# euler calculus & data robert ghrist university of pennsylvania

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euler calculus & data robert ghrist university of pennsylvania
depts. of mathematics & electrical/systems engineering

motivation

tools

euler calculus

euler calculus χ χ = Σ (-1)k # {k-cells} k = 2 χ = Σ (-1)k rank Hk k χ
= 7 χ = 3 χ = 2 χ = 2 χ = Σ (-1)k rank Hk k χ = 3

sheaves

χ(AuB) = χ(A)+ χ(B) – χ(A B) u
lemma: [classical] χ(AuB) = χ(A)+ χ(B) – χ(A B) u

∫ h dχ geometry topology probability networks
χ(AuB) = χ(A)+ χ(B) – χ(A B) u geometry blaschke hadwiger rota chen topology ∫ h dχ kashiwara macpherson schapira viro probability networks adler taylor

integration consider the sheaf of constructible functions
CF(X) = Z-valued functions whose level sets are locally finite and “tame” tools axiomatic approach to tameness in the work on o-minimal structures collections {Sn}n=1,2,... of boolean algebras of sets in Rn closed under projections, products,... elements of {Sn}n=1,2,... are called “definable” or “tame” sets results all definable sets are triangulable & have a well-defined euler characteristic all functions in CF(X) are of the form h = Σci1Ui for Ui definable all functions in CF(X) are integrable with respect to Euler characteristic explicit definition: euler integral ∫ h dχ = ∫ (Σ ci1Ui) dχ = Σ(∫ ci1Ui)dχ = Σci χ(Ui)

integration ∫ dχ ∫ dχ X Y CF(X) CF(Y) X CF(X) CF(pt)=Z X Y CF(X) CF(Y)
[schapira, 1980’s; via kashiwara, macpherson, 1970’s] the induced pushforward on sheaves of constructible functions is the correct way to understand dχ F X Y CF(X) CF(Y) F* X pt CF(X) CF(pt)=Z 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... ∫ dχ F X Y CF(X) CF(Y) pt CF(pt)=Z corollary: [schapira, viro; 1980’s] fubini theorem sheaf-theoretic constructions also give natural convolution operators, duality, integral transforms, ... F* ∫ dχ

problem a network of “minimal” sensors returns target counts without IDs how many targets are there? = 0 = 1 = 2 = 3 = 4

problem

counting theorem: [BG] assuming target supports with uniform χ(Ui)=N
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 Ui in X h:X→Z sensor field on X returns h(x) = #{ i : x lies in Ui } 2 theorem: [BG] assuming target supports with uniform χ(Ui)=N # targets = (1/N) ∫X h dχ N ≠ 0 trivial proof: ∫ h dχ = Σ(∫ 1Ui dχ) = Σ χ(Ui) = ∫ (Σ1Ui) dχ = N # i amazingly, one needs no convexity, no leray (“good cover”) condition, etc. this is a purely topological result.

components of level sets
computation for h in CF(X), integrals with respect to dχ are computable via ∫ h dχ = Σ s χ({ h=s }) s=0 level set = Σ χ({ h>s })-χ({ h<-s }) s=0 upper excursion set = Σ h(V)χ(v) V weighted euler index “chambers” of h components of level sets

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

some applications in minimal sensing

waves consider a sensor modality which counts each wavefronts and increments an internal counter: used to count # events 3 booms… whuh? 2 booms… the resulting target impacts are still nullhomotopic (no echoing) accurate event counts obtained via ad hoc network of acoustic sensors with no clocks, no synchronization, and no localization

wheels consider sensors which count passing vehicles and increment an internal counter acoustic sensors embedded in roads… such target impacts may not be contractible… theorem: [BG] if sensors read h = the total number of time intervals in which some vehicle is nearby, then # vehicles = ∫ h dχ

wheels ∫X h(x) dχ(x) = ∫Y F*h(y) dχ(y) ∫ dχ
supports are the projected image of a contractible subset in space-time F* X Y CF(X) CF(Y) F pt Z ∫ dχ recall: ∫X h(x) dχ(x) = ∫Y F*h(y) dχ(y) F*h(y) = ∫F-1(y) h(x) dχ(x) let X = domain x time ; let Y = domain ; let F = temporal projection map then F*h(y) = total # of (compact) time intervals on which some vehicle is at/near point w = sensor reading at y

numerical integration

ad hoc networks Σ( #comp{ h≥s } - #comp{ h<s } + 1) ∫ h dχ
theorem: [BG] if the function h:R2→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 of h is Σ( #comp{ h≥s } - #comp{ h<s } + 1) s=1 this is a simple application of alexander duality… ∫ h dχ = Σ χ{ h ≥ s } s=1 = Σ b0 {h ≥ s } – b1{h ≥ s } s=1 χ = Σ (-1)k dim Hk k bk ~ = Σ b0{h ≥ s } – b0{h < s } s=1 = Σ b0{h ≥ s } – b0{h < s } + 1 s=1 this works in ad hoc setting : clustering gives fast computation

