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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!