Correlation testing foraffine invariant properties on

Shachar Lovett Institute for Advanced Study

Joint with Hamed Hatami (McGill)

npF

Property testing

• Math: infer global structure from local samples

• CS: Super-fast (randomized) algorithms for approximate decision problems

• Decide if large object approximately has property, while testing only a tiny fraction of it

Graph properties: 3-colorability• Input: graph G• Is G 3-colorable?

• Local test:– Sample (1/)O(1) vertices– Accept if induced subgraph is 3-colorable

• Analysis:– Test always accepts 3-colorable graphs– Test rejects (w.h.p) graphs -far from 3-colorable

[Goldreich-Goldwasser-Ron’96]

Algebraic properties: linearity• Input: function • Is f linear?

• Local test:– Sample– Check if– Repeat 1/O(1) times

• Analysis:– Test always accepts linear functions– Test rejects (w.h.p) functions -far from linear

[Blum-Luby-Rubinfeld’90]

: np pf F F 0 1 3 0 2 5 1 2

, npx yF

( ) ( ) ( )f x y f x f y

Codes: locally testable codes

• Code: distinct elements have large distance

• Input: word

• C is locally testable if there exists a (randomized) test which queries a few coordinates and– Always accepts codewords – Rejects (w.h.p) if w is far from all codewords

• The “mathematical core” of the PCP theorem• Open: can C have constant rate, distance and

testability?

npwF

npC F

Proofs: Probabilistic Checkable Proofs

• PCP Theorem: robust proof system

• Encoding of theorems +randomized local test (queries few bits of proof)– Test always accepts legal proofs of theorems– Test rejects (w.h.p) proofs of false theorems

• Major tool to prove hardness of approximation

Property testing: general framework

• Universe: set of objects (e.g. graphs)• Property: subset of objects (e.g. 3-col graphs)• Test: randomized small sample (e.g. small

subgraph)

• Property is testable if local consistency implies approximate global structure

Which properties are testable?

• Graph properties: well understood

• Algebraic properties: partially understood

• Locally testable codes: major open problems

• PCP / hardness of approximation: whole field

Correlation testing

Correlation testing

Linearity correlation testing

• Function • Correlation of f,g:

• Correlation with linear functions (characters):

: npf F

[ ], ( ) ( )npxf x g xf g

FE

linear:

2 /

ˆ m |a

(

,

)

x |np p

p

i pp

f f

e

‖ ‖F F

Linearity correlation testing

• Linear correlation: global propertyWitnessed by local average

• Identifies functions correlated with linear funcs:– f correlated to linear:

– f is not correlated:

2

, ,

4 4

( ) ( )

ˆ| ( )

(

|

[ ) ( )]npx y z

U

f x y f xf x z fz

f

y x

f

‖ ‖

FE

2ˆ

Uf f ‖ ‖ ‖ ‖

2ˆ

Uf f ‖ ‖ ‖ ‖

Linearity correlation testing• Discrete setting:

Test queries 4 locations, accepts f if

• Acceptance probability:– -correlated with linear: prob. ≥ 1/p + 2 – negligible correlation: prob. ≤ 1/p + o(1)

• Property testing: #queries depends on • Here: #queries=4, acceptance prob. depends on

( ) ( ) ( ) ( ) 0f x y z f x y f x z f x

: np pf F F

Testing correlation with polynomials

• Inverse Gowers Theorem (for finite fields):

Global structure: correlation with low-degree polynomials (Higher-order Fourier coefs)

Witnessed by local average

Testing correlation with polynomials

• Correlation with degree d polynomials:

• Gowers norm: average over 2d+1 points1

11 1

2 | |

, , ,[ 1]

[ ( )]d

d nd p

d IiU x y y

i II d

f f x y

Conjugation

‖ ‖F

E C

C

polynomial degree ( ) :max ||< , n

d p p

Qu Poly pQ d

ff

‖ ‖F F

Testing correlation with polynomials

• Direct theorem [Gowers]

• Inverse Theorem [Bergelson-Tao-Ziegler]

(if p<d then Polyd = non classical polynomials)

1( ) ddu Poly U

f f ‖ ‖ ‖ ‖

1 ( ) ( )ddu PolyU

f f ‖ ‖ ‖ ‖

Main theorem

• Gowers norms: local averages which witness global correlation to low-degree polynomials

• Question: are there other such properties?– Correlation witnessed by local averages

