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Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill) n p F
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Page 1: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

Correlation testing foraffine invariant properties on

Shachar Lovett Institute for Advanced Study

Joint with Hamed Hatami (McGill)

npF

Page 2: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 3: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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]

Page 4: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 5: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 6: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 7: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 8: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

Which properties are testable?

• Graph properties: well understood

• Algebraic properties: partially understood

• Locally testable codes: major open problems

• PCP / hardness of approximation: whole field

Page 9: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

Correlation testing

Page 10: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

Correlation testing

Page 11: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 12: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Uf f ‖ ‖ ‖ ‖

Uf f ‖ ‖ ‖ ‖

Page 13: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 14: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 15: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 16: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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 ‖ ‖ ‖ ‖

Page 17: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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)

Page 18: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 19: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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[

Page 20: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 21: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 22: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 23: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 24: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 25: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 26: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

Proof

Page 27: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 28: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

‖ ‖ ‖ ‖

Page 29: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 30: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 31: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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 ‖ ‖

Page 32: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 33: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

‖ ‖ ‖ ‖

Page 34: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 35: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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 ‖ ‖

Page 36: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 37: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 38: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 39: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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

Page 40: Correlation testing for affine invariant properties on Shachar Lovett Institute for Advanced Study Joint with Hamed Hatami (McGill)

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!


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