Probabilistic existence of regular combinatorial objects - PowerPoint Presentation

Slide1

Probabilistic existence of regular combinatorial objects

Shachar

Lovett (UCSD)

Joint with Greg

Kuperberg

(UC Davis), Ron

Peled

(Tel-Aviv university)Slide2

Overview

Regular combinatorial objectsProbabilistic modelMain Theorem: random walks on lattices

Proof: Fourier analysis and codes

Summary and open problemsSlide3

Overview

Regular combinatorial objectsProbabilistic modelMain Theorem: random walks on lattices

Proof: Fourier analysis and codes

Summary and open problemsSlide4

Regular objects

“highly symmetric” objectsRegular graphsRegular hyper-graphs (aka designs)

K-wise permutations

Orthogonal arrays

q-analogs

…

Constructions

known in some

special cases

This work: First existence proofs for (nearly)

all underlying parameters

by a

probabilistic argumentSlide5

Regular graphs

(n,d) regular graph – n vertices, all of degree dEasy to constructSlide6

Regular hyper-graphs

Also known as designst-(n,k

,

) design

: k-uniform hyper-graph on n vertices; any t vertices belong to exactly  edges

1

-(n,2,d) design: regular graphSlide7

Regular hyper-graphs

t-(n,k,) design: k-uniform hyper-graph on n vertices; any t vertices belong to exactly  edges

Constructions

:

Small values

based on group symmetries

[Teirlinck’87]

first asymptotic construction

of t-(n,t+1,) designs for infinitely many n, 

Few other asymptotic constructionsSlide8

Regular hyper-graphs

t-(n,k,) design: k-uniform hyper-graph on n vertices; any t vertices belong to exactly  edges

Existence unknown

for most parameters:

Steiner systems

: t-(n,k,1) designs, open for t>5

Hadamard

matrices

: 2-(4m-1,2m-1,m-1) designs

In general, constructions (and existence)

unknown

for almost all parametersSlide9

K-wise permutations

Family of permutations acting uniformly on k elementsA set FS

n

is

k-wise

if

for any k distinct elements i

1

,…,

i

k

and j

1

,…,

j

k

125346251643361254Slide10

K-wise permutations

Family of permutations acting uniformly on k elementsConstructions:

Subgroups

of

S

n

: k=1,2,3 (e.g. for k=2 and n prime, subgroup of affine maps {x->

ax+b

})

Fail for k>3

(only

S

n

or A

n

are 4-wise for n>24)

[Finucane-Peled-Yaari’12] Combinatorial constructions for small k of exponential size 125346251643361254Slide11

Other examples

Orthogonal arrays: subsets of [c]n

where any k coordinates get all values equally often

(aka k-wise independent functions [n]->[c])

q-analogs

: Family of k-dimensional subspaces of

F

q

n

which cover uniformly all the t-dimensional subspaces

(

eg

designs for the

Grassmanian)Spherical designs: family of points on Sn which allow to integrate low degree polynomials by summing over the pointsSlide12

Our approach

Probabilistic constructionGeneral technique to prove existence of regular objectsProve

existence

of

designs

,

k-wise permutations

,

orthogonal arrays

, for (nearly)

all underlying parameters

; of

optimal size

up to polynomial overheadSlide13

Overview

Regular combinatorial objectsProbabilistic modelMain Theorem: random walks on lattices

Proof: Fourier analysis and codes

Summary and open problemsSlide14

Probabilistic model

Running example: t-(n,k

,

) designs

k-uniform hyper-graph on n vertices; any t vertices belong to exactly  edges

Random construction:

Sample

N=N(

n,k,t

,) edges uniformly

Analyze probability

that any t vertices covered by exactly  edges

Very unlikely eventSlide15

Probabilistic model

Random constructions unlikely to workBut is probability zero or

tiny but positive

?

How can we analyze “rare events” ?

Standard tools fail

, e.g.

