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euler calculus for data focm : budapest : july : 2011 robert ghrist andrea mitchell university professor of mathematics & electrical/systems engineering the university of pennsylvania

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motivation

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tools

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∫ h d χ geometry probability topology networks kashiwara macpherson schapira viro blaschke hadwiger rota chen adler taylor euler calculus

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χ = Σ (-1) k # { k-cells } k χ = 7 χ = 3 χ = 2 χ = 3 = Σ (-1) k rank k ingredient #1: χ HkHk HcHc k χ (AuB) = χ (A)+ χ (B) – χ (A B) u

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results fix category of “tame” or “definable” sets (semialgebraic, subanalytic, …) all functions in CF(X) are of the form h = Σ c i 1 U i for U i definable all definable sets are triangulable & have a well-defined euler characteristic ingredient #2: CF(X) CF(X) = Z-valued functions whose level sets are locally finite and definable

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explicit definition: euler integral ∫ h d χ = ∫ ( Σ c i 1 U i ) d χ = Σ ( c i ∫ 1 U i ) d χ = Σ c i χ (U i ) integration: explicit χ (AuB) = χ (A)+ χ (B) – χ (A B) u the integral is independent of how the integrand is decomposed… …thanks to mayer-vietoris

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integration: implicit [schapira; via kashiwara, macpherson, 1970’s] the right-derived direct image on CF is the correct way to understand d χ F*F* in the case where Y is a point, CF(Y)=Z, and the pushforward is a homomorphism from CF(X) to Z which respects all the gluings implicit in sheaves... XY CF(X)CF(Y) F X pt CF(X)CF( pt )=Z ∫ d χ corollary: [schapira, viro; 1980’s] fubini theorem F*F* XY CF(X)CF(Y) F pt CF( pt )=Z ∫ d χ

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a network of “anonymous” sensors returns target counts without IDs = 0= 1= 2= 3= 4 problem

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theorem: [BG] assuming target supports with uniform χ (U i )=N # targets = ( 1/N ) ∫ X h d χ ∫ h d χ = ∫ ( Σ 1 U i ) d χ = Σ ( ∫ 1 U i d χ ) = Σ χ (U i ) = N # i let W = “target space” = space where finite # of targets live let X = “sensor space” = space which parameterizes sensors target i is detected on a target support U i in X sensor field on X returns h(x) = #{ i : x lies in U i } h:X → Z 2 N ≠ 0 enumeration

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integrals with respect to d χ are computable via ∫ h d χ = Σ s χ ({ h=s }) s=0 ∞ = Σ χ ({ h>s })- χ ({ h<-s }) s=0 ∞ level set upper excursion set computation

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h>3 : χ = 2 h>2 : χ = 3 h>1 : χ = 3 h>0 : χ = -1 net integral = 2+3+3-1 = 7 = Σ χ { h(x)>s } s=0 ∞ ∫ h d χ example

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numerical integration

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theorem: [BG] if the upper semicontinuous function h:R 2 → N is sampled over a network in a way that correctly samples the connectivity of upper and lower excursion sets, then the exact value of the euler integral is Σ( #comp{ h≥s } - #comp{ h~~
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prop: [BG] if a sum of characteristic functions of convex annuli in general position in the plane is sampled over a sufficiently dense lattice, then #annuli = ½ ∫ theother ∫ oneway h d χ d χ this takes advantage of the errors implicit in numerical integration & fubini w / yuliy baryshnikov exploiting numerical error

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integral transforms

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W X S ∫ X h d χ = N ∫ W 1 T d χ = N #T h = integral transform of 1 T the topological radon transform R S : CF(W) CF(X) R S (R S h) = (μ – λ)h + λ1 W ∫ h d χ theorem: [schapira] if fibers of S are regular, then R S is self-invertible radon transform

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the microlocal fourier(-sato) transform MF : CF(R n ) CF(R n ) ∫ S n-1 ∫ MF(h)(x,ξ) d χ (x) dξ = vol(S n-1 ) ∫ h d χ theorem: [cf. brocker-kuppe] averaging the MF over ξ yields corollary: for X definable in R n the ξ - average of MF(1 X ) is microlocal fourier transform χ (X) = ∫ S n-1 ∫ MF(1 X )(x,ξ) d χ (x) dξ vol(S n-1 ) 1 x ξ ∫X dκ∫X dκ [gauss-bonnet] MF(h)(x,ξ) = lim ∫ 1 ξ ● (x-y)≥0 h(y) d χ (y) B(ε,x) ε → 0 +

