Invariance in Property Testing Madhu Sudan MIT Joint

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Invariance in Property Testing Madhu Sudan MIT Joint work with Tali Kaufman (IAS). Dec.

Invariance in Property Testing Madhu Sudan MIT Joint work with Tali Kaufman (IAS). Dec. 31, 2007 Invariance in Property Testing 1

Property Testing n n Goal: “Efficiently” determine if some “data” “essentially” satisfies some given

Property Testing n n Goal: “Efficiently” determine if some “data” “essentially” satisfies some given “property”. Formalism: ! f : D R given as oracle n Data: D ¯nit e, but huge. R ¯nit e, possibly small n Property: Given by F µ f f : D ! Rg o(D ) queries int o f. Even O(1)! n Efficiently: n Must accept if f 2 F Essentially: Ok t o accept if f ¼ g 2 F. Dec. 31, 2007 Invariance in Property Testing 2

Property Testing n n 6 g(x)] ±(f ; g) = Pr x 2 D

Property Testing n n 6 g(x)] ±(f ; g) = Pr x 2 D [f (x) = Distance: ±(f ; F ) = ming 2 F f ±(f ; g)g f ¼² g if ±(f ; g) · ². Definition F is (q; : ®) locally t est able if 9 a q query t est er t hat accept s f 2 F wit h probability one reject s f 2 6 F wit h probability ¸ ® ¢±(f ; F ). n q locally t est able implies 9® > 0 Notes : locally t est able implies 9 q = O(1) Weaker t est ing: can reject f 2 F wit h small prob. Dec. 31, 2007 Invariance in Property Testing 3

Property Testing (Pictorially( Universe f f : D ! Rg F Must accept Ok

Property Testing (Pictorially( Universe f f : D ! Rg F Must accept Ok t o accept Must reject w. h. p. Dec. 31, 2007 Invariance in Property Testing 4

Example: Pre-election Polling n Domain = Population f 0; 1 g Range = F

Example: Pre-election Polling n Domain = Population f 0; 1 g Range = F = funct ions wit h majority 1 n Property: n Essentially: n Must reject w. h. p. if Pr x 2 D [f (x) = 1] · 1=2 ¡ ² ~ 2 ) queries. Can t est weakly wit h O(1=² Efficiency? Cherno® bounds. Dec. 31, 2007 Invariance in Property Testing 5

Modern Day Example: Testing Linearity n Vect or space Fn 2 Domain = Field

Modern Day Example: Testing Linearity n Vect or space Fn 2 Domain = Field F 2 Range = n F = linear funct ions Property: i. e. , f f (x) = ha; xi ja 2 Fn 2 g n Theorem [Blum, Luby, Rubinfeld ’ 89]: n Pick x; y 2 Fn 2 uniformly. Test: Accept i® f (x) + f (y) = f (x + y) Linearity is 3 locally t est able. Dec. 31, 2007 Invariance in Property Testing 6

Property Testing: Abbreviated History n n n Prehistoric: Statistical sampling n E. g. ,

Property Testing: Abbreviated History n n n Prehistoric: Statistical sampling n E. g. , “Majority = 1? ” Linearity Testing [BLR’ 90], Multilinearity Testing [Babai, Fortnow, Lund ’ 91]. Graph/Combinatorial Property Testing [Goldreich, Goldwasser, Ron ’ 94]. n E. g. , Is a graph “close” to being 3 -colorable. Algebraic Testing [GLRSW, RS, FS, AKKLR, KR, JPSZ] n Is multivariate function a polynomial (of bounded degree). Graph Testing [Alon-Shapira, AFNS, Borgs et al. ] n Characterizes graph properties that are testable. Dec. 31, 2007 Invariance in Property Testing 7

Quest for this talk n n What makes a property testable? In particular for

Quest for this talk n n What makes a property testable? In particular for algebraic properties: n Current understanding: n Low-degree multivariate functions are testable. n Different proofs for different cases. n n Dec. 31, 2007 Linear functions Low-degree polynomials F 2 Higher degree polynomials over other fields Invariance in Property Testing 8

