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

Invariance in Property Testing Goal: “Efficiently” determine if some “data” “essentially” satisfies some given “property”. Formalism: Data: Property: Efficiently: Essentially: f : D ! R g i v e n a s o r c l D ¯ n i t e , b u h g . R p o s l y m a G i v e n b y F µ f : D ! R g o ( D ) q u e r i s n t f . E v O 1 ! M u s t a c e p i f 2 F O k o ¼ g . Dec. 31, 2007 Invariance in Property Testing

Invariance in Property Testing Distance: Definition: Notes: ± ( f ; g ) = P r x 2 D [ 6 ] F m i n ¼ ² · . F i s ( q ; ® ) - l o c a y t e b f 9 u r h p 2 w n j 6 ¸ ¢ ± . q - l o c a y t e s b i m p 9 ® > = O ( 1 ) W k r n g : j f 2 F w h . Dec. 31, 2007 Invariance in Property Testing

Property Testing (Pictorially) U n i v e r s f : D ! R g F M u s t a c e p O k t o a c e p M u s t r e j c w . h p Dec. 31, 2007 Invariance in Property Testing

Example: Pre-election Polling Domain = Population Range = Property: Essentially: Efficiency? f ; 1 g F = f u n c t i o s w h m a j r y 1 M u s t r e j c w . h p i f P x 2 D [ ( ) = 1 ] · ¡ ² C a n t e s w k l y i h ~ O ( 1 = ² 2 ) q u r . o ® b d Dec. 31, 2007 Invariance in Property Testing

Modern Day Example: Testing Linearity Domain = Range = Property: Theorem [Blum,Luby,Rubinfeld ’89]: Test: V e c t o r s p a F n 2 F i e l d 2 F = l i n e a r f u c t o s . , ( x ) h ; j 2 g L i n e a r t y s 3 - l o c b . P i c k x ; y 2 F n u f o r m l . A e p t ® ( ) + = Dec. 31, 2007 Invariance in Property Testing

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

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

Necessary Conditions for Testability One-sided error and testability: Constraint: Conclusion: Testability implies Constraints. ¡ S u p o s e f i r j c t d b y a k - q 1 . n x ; : 2 D L ( ) = ® ¡ T h e n f o r v y u c t i g 2 F , ( x 1 ) ; : b k 6 = ® . C = h x 1 ; : k i S ( R g s a t i ¯ e C f h ( x 1 ) ; : k 2 S F v r y . Dec. 31, 2007 Invariance in Property Testing

Necessary Conditions for Testability One-sided error and testability: Constraint: Conclusion: Testability implies Constraints. ¡ S u p o s e f i r j c t d b y a k - q 1 . n x ; : 2 D L ( ) = ® ¡ T h e n f o r v y u c t i g 2 F , ( x 1 ) ; : b k 6 = ® . C = h x 1 ; : k i S ( R g s a t i ¯ e C f h ( x 1 ) ; : k 2 S F v r y . Dec. 31, 2007 Invariance in Property Testing

Constraints, Characterizations, Testing Strong testing: Conclusion: Testability implies Local Characterizations. Example: E v e r y f 6 2 F j c t d b s o m k - l a n i . S h z 9 C 1 ; : , ¯ f µ F n 2 ! g i s l e a r ® o x ; y , t ¯ C w h = + S 1 . Dec. 31, 2007 Invariance in Property Testing

Characterizations Sufficient? NO! [Ben-Sasson, Harsha, Raskhodnikova] Random 3-locally characterized error-correcting codes (“Expander Codes”) are not o(D)-locally testable. Property: Criticism: Random constraints too “asymmetric”. Perhaps should consider more “symmetric” properties. D = [ n ] ; R f 1 g F s e t o u c i h a y m r d 3 - 2 l . Dec. 31, 2007 Invariance in Property Testing

Invariance & Property testing Invariances (Automorphism groups): Hope: If Automorphism group is “large” (“nice”), then property is testable. F o r p e m u t a i n ¼ : D ! , s - v f 2 l ± . A u t ( F ) = f ¼ j i s - n v a r g o m p d e c . Dec. 31, 2007 Invariance in Property Testing

Invariance in Property Testing Examples Majority: Graph Properties: Matrix Properties: Have lots of symmetries – do they suffice? Algebraic Properties: What symmetries do they have? Will focus on this today. ¡ A u t g r o p = S D ( f l ) . ¡ E a s y F c t : I f A u ( ) = S D h e n i p o l R ; 1 ² - b . ¡ A u t . g r o p i v e n b y a m f c s [ F N S , B l ] h Dec. 31, 2007 Invariance in Property Testing

