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

2. 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 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 g .
Dec. 31, 2007
Invariance in Property Testing

3. Invariance in Property Testing
Distance:
Definition:
Notes: ( f ; g ) = P r x 2 D [ 6 ] F m i n . 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

4. 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

5. 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 ] C a n t e s w k l y i h ~ O ( 1 = 2 ) q u r . o b d
Dec. 31, 2007
Invariance in Property Testing

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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 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

12. 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

13. 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 : D ! , s - v f 2 l . A u t ( F ) = f j i s - n v a r g o m p d e c .
Dec. 31, 2007
Invariance in Property Testing

14. 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 - 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

15. 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 . ( 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

16. 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

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

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

19. Examples of Linear-Invariant FamiliesP 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

20. 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

21. 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

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

23. Key Notion: Formal CharacterizationF i s f o r m a l y c h t e z d 9 n g C = ( x 1 ; : k S ) u 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

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

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

26. Invariance in Property Testing
BLR Analysis: Step 0 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

27. Invariance in Property Testingx ( y )
BLR Analysis: Step 1 D e n g ( x ) = m o s t l i k y f + . 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 ? 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

28. Invariance in Property Testingx ( y )
BLR Analysis: Step 1 D e n g ( x ) = m o s t l i k y f + . 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 ? 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

29. Invariance in Property Testing
BLR Analysis: Step 1 V o t e x ( y ) 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

30. Invariance in Property Testing
BLR Analysis: Step 1 V o t e x ( y ) 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

31. 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

32. Our Analysis: Outline ² f s . t P r [ h ( x ) ; : i 2 V ] = ± ¿ ² D e1 ) ; : k i 2 V ] = D e n g ( x ) = t h a m i z s P r f L j 1 [ ; 2 : k V ] 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

33. Our Analysis: Outline ² f s . t P r [ h ( x ) ; : i 2 V ] = ± ¿ ² D e1 ) ; : k i 2 V ] = 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 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

34. Invariance in Property Testingx ( L )
Matrix Magic? 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

35. ? Matrix Magic? L ( x ) ¢ L ( x ) x K ( x ) . K ( x ) ² W a n t m r k2 ) L ( x k ) x K ( x 2 ) ? . K ( x k ) W a n t m r k e d o w s b c i . S u p o s e x 1 ; : ` l i n a r y d t h m .

36. 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 ) S u p o s e x 1 ; : ` l i n a r y d t h m .
Dec. 31, 2007

37. 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 ) S u p o s e x 1 ; : ` l i n a r y d t h m .
Dec. 31, 2007

38. 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