Necessary Conditions for Testability n n n One-sided error and testability: ¡ Suppose f

Necessary Conditions for Testability n n n One-sided error and testability: ¡ Suppose f is reject ed by a k query 1 sided t est er. Suppose queried point s are x 1 ; : : : ; x k 2 D. Let (x i ) = ®i. ¡ T hen for every funct ion g 2 F , 6 h® 1 ; : : : ; ®k i. hg(x 1 ); : : : ; b(x k )i = C = hx 1 ; : : : ; x k i ; S ( R k Constraint: g sat is¯es C if hg(x 1 ); : : : ; g(x k )i 2 S F sat is¯es C if every g 2 F sat is¯es C. Conclusion: Testability implies Constraints. Dec. 31, 2007 Invariance in Property Testing 9

Necessary Conditions for Testability n n n One-sided error and testability: ¡ Suppose f

Necessary Conditions for Testability n n n One-sided error and testability: ¡ Suppose f is reject ed by a k query 1 sided t est er. Suppose queried point s are x 1 ; : : : ; x k 2 D. Let (x i ) = ®i. ¡ T hen for every funct ion g 2 F , 6 h® 1 ; : : : ; ®k i. hg(x 1 ); : : : ; b(x k )i = C = hx 1 ; : : : ; x k i ; S ( R k Constraint: g sat is¯es C if hg(x 1 ); : : : ; g(x k )i 2 S F sat is¯es C if every g 2 F sat is¯es C. Conclusion: Testability implies Constraints. Dec. 31, 2007 Invariance in Property Testing 10

Constraints, Characterizations, Testing Strong testing: 6 F reject ed Every f 2 n by

Constraints, Characterizations, Testing Strong testing: 6 F reject ed Every f 2 n by some k local const raint. Set of k local const raint s charact erize F. 9 C 1 ; : : : ; Cm s. t. f 2 F , f sat is¯es Cj for every j. n n Conclusion: Testability implies Local Characterizations. Example: n ! f µ f F 2 g is linear i® for all x; y 2 Fn 2 , f sat is¯es Cx ; y where Cx ; y = hx; y; x + yi ; S = f 000; 011; 101; 110 g. Dec. 31, 2007 Invariance in Property Testing 11

Characterizations Sufficient? n NO! [Ben-Sasson, Harsha, Raskhodnikova] n Random 3 -locally characterized errorcorrecting codes

Characterizations Sufficient? n NO! [Ben-Sasson, Harsha, Raskhodnikova] n Random 3 -locally characterized errorcorrecting codes (“Expander Codes”) are not o(D)-locally testable. n Property: D = [n]; R = f 0; 1 g; F = set of funct ions t hat sat isfy some random 3 ary F 2 linear const raint s. n n Criticism: Random constraints too “asymmetric”. Perhaps should consider more “symmetric” properties. Dec. 31, 2007 Invariance in Property Testing 12

Invariance & Property testing n n Invariances (Automorphism groups: ( For permut at ion

Invariance & Property testing n n Invariances (Automorphism groups: ( For permut at ion ¼: D ! D , F is ¼ invariant if f 2 F implies f ± ¼ 2 F. Aut (F ) = f ¼ j F is ¼ invariant g Forms group under composit ion. Hope: If Automorphism group is “large” (“nice”), then property is testable. Dec. 31, 2007 Invariance in Property Testing 13

Examples n Majority: ¡ Aut group = SD (full group). ¡ Easy Fact :

Examples n Majority: ¡ Aut group = SD (full group). ¡ Easy Fact : If Aut (F ) = SD t hen F is poly(R; 1=²) locally t est able. n Graph Properties: ¡ Aut. group given by renaming of vert ices ¡ [AFNS, Borgs et al. ] implies regular graph propert ies t est able. n n Matrix Properties: Have lots of symmetries – do they suffice? Algebraic Properties: What symmetries do they have? Will focus on this today. Dec. 31, 2007 Invariance in Property Testing 14