Limited dependence (e.g.

Lovasz

Local lemma) doesn’t hold

Spencer’s method not relevantSlide16

Probabilistic model

Another viewpoint: sum of matrix rows

0100101001

1001011000

0000101110

…

0010110100

Edges:

k-subsets of [n]

t-subsets of [n]

Sample few rows

Analyze probability that sum is (

,…,)

Pr[sum=expected sum]

Incidence matrixSlide17

Probabilistic model

Yet another viewpoint: short random walk on a latticeLattice spanned by rows

Steps: rows

Probability that a

short random

walk

will

end in a specific point

Can we guarantee fast convergence?

0100101001

1001011000

0000101110

…

0010110100Slide18

Overview

Regular combinatorial objectsProbabilistic modelMain Theorem: random walks on lattices

Proof: Fourier analysis and codes

Summary and open problemsSlide19

General setup

Matrix Sample N rowsPr[

sum of rows=

expected sum of rows

]

When can we guarantee it is positive?

0100101001

1001011000

0000101110

…

0010110100Slide20

General setup

[Alon-Vu’97] example of regular

hyper-graphs

on n vertices,

~

n

n

/2

edges, with

no regular

sub-

hypergraphs

Pr[

sum of rows= expected sum of rows]=0Conclusion: need to assume some structure010010100110010110000000101110…0010110100Slide21

Main Theorem

Main theorem: set of conditions that guarantee that

N is

polynomial

in underlying

parameters

In

our applications

we get

optimal N

(up to polynomial factors)

Can

approximate probability

up to 1+o(1)

010010100110010110000000101110…0010110100Pr[sum of N rows= expected sum of rows]>0Slide22

Main Theorem

Some notationA – set of columns

B – set of rows (|B| >> |A|)

V –

linear space

spanned by columns

row(b) – row in index

b

B

We want SB of size |S|=N such that

0100101001

1001011000

0000101110

…

0010110100

A

BSlide23

Main Theorem

Pre-condition: divisibilityWe want |S|=N for which

Let c

1

be minimal integer such that

N must be a

multiple

of c

1

0100101001

1001011000

0000101110

…

0010110100

A

BSlide24

Main Theorem

Pre-condition: divisibilityExample:

t-(

n,k

,

) designs

[Wilson’73, Graver-Jurkat’73]

analyze divisibility of incidence matrices

N multiple of

0100101001

1001011000

0000101110

…

0010110100

A

BSlide25

Main Theorem

Main condition: column spanV = space spanned by columns

Need:

(

a) V has transitive symmetry group

(b) V spanned by short integer vectors in l



(c) V



spanned by short integer vectors in l

1

(in coding terms, V is an LDPC)

(d) V contains the constant vectors

0100101001

10010110000000101110…0010110100

ABSlide26

Main Theorem

V = space spanned by columnsExample: t-(

n,k

,

) designs

(a) V has transitive symmetry group

S

n

acts as permutations on k-subsets (rows) and t-subsets (columns)

Acts transitively on

rows (e.g. B)

0100101001

1001011000

0000101110

…

0010110100

A

BSlide27

Main Theorem

V = space spanned by columnsExample: t-(

n,k

,

) designs

(b)

V spanned by short integer vectors in l



Immediate since matrix has small elements, so columns are such a basis for V

0100101001

1001011000

0000101110

…

0010110100

A

BSlide28

Main Theorem

V = space spanned by columnsExample: t-(

n,k

,

) designs

(c)

V



spanned by short integer vectors in l

1

Usually the hardest condition to verify; for designs, requires some combinatorial calculations

0100101001

1001011000

0000101110

…

0010110100

A

BSlide29

Main Theorem

V = space spanned by columnsExample: t-(

n,k

,

) designs

(d)

