigzag Persistence
Notices most use cases do not use the structure from a 𝕜[t]-module: finite diagram most common.
Recalls Gabriel (1975), classifying tame representations of quivers
Recognizes persistence as a representation of the quiver An
Suggests strict filtration not necessary: arrows can go both ways
Carlsson and de Silva (2008)
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Persistence for circle valued maps
Parameters on S1 instead of in ℝ. Connect level-sets.
Discretizes to working with cyclic quiver representations: no longer tame, requires Jordan blocks for an invariant.
Burghelea and Dey (2011)
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Persistent homology over a DAG
Uses a potentially branching directed acyclic graph as underlying parameter space for persistent homology.
Chambers (2014)
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• Topological Data Analysis and Persistent Homology
• The two persistences
• Algebraic foundations: the more we know…
• Sheaves
Outline
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Sheaves, cosheaves and applications
Observes that persistent (co)homology has a natural structure as cosheaves over a discretization of ℝ.
Curry (2013)
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• Edelsbrunner, Letscher and ZomorodianRestricted to ℤ2 coefficients.
• 𝕜[t]-modules Requires discretization
• Quiver representations Requires finite discretization
• Multidimensional, circles and DAGs Indicate more shapes are interesting
Why sheaves?
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Sheaves allow for very high generality in the shape of persistence: picking a base space fixes the shape
The original definition, producing a vector space for any specified interval [b,d], echoes the definition of a (pre)sheaf.
Why sheaves?
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• Curry makes a case for cosheaves of vector spaces
• Vejdemo-Johansson, Škraba and Pita-Costa (in preparation) make the case for topoi: by introducing the parameter dependency at the set theory level persistence emerges as semi-simplicial homology over this new set theory
How sheaves?
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• Patel argues for using sheaves over the classical (Borel) topology on ℝ for persistence This invites wild representation theories: removing the hope for discrete and field-agnostic invariants
• VJ-Š-PC argue for building a new “topology” on ℝ using only connected intervals
What sheaves?
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Heyting Algebras are partial orders similar enough to topologies for the sheaf axioms
For persistence, we wish to design a Heyting algebra with:
• Elements: intervals [b,d] in ℝ
• Meet (intersection): usual intersection
• Join (union): covering interval
The Persistence Topos
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Naïve approach fails: the structure is not distributive Key issue: empty intersections
Solution: orient all intervals, introduce «negative» intervals for empty intersections.
[a,b] ∪ [c,d] = [min(a,c), max(b,d)] [a,b] ∩ [c,d] = [max(a,c), min(b,d)]
The Persistence Topos
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These oriented intervals occur in Cohen-Steiner, Edelsbrunner and Harer (2009) Extending Persistence using Poincaré and Lefschetz Duality
Negative intervals can usefully represent
• Relative homology: Hk(𝕏, 𝕏∖𝕏n)
• Existence of features: the set of all homology classes with any presence
The Persistence Topos
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• We (VJ-Š-PC) also have Heyting algebras developed for zigzag, 1-critical multidimensional, 1-critical multidimensional zigzag, convex footprint multidimensional, circular persistent homologies.
Other shapes
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Thank you!