Algebraic Properties & Invariances n Properties : = n = D F , R

Algebraic Properties & Invariances n Properties : = n = D F , R F (Linearity, Low degree, Reed Muller) Or D = K ¶ F, R = F (Dual BCH) (K ; F ¯nit e ¯elds) n Automorphism groups? Linear t ransformat ions of domain. ¼(x) = Ax where A 2 Fn £ n (Linear-Invariant) n n Additional restriction: Linearity f ; g 2 F and ®; ¯ 2 F implies ®f + ¯g 2 F Question: Are Linear, Linear-Invariant, Locally Characterized Properties Testable? Dec. 31, 2007 Invariance in Property Testing 15

Linear-Invariance & Testability n Question: Are Linear, Linear-Invariant, Locally Characterized Properties Testable? n Why?

Linear-Invariance & Testability n Question: Are Linear, Linear-Invariant, Locally Characterized Properties Testable? n Why? n Unifies previous results on Prop. Testing. n (Will show it also is non-trivial extension) n Nice family of 2 -transitive group of symmetries. n Conjecture [Alon, Kaufman, Krivelevich, Litsyn, Ron] : Linear code with k-local constraint and 2 transitive group of symmetries must be testable. Dec. 31, 2007 Invariance in Property Testing 16

Our Results n n n F µ f K n ! Fg linear, linear

Our Results n n n F µ f K n ! Fg linear, linear invariant , Theorem 1: k locally charact erized implies F is f (K ; k) locally t est able. F µ f K n ! Fg linear, a± ne invariant , Theorem 2: has k local constraint implies F is f (K ; k) locally t est able. Other stuff: Study of Linear-invariant Properties. Dec. 31, 2007 Invariance in Property Testing 17

Linear Invariant Properties Dec. 31, 2007 Invariance in Property Testing 18

Linear Invariant Properties Dec. 31, 2007 Invariance in Property Testing 18

Examples of Linear-Invariant Families ¡ Polynomials in F[x 1 ; : : : ;

Examples of Linear-Invariant Families ¡ Polynomials in F[x 1 ; : : : ; x n ] of degree at most d ¡ Traces of Poly in K [x 1 ; : : : ; x n ] of degree at most d ¡ (Traces of ) Homogenous polynomials of degree d ¡ F 1 + F 2 , where F 1 , F 2 are linear invariant. Polynomials support ed by degree 2; 3; 5; 7 monomials. Dec. 31, 2007 Invariance in Property Testing 19

What Dictates Locality of Characterizations? ¡ Precise locality not yet underst ood: Depends on

What Dictates Locality of Characterizations? ¡ Precise locality not yet underst ood: Depends on p ary represent at ion of degrees. Example: F support ed by monomials x pi + pj behaves like degree two polynomial ¡ For a± ne invariant family dict at ed (coarsely) by highest degree monomial in family ¡ For some linear invariant families, can be much less t han t he highest degree monomial. Example: K = F 7 ; F = F 1 + F 2 F 1 = poly of degree at most 16 F 2 = poly support ed on monomials of degree 3 mod 6. Degree(F ) = (n); Locality(F ) · 49. Dec. 31, 2007 Invariance in Property Testing 20

Analysis Ingredients n Monomial Extraction: n Monomial Spread: E. g. , xy 2 +

Analysis Ingredients n Monomial Extraction: n Monomial Spread: E. g. , xy 2 + xyz + x 4 2 F implies xyz 2 F x 5 2 F implies x 4 y; x 3 y 2 also in F (if char(F) large) Su± ces for a± ne invariant families. For linear invariant families, need t o de¯ne t he right paramet er and bound locality weakly in t erms of it. Dec. 31, 2007 Invariance in Property Testing 21

Local Testing Dec. 31, 2007 Invariance in Property Testing 22

Local Testing Dec. 31, 2007 Invariance in Property Testing 22

Key Notion: Formal Characterization ¡ F is formally charact erized if 9 a single

Key Notion: Formal Characterization ¡ F is formally charact erized if 9 a single const raint C = (hx 1 ; : : : ; x k i ; S) such t hat f C ± ¼g¼ 2 A ut ( F ) charact erize F. T heorem: If F is formally charact erized by a k local const raint (wit h some rest rict ions) t hen it is k locally t est able. Dec. 31, 2007 Invariance in Property Testing 23

