euler calculus
& data
robert ghrist
university of pennsylvania
depts. of mathematics &
electrical/systems engineering
machine learning summer school : june 2009
motivation
tools
euler calculus
euler calculus
χ =
χ
χ =
Σk (-1)k # {k-cells}
=2
Σk (-1)k rank Hk
χ
=7
χ
=3
χ
=2
χ
=3
sheaves
u
geometry
χ(AuB) = χ(A)+ χ(B) – χ(A B)
blaschke
hadwiger
rota
chen
topology
∫ h dχ
kashiwara
macpherson
schapira
viro
probability
adler
taylor
networks
integration
consider the sheaf of constructible functions
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
CF(X) = Z-valued functions whose level sets are locally finite and “tame”
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
∫ h dχ = ∫ (Σ ci1Ui) dχ = Σ(∫ ci1Ui)dχ = Σci χ(Ui)
euler integral
explicit definition:
integration
[schapira, 1980’s; via kashiwara, macpherson, 1970’s]
X
the induced pushforward on sheaves of constructible
functions is the correct way to understand dχ
CF(X)
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
CF(X)
F
F*
Y
CF(Y)
pt
∫
F
F*
X
CF(X)
Y
CF(Y)
pt
∫
CF(pt)=Z
dχ
corollary: [schapira, viro; 1980’s]
fubini theorem
CF(pt)=Z
dχ
sheaf-theoretic constructions also give natural
convolution operators, duality, integral transforms, ...
problem
a network of “minimal” sensors returns target counts without IDs
how many targets are there?
=0
=1
=2
=3
=4
problem
counting
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
N≠0
h dχ
trivial proof:
∫ h dχ = ∫ (Σ1Ui) dχ = Σ(∫ 1Ui dχ) = Σ χ(U ) = N # i
i
amazingly, one needs no convexity, no leray (“good cover”) condition, etc.
this is a purely topological result.
computation
for h in CF(X), integrals with respect to dχ are computable via
∞
∫ h dχ = s=0
Σ s χ({ h=s })
level set
∞
= s=0
Σ χ({ h>s })-χ({ h<-s })
= ΣV h(V)χ(v)
“chambers” of h
components of level sets
upper excursion set
weighted euler index
example
∞
∫ h dχ = s=0
Σ χ {h(x)>s}
h>3 : χ = 2
h>2 : χ = 3
h>1 : χ = 3
h>0 : χ = -1
net integral = 2+3+3-1 = 7
example
∞
∫ h dχ = s=0
Σ χ {h(x)>s} = Σ h(V)χ(V)
V
h=4 : Σ = 2
h=3 : Σ = 1
h=2 : Σ = 0
h=1 : Σ = -4
net integral = 4(1+1)+3(1+1-1)+2(1+1+1-1-1-1)+1(1-1-1-1-2) = 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
17
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
supports are the projected image of a contractible subset in space-time
recall:
F
X
CF(X)
∫
X
F*
Y
pt
CF(Y)
Z
∫ dχ
h(x) dχ(x) = ∫Y F*h(y) dχ(y)
F*h(y) =
∫
-1
F (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
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…
∞
∞
χ{ h ≥ s } = Σ b0 {h ≥ s } – b1{h ≥ s }
∫ h dχ = Σ
s=1
∞
s=1
~
=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
bk
χ = Σ (-1)k dim Hk
k
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
unfortunately, ∫ _ dχ ● &
however, ∫ _ dχ ● &
∫ h dχ● = lim 1/n∫ ceil(nh) dχ
∫ _ dχ● are no longer homomorphisms Def(X)→R
∫ _ dχ● have an interpretation in o-minimal category
lemma
if h is affine on an open k-simplex, then
∫ h dχ●
= (-1)k inf (h)
∫ h dχ●
h
= (-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
∫ h dχ∙ = Σ (-1)n-μ(p) h(p)
crit(h)
μ = morse index
corollary: [BG] if h : X → R is morse on an n-manifold, then
∫ h dχ∙ = Σ (-1)μ(p) h(p)
crit(h)
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● = limε→0+∫R s χ{s ≤ h < s+ε} ds
∫ h dχ● = ∫R χ{h>s} - χ{h≤-s} ds
∫ h dχ● = limε→0+∫R s χ{s < h ≤ s+ε} ds
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
consider the following relative problem:
D
given h on the complement of a hole D,
∫
estimate h dχ over the entire domain
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
∫R h dχ ≤ ∫R h dχ ≤ ∫R h dχ
2
h = fill in D with maximum of h on ∂D
2
2
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
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
∫R h dχ ≤ ∫R
2
2
f dχ ≤
∫R h dχ
2
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
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
inversion
X
S
W
∫
X
h dχ = N
∫
W
1T dχ = N #T
h = integral transform of 1T with kernel S
fourier transform
radon transform
bessel transform
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
professional support
university of pennsylvania
a. mitchell
java code
david lipsky, uillinois, urbana
a.j. friend, stanford
naveen kasthuri, penn