• Theorem [today]: no (affine invariant properties, in large fields)

Correlation with property

• Property(can also consider )

• Function

• Correlation of f with property P:

{ : }np pP g F F

: np pf F F

( ) max | , |f gu P g Pf ‖ ‖

{ : }npP g F

Local test

• Local test (with q queries):– Distribution over– Local test

• T tests correlation with property P if such that

1 }, ,{ nq pxx F

{: 0,1}qpT F

( ) ( )u Pf T f ‖ ‖

(0, ),

( ) ( ) ( )u Pf T f ‖ ‖

1[ ( ( ), ,( ))]) ( qT f xf fT xE[

Affine invariant properties

• Property• P is affine invariant if

• Examples:– Linear functions; degree-d polynomials– Functions with sparse / low-dim. Fourier representation

• Local tests for affine invariant properties are w.l.o.g local averages over linear forms

{ : }np pP f F F

( )( ) ( )f P g x fx Ax b P

Local average over linear forms

• Variables• Linear form• System of linear forms– E.g.

• Average over linear forms:

1 (( ) ), , n kk pX XX F

1 1( ) ( )k k i pL X X X F

1{ , }, qL L L

1 1( ( )) ... ( ( )), ( )

( ) [ ]q qn kp

f L X f L XpX

T f

FEL

( )qp F

{ , , , }X Y Z X Y X Z X L

Local tests: affine invariant properties

• Local tests for affine invariant properties are w.l.o.g averages over homogenous linear forms

–

• systems of linear forms such that the sets

are disjoint

,i iL

1 1, , ( )( ),{( , ( )) : }m m u PfT Tf f ‖ ‖L L

1 1, , ( )( ), ( )) :{( , ( )}m m u PT T ff f ‖ ‖L L

1 12

} if homogenous{ , , ( )k

q i i ii

LL L X X X

L

Local tests: affine invariant properties

• Claim: any local test local averages

• Proof: P affine invariant, so

• Choosing A,b uniformly: – transform each query – to a homogeneous system

( ) ( )( )u P u Pf f Ax b‖ ‖ ‖ ‖

1, , )( qx x1 ,( , )qAx b Ax b

,A b

Main theorem (1)

• Property– Consistent– Affine invariant– Sparse

• Thm: If P is locally testable with q queries (p>q) then such that for any sequence of functions which are unbiased

:( }){ nn p p ngP P F F

1n nP P

( )| |no p

n pP

d q )( : n

n p p nf F F

( )lim 0 lim 0dn u P n Un nf f

‖ ‖ ‖ ‖

l 0im nfp

n

E

Main theorem (2)

• Consistent property

• Thm: If P is testable by systems of q linear forms (p>q) then , for any bounded functions

• Q: Is this true for any norm defined by linear forms?

:( }, 1 ){ nn p ngP gP ‖ ‖F

d q : )( nn p nf F

( )lim 0 lim 0dn n u P n n Un nf f f f

‖ - ‖ ‖ - ‖E E

Proof

Main theorem

•

• P testable by systems of q linear forms (q<p)

• Thm: u(P) norm equivalent to some Ud norm:if then

:( }, 1 ){ nn p ngP gP ‖ ‖F

( )lim 0 lim 0dn u P n Un nf f

‖ ‖ ‖ ‖

lim 0nn

f

E

Proof idea

• Dfn: S = {degrees d: large n degree-d poly Qn

1. Qn correlated with property P2. Qn has “high enough” rank}

• D=Max(S) – D is bounded (bound depends on the linear systems)

• Lemma 1:

• Lemma 2:

1 ( )lim 0 lim 0Dn n u PUn nf f

‖ ‖ ‖ ‖

1( )lim 0 lim 0Dn u P n Un nf f

‖ ‖ ‖ ‖

Polynomial rank

• Q – degree d polynomial

• Rank(Q) – minimal number of lower-degree polynomials R1,…,Rc needed to compute Q–

• Thm [Green-Tao, Kaufman-L.]If P has high enough rank, it has negligible correlation with lower degree polynomials

1( ( ), , ( ))( ) cR xx RQ x

Polynomial factors

• Polynomial factor: – Sigma-algebra defined by Q1,…,QC

– : average over B,

• Complexity(B): C = number of basis polys• Degree(B): max degree of Q1,…,QC

• Rank(B): min. rank of linear comb. of Q1,…,QC

– Large rank: Q1(x),…,QC(x) are nearly independent

1 ,{ }, : nC p pB Q Q F F

: npf F [ | ]f BE

Decomposition theorems

• Fix d<p

• can be decomposed as

– B has degree d, high rank, bounded complexity

–

: npf F 1 2f ff

1 [ | ]Bf fE

12 1dUf ‖ ‖

Complexity of linear systems

• Linear form: • Linear system:

• Average:

• Complexity: min. d, if then

• C-S complexity [Green-Tao]• True complexity [Gowers-Wolf, Hatami-L.]