BLR (and our) analysis Dec. 31, 2007 Invariance in Property Testing 24

BLR (and our) analysis Dec. 31, 2007 Invariance in Property Testing 24

BLR Analysis: Outline ² Have f s. t. Pr x ; y [f (x)

BLR Analysis: Outline ² Have f s. t. Pr x ; y [f (x) + f (y) = 6 f (x + y)] = ± < 1=20. Want t o show f close t o some g 2 F. ² De¯ne g(x) = most likely f f (x + y) ¡ f (y)g. y ² If f close t o F t hen g will be in F and close t o f. ² But if f not close? g may not even be uniquely de¯ned! ² St eps: ¡ St ep 0: Prove f close t o g ¡ St ep 1: Prove most likely is overwhelming majority. ¡ St ep 2: Prove t hat g is in F. Dec. 31, 2007 Invariance in Property Testing 25

BLR Analysis: Step 0 ² De¯ne g(x) = most likely y f f (x

BLR Analysis: Step 0 ² De¯ne g(x) = most likely y f f (x + y) ¡ f (y)g. = g(x)] · 2± Claim: Pr x [f (x) 6 = f (x + y)f (y)] ¸ ¡ Let B = f xj Pr y [f (x) 6 ¡ Pr x ; y [linearity t est reject s jx 2 B ] ¸ 1 g 2 1 2 ) Pr x [x 2 B ] · 2± ¡ If x 2 6 B t hen f (x) = g(x) Dec. 31, 2007 Invariance in Property Testing 26

Vot ex (y) BLR Analysis: Step 1 ² De¯ne g(x) = most likely y

Vot ex (y) BLR Analysis: Step 1 ² De¯ne g(x) = most likely y f f (x + y) ¡ f (y)g. ² Suppose for some x, 9 two equally likely values. Presumably, only one leads t o linear x, so which one? ² If we wish t o show g linear, t hen need t o rule out t his case. Lemma: 8 x, Pr y ; z [Vot ex (y) = 6 Vot ex (z))] · 4± Dec. 31, 2007 Invariance in Property Testing 27

Vot ex (y) BLR Analysis: Step 1 ² De¯ne g(x) = most likely y

Vot ex (y) BLR Analysis: Step 1 ² De¯ne g(x) = most likely y f f (x + y) ¡ f (y)g. ² Suppose for some x, 9 two equally likely values. Presumably, only one leads t o linear x, so which one? ² If we wish t o show g linear, t hen need t o rule out t his case. Lemma: 8 x, Pr y ; z [Vot ex (y) = 6 Vot ex (z))] · 4± Dec. 31, 2007 Invariance in Property Testing 28

Vot ex (y) BLR Analysis: Step 1 ² De¯ne g(x) = most likely y

Vot ex (y) BLR Analysis: Step 1 ² De¯ne g(x) = most likely y f f (x + y) ¡ f (y)g. = Vot ex (z))] · 4± Lemma: 8 x, Pr y ; z [Vot ex (y) 6 ? f (z) f (y) ¡ f (x + y) f (y + z) ¡ f (y + 2 z) ¡ f (x + z) ¡ f (2 y + z) f (x + 2 y + 2 z) Prob. Row/ column sum non zero · ±. Dec. 31, 2007 Invariance in Property Testing 29

Vot ex (y) BLR Analysis: Step 1 ² De¯ne g(x) = most likely y

Vot ex (y) BLR Analysis: Step 1 ² De¯ne g(x) = most likely y f f (x + y) ¡ f (y)g. = Vot ex (z))] · 4± Lemma: 8 x, Pr y ; z [Vot ex (y) 6 ? f (z) f (y) ¡ f (x + y) f (y + z) ¡ f (y + 2 z) ¡ f (x + z) ¡ f (2 y + z) f (x + 2 y + 2 z) Prob. Row/ column sum non zero · ±. Dec. 31, 2007 Invariance in Property Testing 30