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get real…

real-valued integrands
it’s helpful to have a well-defined integration theory for R-valued integrands: Def(X) = R-valued functions whose graphs are “tame” (definable in o-minimal) take a riemann-sum approach ∫ h dχ● = lim 1/n∫ floor(nh) dχ ∫ h dχ● = lim 1/n∫ ceil(nh) dχ unfortunately, ∫ _ dχ ● & ∫ _ dχ● are no longer homomorphisms Def(X)→R however, ∫ _ dχ ● & ∫ _ dχ● have an interpretation in o-minimal category lemma if h is affine on an open k-simplex, then h ∫ h dχ● = (-1)k inf (h) ∫ h dχ● = (-1)k sup (h)

real-valued integrands
intuition: the two measures correspond to the stratified morse indices of the graph of h in Def(X) with respect to two graph axis directions… I*, I* : Def(X)→CF(X) theorem: [BG] for h in Def(X) ∫ h dχ∙ = ∫ h I*h dχ ∫ h dχ∙ = ∫ h I*h dχ corollary: [BG] if h : X → R is morse on an n-manifold, then ∫ h dχ∙ = Σ (-1)n-μ(p) h(p) crit(h) = Σ (-1)μ(p) h(p) μ = morse index corollary: [BG] if h is univariate, then ∫ h dχ∙ = totvar(h)/2 = - ∫ h dχ∙

real-valued integrands
Lebesgue ∫ h dχ● = ∫R χ{h≥s} - χ{h<-s} ds ∫ h dχ● = ∫R χ{h>s} - χ{h≤-s} ds ∫ h dχ● = limε→0+∫R s χ{s ≤ h < s+ε} ds ∫ h dχ● = limε→0+∫R s χ{s < h ≤ s+ε} ds Blaschke, Hadwiger, ... Morse ∫ h dχ● = Σ (-1)n-μ(p) h(p) crit(h) ∫ h dχ● = Σ (-1)μ(p) h(p) crit(h) Duality ∫ h dχ● = - ∫ - h dχ● (Dh)(x) = limε→0+∫ h 1B(ε,x) dχ D(Dh) = h Fubini ∫X h dχ●(x) = ∫Y ∫ {F(x)=y} h(x) dχ●(x)dχ●(y) F:X→Y with h∙F=h

incomplete data ∫R2 h dχ ≤ ∫R2 h dχ ≤ ∫R2 h dχ D
consider the following relative problem: given h on the complement of a hole D, estimate ∫ h dχ over the entire domain D theorem: [BG] for h:R2→Z a sum of indicator functions over homotopically trivial supports, none of which lies entirely within a contractible hole D, then ∫R2 h dχ ≤ ∫R2 h dχ ≤ ∫R2 h dχ h = fill in D with maximum of h on ∂D h = fill in D with minimum of h on ∂D reminder: f < g does not imply that ∫ f dχ < ∫ g dχ ...in this case the opposite occurs…

incomplete data ∫R2 h dχ ≤ ∫R2 f dχ ≤ ∫R2 h dχ
but what to choose in between upper and lower bounds? claim: a harmonic extension over a hole is a “best guess”... theorem: [BG] For h:R2→Z a sum of indicator functions over homotopically trivial supports, none of which lies entirely within a contractible hole D, then ∫R2 h dχ ≤ ∫R2 f dχ ≤ ∫R2 h dχ for f any “harmonic” extension of h over D (weighted average of h rel ∂D) the proof is surprisingly easy using morse theory: a “harmonic” extension has no local maxima or minima within D... # saddles in D - # maxima on ∂D = χ(D)=1 the integral over D is the heights of the maxima minus the heights of the saddles

expected values ∫ h dχ = 1+1-c
in practice, harmonic extensions lead to non-integer target counts ∫ h dχ = 1+1-c this is an “expected” target count weights for the laplacian can be chosen based on confidence of data points toward a general theory of expected integrals

integral transforms

sensing relations ∫X h dχ = N ∫W 1T dχ = N #T X S W
h = integral transform of 1T with kernel S

fourier transform

bessel transform

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open questions how to correct “side lobes” and energy loss in integral transforms? what is the appropriate integration theory for multi-modal and logical-valued data? how to efficiently compute integral transforms given discrete (sparse) data? …and, well, numerical analysis in general

topological network topology

closing credits… research sponsored by darpa (stomp program)
national science foundation office of naval research primary collaborator yuliy baryshnikov, bell labs work in progress with michael robinson, penn matthew wright, penn professional support university of pennsylvania a. mitchell java code david lipsky, uillinois, urbana a.j. friend, stanford naveen kasthuri, penn