Algebraic Properties & Invariances Automorphism groups? Additional restriction: Linearity Question: Are Linear, Linear-Invariant, Locally Characterized Properties Testable? D = F n , R ( L i e a r t y o w - d g M u l ) O r D = K ¶ F , R ( u a l - B C H ) ( K ; F ¯ n i t e l d s ) L i n e a r t s f o m d . ¼ ( x ) = A w h 2 F £ (Linear-Invariant) f ; g 2 F a n d ® ¯ i m p l e s + Dec. 31, 2007 Invariance in Property Testing

Linear-Invariance & Testability Question: Are Linear, Linear-Invariant, Locally Characterized Properties Testable? Why? Unifies previous results on Prop. Testing. (Will show it also is non-trivial extension) Nice family of 2-transitive group of symmetries. 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

Invariance in Property Testing Our Results Theorem 1: Theorem 2: Other stuff: Study of Linear-invariant Properties. F µ f K n ! g l i e a r , - v t k o c y h z d m p s ( ; ) b . F µ f K n ! g l i e a r , ± - v t h s k o c m p ( ; ) y b . Dec. 31, 2007 Invariance in Property Testing

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

Examples of Linear-Invariant Families ¡ P o l y n m i a s F [ x 1 ; : ] f d e g r t ¡ T r a c e s o f P l y i n K [ x 1 ; : ] d g t m ¡ ( T r a c e s o f ) H m g n u p l y i d ¡ F 1 + 2 , w h e r a l i n - v t . P o y m s u p d b g ; 3 5 7 Dec. 31, 2007 Invariance in Property Testing

What Dictates Locality of Characterizations? ¡ P r e c i s l o a t y n u d : D p - f g . E x m F b + j h v k w ¡ F o r a ± n e - i v t f m l y d c ( s ) b h g ¡ F o r s m e l i n a - v t f , c b u h g d . E x a m p l e : K = F 7 ; 1 + 2 o y f d g r t s 6 u n i 3 . D ( ) ­ L c · 4 9 Dec. 31, 2007 Invariance in Property Testing

Invariance in Property Testing Analysis Ingredients Monomial Extraction: Monomial Spread: E . g , x y 2 + z 4 F i m p l e s x 5 2 F i m p l e s 4 y ; 3 a o n ( f c h r ) g S u ± c e s f o r a n - i v t m l . F , d ¯ h g p b y w k Dec. 31, 2007 Invariance in Property Testing

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

Key Notion: Formal Characterization ¡ F i s f o r m a l y c h t e z d 9 n g C = ( x 1 ; : k S ) u ± ¼ 2 A . T h e o r m : I f F i s a l y c t z d b k - n ( w ) . Dec. 31, 2007 Invariance in Property Testing

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

Invariance in Property Testing BLR Analysis: Outline ² H a v e f s . t P r x ; y [ ( ) + 6 = ] ± < 1 2 W n o h w c l m g F ² D e ¯ n g ( x ) = m o s t l i k y f + ¡ . ² I f c l o s e t F h n g w i b a d . ² B u t i f n o c l s e ? g m a y v b q d ¯ ! ² S t e p s : ¡ S t e p : P r o v f c l s g ¡ S t e p 1 : P r o v m s l i k y w h n g a j . ¡ S t e p 2 : P r o v h a g i s n F . Dec. 31, 2007 Invariance in Property Testing

Invariance in Property Testing BLR Analysis: Step 0 ² D e ¯ n g ( x ) = m o s t l i k y f + ¡ . C l a i m : P r x [ f ( ) 6 = g ] · 2 ± ¡ L e t B = f x j P r y [ ( ) 6 + ] ¸ 1 2 g ¡ P r x ; y [ l i n e a t s j c 2 B ] ¸ 1 ) P r x [ 2 B ] · ± ¡ I f x 6 2 B t h e n ( ) = g Dec. 31, 2007 Invariance in Property Testing

Invariance in Property Testing x ( y ) BLR Analysis: Step 1 ² D e ¯ n g ( x ) = m o s t l i k y f + ¡ . ² S u p o s e f r m x , 9 t w q a l y i k v . P b n d h c ? ² I f w e i s h t o g l n a r , d u c . L e m a : 8 x , P r y ; z [ V o t ( ) 6 = ] · 4 ± Dec. 31, 2007 Invariance in Property Testing