1{ , }, qL L L

( )1

( ) ( ( ))n kp

q

iXi

f XT f L

FE_L

1 1( ) k kX XL X

11 2 2 1, dUf ff f ‖ ‖

1( ) ( )f T fT L L

Proof idea

• Dfn: S = {degrees d: large n degree-d poly Qn

1. Qn correlated with property P2. Qn has “high enough” rank}

• D=Max(S) – D is bounded (≤ complexity of linear systems)

• Lemma 1:

• Lemma 2:

1 ( )lim 0 lim 0Dn n u PUn nf f

‖ ‖ ‖ ‖

1( )lim 0 lim 0Dn u P n Un nf f

‖ ‖ ‖ ‖

Lemma 1: Small UD+1 small u(P)• D: max deg of high rank polys correlate with P• Assume

• Step 1: reduce to “structured function”– Linear system of complexity S (S>D)– Decompose:

• Reduce to studying f1 - func. of deg ≤S polys:– –

1 ( )1 but D u PUf f ‖ ‖ ‖ ‖

L

11 2[ | ], 1SUf B ff ‖ ‖E

11 1DUf ‖ ‖

1 1 ( )( ) ( ) u PT f T f f ‖ ‖L L

1 2f ff

Lemma 1: Small UD+1 small u(P)

• D: max deg of high rank polys correlate with P• Structured function: –

–

• Will show: • Use the structure:– –

11 1DUf ‖ ‖

1 [ | ], deg( )f B Sf B E

1 ( ) 'u Pf ‖ ‖

(1

)( ) de, gi xQi p if x Q S

11deg( ) because 0 1Di i UQ D f ‖ ‖

( )deg( ) by def o0 f DiQp u PiQ D ‖ ‖

1 ( ) 0u Pf ‖ ‖

Lemma 2: small u(P) small UD+1

• Key ingredient: invariance principle– High rank polynomials “look the same” to averages

•

Then local averages cannot distinguish f,f’:

1( ( ), , ( ))( ) cQ xx Qf x 1 1{ },{ ' } high rank deg( ) de, g(, , , , )c c i iQ Q Q QQ Q

1( ( ), ,'( ) ' ( )' )cQ x Qf x x

( ) ( ')T Tf fL L

Part 2: small u(P) small UD+1

• D: max deg of high rank polys correlate with P• Assume– Reduce to structured function,

• f1 correlated with high-rank Q of degree ≤D– Assume for now: deg(Q)=D

• Dfn of D: Exists high rank poly Q’, deg(Q’)=D, Q’ correlated with some function gP

• Contradiction: Define f’1 = f1 with Q replaced by Q’– Invariance principle: – f’1 is correlated with g P

1( ) 1 but Du P Uf f ‖ ‖ ‖ ‖

1 [ | ]Bf fE

1 1( ) ( ')T f T fL L

Part 2: small u(P) small UD+1

• Problem: what if f1’ correlated with high rank poly of degree < D? – Solution: can find Q’ correlated with property P for of

all degrees ≤ D– Reason: systems of averages are robust

• Thm: for any family of linear systems, the set

has a non-empty interior for some finite n(unless not for trivial reasons)– analog of [Erdos-Lovasz-Spencer] for additive settings

1( ), , ( )) :( , 1}{ :

k

n kpf f fT fT ‖ ‖FL L

Summary

• Property testing: witness strong structure by local samples

• Correlation test: witness weak structure

• Main result: any affine invariant property which is correlation testable, is essentially equivalent to low-degree polynomials

Open problems

• Which norms can be defined by local averages– Are always equivalent to some Ud norm?

• Testing in low characteristics

• Is it possible to test if a function is correlated with cubic polynomials?– U4 norm doesn’t work– Unknown even if #queries depends on correlation

2 2: nf F F

THANK YOU!