Algebraic Property Testing Madhu Sudan MIT CSAIL Joint work with Tali Kaufman (IAS).

Classical Data Processing Big computers Small Data

Modern Data Processing Small computers Enormous Data

Needs new algorithmic paradigm Imbalance not a question of technology : I.e., not because computing speeds are growing less fast than memory capacity. Imbalance is a function of expectations: E.g., Users expect to be able to “analyze” the WWW, using a laptop. But WWW includes millions other such laptops. Need: Sublinear time algorithms That “estimate” rather than “compute” some given function.

Property Testing D a t : f ! R f g i v e n b y a s m p l o x P r o p e f(x) P r o p e t y : F µ f D ! R g q - u e r y T s t : S a m p l f b o x i . A c e p t s i f 2 F Hope: I m p o s i b l e w t h q ¿ j D R e j c t s i f 6 2 F R e j c t s i f ± - a r o m F f ± - c l o s e t F i 9 g 2 . P r x D [ ( ) 6 = ] ·

Property Testing D a t : f ! R f g i v e n b y a s m p l o x P r o p e f(x) P r o p e t y : F µ f D ! R g q - u e r y T s t : S a m p l f b o x i . A c e p t s i f 2 F Hope: I m p o s i b l e w t h q ¿ j D R e j c t s i f 6 2 F R e j c t s i f ± - a r o m F f ± - c l o s e t F i 9 g 2 . P r x D [ ( ) 6 = ] ·

Example: Linearity Testing f : F n 2 ! g [Blum, Luby, Rubinfeld ’90] Property = Linearity Test: Non-trivial analysis: D = F n 2 ; R F = f j 8 x ; y ( ) + g P i c k r a n d o m x ; y A c e p t i f ( x ) + y = F f ± - a r o m F ) e j c t w . p 2 = 9

Example: Linearity Testing f : F n 2 ! g [Blum, Luby, Rubinfeld ’90] Property = Linearity Test: Non-trivial analysis: D = F n 2 ; R F = f j 8 x ; y ( ) + g P i c k r a n d o m x ; y A c e p t i f ( x ) + y = F f ± - a r o m F ) e j c t w . p 2 = 9

Property Testing: Abbreviated History Prehistoric: Statistical sampling E.g., “Is mean/median at least X”. 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.

This Talk I f F µ ! i s c l o e d u r a t , ± m h y z b . Abstracting Algebraic Property Testing Generic Theorem: Motivations: Generalizes, unifies previous algebraic works Initiates systematic study of testability for algebraic properties Sheds light on testing and invariances of properties. I f F µ n ! i s c l o e d u r a t , ± m h y z b .

Property Testing vs. “Statistics” Classical Statistics (Mean, Median, Quantiles): Also run in sublinear time. So what’s special about “linearity testing”? Classical statistics work on “symmetric” properties: Linear functions closed under much smaller group of permutations: Simlarly, graph properties have nice invariances F c l o s e d u n r a b i t y p m D . F c l o s e d u n r i a m p L : 2 ! j D l o g s u c h m a p v . !

Is testing a corollary of invariance? Example 1: Invariant group trivial. Testing easy. Example 2: Invariant group still trivial. No O(1) local tests. Conclusion: Testing not necessarily a consequence of invariance. If we believe this to be the case for the linearity test, must prove it!! D = R ; F 1 f I d j ( x ) g D = f 1 ; : n g R 9 F 2 j x < y ) (

Is testing a corollary of invariance? Example 1: Invariant group trivial. Testing easy. Example 2: Invariant group still trivial. No O(1) local tests. Conclusion: Testing not necessarily a consequence of invariance. If we believe this to be the case for the linearity test, must prove it!! D = R ; F 1 f I d j ( x ) g D = f 1 ; : n g R 9 F 2 j x < y ) (

Part II: Formal Definitions & Results

Linear Invariance Examples: F i s L n e a r I v t f ² l u b p c ( o ) j ) 2 d : ! ± ( A ± n e I v a r i c d ¯ s m l y ) ² L i n e a r f u c t o s , - v p l y m d g · h F 1 + 2

Testing, constraints, characterizations ² S u p o s e F h a k - q r y t . ² T h e n m b r s o f F a t i y k - l c . C o n s t r a i : = ( x 1 ; k 2 F u b p c e V µ ) 8 f ¯ . , h ² E . g , i n t h e l a r c s : f ( ® ) + ¯ = C = ( ® ; ¯ + V f 1 g ) C h a r c t e i z o n : = f 1 ; m g 2 F , s ¯

(Linear-Invariant) Algebraic Characterizations Characterizations require many constraints! Linear (affine) invariance turns one constraint into many. (Linearity) Example: C = ( x 1 ; : k V ) c o n s t r a i d L l e ± . ( L ® ) ; ¯ + V c o n s t r a i f l e y A l g e b r a i c C h t z o n : S s . f ± L j ( ) F

Main Theorems T h e o r m : F a ± n - i v t d s k l c g b z , p q u y . ² U n i ¯ e s , m p l a d x t r v o u g b c T h e o r m : F a ± n - i v t d s k l c ) f ( g b z .

X Pictorially F o r a ± n e - i v t p s k - q u e r y t s k ! k - l o Theorem 1 By defn. k ! k - l o c a h r t e i z n k - l o c a g . h r By defn. Theorem 2 k ! f F ( ) X k - l o c a n s t r i k ! (with Grigorescu)

X Pictorially F o r a ± n e - i v t p s k - q u e r y t s k ! k - l o Theorem 1 By defn. k ! k - l o c a h r t e i z n k - l o c a g . h r By defn. Theorem 2 k ! f F ( ) X k - l o c a n s t r i k ! (with Grigorescu)

Part III: BLR (and our) analysis

Step 1: Prove “most likely” is overwhelming majority. BLR Analysis: Outline ² H a v e f s . t P r x ; y [ ( ) + 6 = ] ± < 2 9 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 Step 1: Prove “most likely” is overwhelming majority. ¡ ¡ S t e p 2 : P r o v h a g i s n F .

BLR Analysis: Step 0 ² D e ¯ n g ( x ) = m o s t l i k y f + ¡ . C l a [ 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

BLR Analysis: Step 1 V o t e ( y ) 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 ±

BLR Analysis: Step 1 V o t e ( y ) 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 ±

BLR Analysis: Step 1 V o t e ( 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 · ±

BLR Analysis: Step 1 V o t e ( 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 · ±

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 )

Step 1: Prove “most likely” is overwhelming majority. Our Analysis: Outline ² f s . t P r L [ h ( x 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 .

Step 1: Prove “most likely” is overwhelming majority. Our Analysis: Outline ² f s . t P r L [ h ( x 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 .

? Matrix Magic? V o t e ( L ) 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 )

? 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 make 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 .

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 make 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 .

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)?