V contains the constant vector

Sum of columns is constant

0100101001

1001011000

0000101110

…

0010110100

A

BSlide30

Main Theorem

B x A matrix, V=span(columns)Assumec

1

divisibility constant

V spanned by integer vectors with l



bound c

2

V



spanned by integer vectors with l

1

bound c

3

V has transitive symmetry group

V contains the constant vectors

Then for N=poly(|A|,c1,c2,c3), Pr[sum of N rows= expected sum]>0 In fact, we approximate the probability up to 1+o(1)010010100110010110000000101110…

0010110100ABSlide31

Main Theorem

B x A matrix, V=span(columns)Assumec

1

divisibility constant

V spanned by integer vectors with l



bound c

2

V



spanned by integer vectors with l

1

bound c

3

V has transitive symmetry group

V contains the constant vectors

Then for N=poly(|A|,c1,c2,c3), Pr[sum of N rows= expected sum]>0 In fact, we approximate the probability up to 1+o(1)010010100110010110000000101110…

0010110100ABConditions on V as a code (over the rationals)Slide32

Applications

Optimal size up to polynomial overheadCan

count

the number of objects (up to 1+o(1))

Existence of

t-(

n,k

,

) designs

with N=(n/t)

O(t)

Verification of conditions relatively simple

Existence of

k-wise permutations

with N=

n

O(k) permutationsVerification of conditions was harder; required nontrivial representation theory of SnOrthogonal arrays Slide33

Overview

Regular combinatorial objectsProbabilistic modelMain Theorem: random walks on lattices

Proof: Fourier analysis and codes

Summary and open problemsSlide34

Proof of main theorem

N - Target #rowsChoose each row with prob p=N/|B|

X = sum of selected rows (random

var

)

Pr[X=E[X]] = ?

0100101001

1001011000

0000101110

…

0010110100

A

BSlide35

Proof of main theorem

X = sum of selected rowsPr[X=E[X]] = ?Main tool: Fourier analysis

Fourier coefficients of X:

0100101001

1001011000

0000101110

…

0010110100

A

BSlide36

Proof of main theorem

X = sum of selected rowsFourier coefficients of X:

Maximal Fourier

coefs

form a lattice:

0100101001

1001011000

0000101110

…

0010110100

A

BSlide37

Proof of main theorem

Fourier spaceMaximal Fourier coefsSlide38

Proof of main theorem

Fourier spaceMaximal Fourier coefs

Step I: approximate F.C. near

maximas

Gaussian approximation

Relatively straight-forwardSlide39

Proof of main theorem

Fourier spaceMaximal Fourier coefs

Step II: bound F.C. far from

maximas

Most Fourier space

Harder step, requires all the conditions on V

Develop new connections between

coding properties

of V and the Fourier

coefsSlide40

Proof of main theorem

End result: Gaussian approximation, restricted to the lattice

in which X lives

X = sum of N rows

Y = Gaussian with same covariance as X

Pr[X=E[X]]

 density of Y at E[X]

times some lattice related factorsSlide41

Overview

Regular combinatorial objectsProbabilistic modelMain Theorem: random walks on lattices

Proof: Fourier analysis and codes

Summary and open problemsSlide42

Summary

New probabilistic techniqueTheorem: can prove existence of regular structures by verification of a

few conditions

, which are

verified explicitly

This is in contrast to the

existence result

which is

non-explicitSlide43

Summary

Proof technique: Fourier analysisMake new connections between coding theory and

Fourier analysis

in order to bound Fourier coefficientsSlide44

Open problems

ApplicationsWork in progress (with Kuperberg and

Peled

):

spherical designs

Work in progress (with Vardy):

q-analogsSlide45

Open problems

AlgorithmsCurrent proof is purely existential

Don’t know how to find objects efficiently, even using randomness

Other probabilistic techniques for rare events were made algorithmic, so

there is hope

Lovasz

local lemma: Moser, Moser-

Tardos

Spencer’s theorem:

Bansal

, L-

MekaSlide46