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toward numerical analysis…

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it’s helpful to have a well-defined integration theory for R-valued integrands: Def(X) = R-valued functions whose graphs are “tame” (definable) a riemann-sum definition ∫ h d χ = lim 1/n ∫ nh d χ ∫ h d χ = lim 1/n ∫ nh d χ real-valued integrands ≠ w / yuliy baryshnikov

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if h is affine on an open simplex σ, then ∫ h d χ = χ (σ) inf h ∫ h d χ = χ (σ) sup h h lemma real-valued integrands ∫ x d χ + ∫ 1-x d χ (0,1) =0 + 0 ∫ 1 d χ (0,1) = –1 ∫ d χ nonlinear fubini fails however… w / yuliy baryshnikov there is a strong connection to morse theory…

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intuition: the two measures correspond to the stratified morse indices of the graph of h:X→R with respect to two graph axis directions… ∫ h d χ = ∫ h Ind ↑ (h) d χ theorem: [BG] for h in Def(X) ∩ C(X) ∫ h d χ = ∫ h Ind ↓ (h) d χ morse interpretation Ind ↑, Ind ↓ : Def(X)→CF(X) Ind ↑ (h)(x) = lim χ ( B ε (x) ∩ {h

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intuition: the two measures correspond to the stratified morse indices of h or -h ∫ h d χ = ∫ h LMDI(-h) d χ theorem: [BG] for h in Def(X) ∩ C(X) ∫ h d χ = ∫ h LMDI(h) d χ morse interpretation LMDI : Def(X)→CF(X) LMDI(h)(x) = lim χ ( B ε (x) ∩ {h>h(x) – ε’ } ) ε’<< ε → 0 + w / yuliy baryshnikov

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∫ h d χ = Σ h(p) (-1) n- μ (p) crit(h) = Σ h(p) (-1) μ (p) crit(h) ∫ h d χ corollary: [BG] if h : X → R is morse on an n-manifold, then morse interpretation corollary: [BG] if h is univariate, then ∫ h d χ = totvar(h)/2 = – ∫ h d χ w / yuliy baryshnikov ∫ h d χ = 1+1-c

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integral transforms

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euler-fourier transform

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euler-bessel transform

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the euler-bessel transform B: CF(R n ) Def(R n ) Bh(x) = ∫ ∫ d(x,y)=s h(y) d χ ds 0 ∞ B1 A (x)= ∫ ∂A+ d x d χ - ∫ ∂A- d x d χ theorem: [GR] for A codimension-0 submanifold (w/corners) B1 A (x)= ∫ ∂A d x d χ cor: in even dimensions, this yields an index theory for computing B euler-bessel transform w / michael robinson

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the euler-bessel transform B: CF(R n ) Def(R n ) Bh(x) = ∫ ∫ d(x,y)=s h(y) d χ ds 0 ∞ euler-bessel transform w / michael robinson = ∫ ∫ h(y) ds d χ 0 ∞ this yields index-theoretic formulae for fast computation of the transform

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shape discrimination an L 2 bessel transform detects circular targets from anonymous counts w / michael robinson

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shape discrimination an L ∞ bessel transform + SVA detects square targets up to rotation w / michael robinson

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concluding postscript…

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w / yasu hiraoka network coding sheaves w / michael robinson 0 → H 0 (X,A;S) → H 0 (X;S) → H 0 (A;S) → H 1 (X,A;S) → H 1 (X;S) → sheaf cohomology → global information flows for a network flow sheaf S on X with cut A, H 1 (X,A;S) is the obstruction to max-cut/min-flow constructible sheaves data enable output network switching sheaves

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applied topology

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closing credits… research sponsored by professional support a.j. friend, stanford university of pennsylvania andrea mitchell darpa (stomp program) air force office of scientific research office of naval research primary collaboratorsyuliy baryshnikov, univ. illinois java codedavid lipsky, penn naveen kasthuri, penn michael robinson, penn david lipsky, penn

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