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euler calculus & data. robert ghrist university of pennsylvania depts. of mathematics & electrical/systems engineering. motivation. tools. euler calculus. χ = Σ (-1) k # { k-cells }. χ = Σ (-1) k rank H k. k. k. euler calculus. χ. χ. χ. χ. χ. = 7. - PowerPoint PPT Presentation
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robert ghristuniversity of pennsylvaniadepts. of mathematics &
electrical/systems engineering
euler calculus & data
motivation
tools
euler calculus
χ = Σ (-1)k # {k-cells} k
χ = 2
χ = 7
χ = 3
χ = 2
χ = 3
euler calculus
χ = Σ (-1)k rank Hk k
sheaves
lemma: [classical]
χ(AuB) = χ(A)+ χ(B) – χ(A B)
u
χ(AuB) = χ(A)+ χ(B) – χ(A B)
u
∫ h dχ
geometry
probability
topology
networks
kashiwaramacpherson
schapiraviro
blaschkehadwigerrotachen
adlertaylor
resu
lts
axiomatic approach to tameness in the work on o-minimal structures
consider the sheaf of constructible functions
CF(X) = Z-valued functions whose level sets are locally finite and “tame”
collections {Sn}n=1,2,... of boolean algebras of sets in Rn closed under projections, products,...
all functions in CF(X) are of the form h = Σci1Ui for Ui definable
elements of {Sn}n=1,2,... are called “definable” or “tame” sets
all definable sets are triangulable & have a well-defined euler characteristic
all functions in CF(X) are integrable with respect to Euler characteristic
tool
s
explicit definition:
euler integral
∫ h dχ = ∫ (Σ ci1Ui) dχ = Σ(∫ ci1Ui
)dχ = Σci χ(Ui)
integration
[schapira, 1980’s; via kashiwara, macpherson, 1970’s]
the induced pushforward on sheaves of constructible functions is the correct way to understand dχ
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...
X Y
CF(X) CF(Y)
F
X pt
CF(X) CF(pt)=Z∫ dχ
corollary: [schapira, viro; 1980’s] fubini theorem
F*
X Y
CF(X) CF(Y)
Fpt
CF(pt)=Z∫ dχ
sheaf-theoretic constructions also give naturalconvolution operators, duality, integral transforms, ...
integration
a network of “minimal” sensors returns target counts without IDshow many targets are there?
= 0 = 1 = 2 = 3 = 4
problem
problem
theorem: [BG] assuming target supports with uniform χ(Ui)=N
# targets = (1/N) ∫X h dχ
trivial proof:
∫ h dχ = ∫ (Σ1Ui) dχ = Σ(∫ 1Ui
dχ) = Σ χ(Ui) = 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 Ui in X
sensor field on X returns h(x) = #{ i : x lies in Ui }
amazingly, one needs no convexity, no leray (“good cover”) condition, etc.this is a purely topological result.
h:X→Z
2
N ≠ 0
counting
for h in CF(X), integrals with respect to dχ are computable via
∫ h dχ = Σ s χ({ h=s }) s=0
∞
= Σ χ({ h>s })-χ({ h<-s }) s=0
∞
= Σ h(V)χ(v) V
level set
upper excursion set
weighted euler index
“chambers” of h components of level sets
computation
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 dχ
example
some applications in
minimal sensing
17
the resulting targetimpacts are stillnullhomotopic (no echoing)
3 booms…
whuh?