Invariance in Property Testing x ( y ) BLR Analysis: Step 1 ² D e ¯ n g ( x ) = m o s t l i k y f + ¡ . ² S u p o s e f r m x , 9 t w q a l y i k v . P b n d h c ? ² I f w e i s h t o g l n a r , d u c . L e m a : 8 x , P r y ; z [ V o t ( ) 6 = ] · 4 ± Dec. 31, 2007 Invariance in Property Testing

Invariance in Property Testing BLR Analysis: Step 1 V o t e x ( y ) ² D e ¯ n g ( x ) = m o s t l i k y f + ¡ . L e m a : 8 x , P r y ; z [ V o t ( ) 6 = ] · 4 ± f ( y ) ¡ f ( x + y ) ? f ( z ) f ( y + z ) ¡ f ( y + 2 z ) ¡ f ( x + z ) ¡ f ( 2 y + z ) f ( x + 2 y z ) P r o b . R w / c l u m n s - z e · ± Dec. 31, 2007 Invariance in Property Testing

Invariance in Property Testing BLR Analysis: Step 1 V o t e x ( y ) ² D e ¯ n g ( x ) = m o s t l i k y f + ¡ . L e m a : 8 x , P r y ; z [ V o t ( ) 6 = ] · 4 ± f ( y ) ¡ f ( x + y ) ? f ( z ) f ( y + z ) ¡ f ( y + 2 z ) ¡ f ( x + z ) ¡ f ( 2 y + z ) f ( x + 2 y z ) P r o b . R w / c l u m n s - z e · ± Dec. 31, 2007 Invariance in Property Testing

BLR Analysis: Step 2 (Similar) f ± < 1 2 , t h n 8 x ; y g ( ) + = P r o b . R w / c l u m n s - z e · 4 ± g ( x ) g ( y ) ¡ g ( x + y ) f ( z ) f ( y + z ) ¡ f ( y + 2 z ) ¡ f ( x + z ) ¡ f ( 2 y + z ) f ( x + 2 y z ) Dec. 31, 2007 Invariance in Property Testing

Our Analysis: Outline ² f s . t P r [ h ( x ) ; : i 2 V ] = ± ¿ ² D e 1 ) ; : k i 2 V ] = ± ¿ ² D e ¯ n g ( x ) = ® t h a m i z s P r f L j 1 [ ; 2 : k V ] ² S t e p s : ¡ S t e p : P r o v f c l s g Step 1: Prove “most likely” is overwhelming majority. ¡ ¡ S t e p 2 : P r o v h a g i s n F . Dec. 31, 2007 Invariance in Property Testing

Our Analysis: Outline ² f s . t P r [ h ( x ) ; : i 2 V ] = ± ¿ ² D e 1 ) ; : k i 2 V ] = ± ¿ ² D e ¯ n g ( x ) = ® t h a m i z s P r f L j 1 [ ; 2 : k V ] S a m e s b f o r ² S t e p s : ¡ S t e p : P r o v f c l s g Step 1: Prove “most likely” is overwhelming majority. ¡ ¡ S t e p 2 : P r o v h a g i s n F . Dec. 31, 2007 Invariance in Property Testing

Invariance in Property Testing x ( L ) Matrix Magic? ² D e ¯ n g ( x ) = ® t h a m i z s P r f L j 1 [ ; 2 : k V ] L e m a : 8 x , P r ; K [ V o t ( ) 6 = ] · 2 k ¡ 1 ± L ( x 2 ) L ( x k ) ¢ x K ( x 2 ) ? . K ( x k ) Dec. 31, 2007 Invariance in Property Testing

? Matrix Magic? L ( x ) ¢ L ( x ) x K ( x ) . K ( x ) ² W a n t m r k 2 ) ¢ L ( x k ) x K ( x 2 ) ? . K ( x k ) ² W a n t m r k e d o w s b c i . ² S u p o s e x 1 ; : ` l i n a r y d t h m .

Matrix Magic? ` L ( x ) ¢ L ( x ) x K ( x ) ` . K ( x ) ² S u p o s e Fill with random entries Matrix Magic? Fill so as to form constraints Linear algebra implies final columns are also constraints. ` L ( x 2 ) ¢ L ( x k ) x K ( x 2 ) ` . K ( x k ) ² S u p o s e x 1 ; : ` l i n a r y d t h m . Dec. 31, 2007

Matrix Magic? ` L ( x ) ¢ L ( x ) x K ( x ) ` . K ( x ) ² S u p o s e Fill with random entries Matrix Magic? Fill so as to form constraints Linear algebra implies final columns are also constraints. ` L ( x 2 ) ¢ L ( x k ) x K ( x 2 ) ` . K ( x k ) ² S u p o s e x 1 ; : ` l i n a r y d t h m . Dec. 31, 2007

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