BLR Analysis: Step 2 (Similar( Lemma: If ± < g(x) f (z) 1 ,

BLR Analysis: Step 2 (Similar( Lemma: If ± < g(x) f (z) 1 , 20 t hen 8 x; y, g(x) + g(y) = g(x + y) g(y) ¡ g(x + y) Prob. Row/ column sum non zero · 4±. f (y + z) ¡ f (y + 2 z) ¡ f (x + z) ¡ f (2 y + z) f (x + 2 y + 2 z) Dec. 31, 2007 Invariance in Property Testing 31

Our Analysis: Outline ² f s. t. Pr L [hf (L (x 1 );

Our Analysis: Outline ² f s. t. Pr L [hf (L (x 1 ); : : : ; f (L (x k ))i 2 V ] = ± ¿ 1. ² De¯ne g(x) = ® t hat maximizes Pr f L j L ( x ) = x g [h®; f (L (x 2 )); : : : ; f (L (x k ))i 2 V ] 1 ² St eps: ¡ St ep 0: Prove f close t o g ¡ Step 1: Prove “most likely” is overwhelming majority. ¡ St ep 2: Prove t hat g is in F. Dec. 31, 2007 Invariance in Property Testing 32

Our Analysis: Outline ² f s. t. Pr L [hf (L (x 1 );

Our Analysis: Outline ² f s. t. Pr L [hf (L (x 1 ); : : : ; f (L (x k ))i 2 V ] = ± ¿ 1. ² De¯ne g(x) = ® t hat maximizes Pr f L j L ( x ) = x g [h®; f (L (x 2 )); : : : ; f (L (x k ))i 2 V ] 1 Same as before ² St eps: ¡ St ep 0: Prove f close t o g ¡ Step 1: Prove “most likely” is overwhelming majority. ¡ St ep 2: Prove t hat g is in F. Dec. 31, 2007 Invariance in Property Testing 33

Vot ex (L ) Matrix ? ² De¯ne Magic g(x) = ® t hat

Vot ex (L ) Matrix ? ² De¯ne Magic g(x) = ® t hat maximizes Pr f L j L ( x = 1) xg [h®; f (L (x 2 )); : : : ; f (L (x k ))i 2 V ] 6 Vot ex (K ))] · 2(k ¡ 1)± Lemma: 8 x, Pr L ; K [Vot ex (L ) = x K (x 2 ). . . L (x 2 ) ? ¢¢¢ L (x k ) K (x k ) Dec. 31, 2007 Invariance in Property Testing 34

Matrix Magic? x K (x 2 ). . . L (x 2 ) ?

Matrix Magic? x K (x 2 ). . . L (x 2 ) ? ¢¢¢ L (x k ) K (x k ) ² Want marked rows t o be random const raint s. ² Suppose x 1 ; : : : ; x ` linearly independent ; and rest dependent on t hem. 35

Matrix Magic? n ` x ` L (x 2 ) Fill with random entries

Matrix Magic? n ` x ` L (x 2 ) Fill with random entries n n ¢¢¢ Fill so as to form constraints Linear algebra implies final columns are also constraints. L (x k ) K (x 2 ). . . K (x k ) ² Suppose x 1 ; : : : ; x ` linearly independent ; and rest dependent on t hem. Dec. 31, 2007 36

Matrix Magic? n ` x ` L (x 2 ) Fill with random entries

Matrix Magic? n ` x ` L (x 2 ) Fill with random entries n n ¢¢¢ Fill so as to form constraints Linear algebra implies final columns are also constraints. L (x k ) K (x 2 ). . . K (x k ) ² Suppose x 1 ; : : : ; x ` linearly independent ; and rest dependent on t hem. Dec. 31, 2007 37

Conclusions n n n Linear/Affine-invariant properties testable if they have local constraints. Gives clean

Conclusions n n n Linear/Affine-invariant properties testable if they have local constraints. Gives clean generalization of linearity and lowdegree tests. Future work: What kind of invariances lead to testability (from characterizations? ( Dec. 31, 2007 Invariance in Property Testing 38