2 booms…
consider a sensor modality which counts each wavefronts andincrements an internal counter: used to count # events
accurate event counts obtained via ad hoc network of acoustic sensorswith no clocks, no synchronization, and no localization
waves
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
supports are the projected image of a contractible subset in space-time
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
F*
X Y
CF(X) CF(Y)
Fpt
Z∫ dχ
wheels
numerical integration
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…
= Σ χ{ h ≥ s } s=1
∞
∫ h dχ = Σ b0 {h ≥ s } – b1{h ≥ s } s=1
∞
this works in ad hoc setting : clustering gives fast computation
= Σ b0{h ≥ s } – b0{h < s } + 1s=1
∞
~= Σ b0{h ≥ s } – b0{h < s }s=1
∞ χ = Σ (-1)k dim Hk
k
bk
ad hoc networks
eucharis
eucharis
eucharis
eucharis
eucharis
eucharis
eucharis
get real…
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)
unfortunately, ∫ _ dχ ● & ∫ _ dχ● are no longer homomorphisms Def(X)→R
take a riemann-sum approach
∫ h dχ● = lim 1/n∫ floor(nh) dχ ∫ h dχ● = lim 1/n∫ ceil(nh) dχ
however, ∫ _ dχ ● & ∫ _ dχ● have an interpretation in o-minimal category
if h is affine on an open k-simplex, then
∫ h dχ● = (-1)k inf (h) ∫ h dχ● = (-1)k sup (h)
h
lemma
real-valued integrands
I*, I* : Def(X)→CF(X)
intuition: the two measures correspond to the stratified morse indices ofthe graph of h in Def(X) with respect to two graph axis directions…
∫ h dχ∙ = Σ (-1)n-μ(p) h(p)
crit(h)
= Σ (-1)μ(p) h(p) crit(h)
μ =
mor
se in
dex
∫ h dχ∙
corollary: [BG] if h : X → R is morse on an n-manifold, then∫ h dχ∙ = ∫ h I*h dχ
theorem: [BG] for h in Def(X)
real-valued integrands
∫ h dχ∙ = ∫ h I*h dχ
corollary: [BG] if h is univariate, then ∫ h dχ∙ = totvar(h)/2 = - ∫ h dχ∙
∫ 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
Lebesgue
Morse
∫ h dχ● = Σ (-1)n-μ(p) h(p)
crit(h)
∫ h dχ● = Σ (-1)μ(p) h(p) crit(h)
∫ h dχ● = - ∫ - h dχ● (Dh)(x) = limε→0+∫ h 1B(ε,x) dχ
Duality
D(Dh) = h
∫X h dχ●(x) = ∫Y ∫ {F(x)=y} h(x) dχ
●(x)dχ
●(y)
Fubini
F:X→Y with h∙F=h
real-valued integrands
consider the following relative problem:
given h on the complement of a hole D,
estimate ∫ h dχ over the entire domain
reminder: f < g does not imply that ∫ f dχ < ∫ g dχ ...in this case the opposite occurs…
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 ≤ ∫R
2 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
D
incomplete data
but what to choose in between upper and lower bounds?
claim: a harmonic extension over a hole is a “best guess”...
the proof is surprisingly easy using morse theory:
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
for f any “harmonic” extension of h over D (weighted average of h rel ∂D)
the integral over D is the heights of the maxima minus the heights of the saddles
a “harmonic” extension has no local maxima or minima within D... # saddles in D - # maxima on ∂D = χ(D)=1
∫R2 h dχ ≤ ∫R
2 f dχ ≤ ∫R2 h dχ
incomplete data
in practice, harmonic extensions lead to non-integer target counts
this is an “expected” target count
∫ h dχ = 1+1-c
weights for the laplacian can be chosen based on confidence of data
points toward a general theory of expected integrals
expected values
integral transforms
W
X
S
sensing relations
∫X h dχ = N ∫W 1T dχ = N #T
h = integral transform of 1T with kernel S
fourier transform
radon transform
bessel transform
eucharis
eucharis
eucharis
eucharis
eucharis
eucharis
eucharis
how to correct “side lobes” and energy loss in integral transforms?
open questions
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
professional support
a.j. friend, stanford
university of pennsylvaniaa. mitchell
darpa (stomp program)national science foundation
office of naval research
primary collaborator yuliy baryshnikov, bell labs
java code david lipsky, uillinois, urbana
naveen kasthuri, penn
work in progress with michael robinson, pennmatthew wright, penn