1. Algebraic Property Testing:
A Survey
Madhu Sudan
MIT
April 1, 2009
Algebraic Property DIMACS

2. Algebraic Property Testing:
Personal Perspective
Madhu Sudan
MIT
April 1, 2009
Algebraic Property DIMACS

3. Algebraic Property Testing:
Personal Perspective
Madhu Sudan
MIT
April 1, 2009
Algebraic Property DIMACS

4. Algebraic Property Testing @ DIMACS
Distance:
Definition:
Notes: ( f ; g ) = P r x 2 D [ 6 ] F m i n . F i s ( k ; ) - l o c a y t e b f 9 q u r T . 2 p w 1 j k - l o c a y t e s b i m p 9 ;
> = O ( 1 ) n d r : A f 2 F w .
April 1, 2009
Algebraic Property DIMACS

5. Algebraic Property Testing @ DIMACS
Brief History
[Blum,Luby,Rubinfeld – S’90]
Linearity + application to program testing
[Babai,Fortnow,Lund – F’90]
Multilinearity + application to PCPs (MIP).
[Rubinfeld+S.]
Low-degree testing + Formal Definition
[Goldreich,Goldwasser,Ron]
Graph property testing.
Since then … many developments
Graph properties
Statistical properties
More algebraic properties
April 1, 2009
Algebraic Property DIMACS

6. Specific Directions in Algebraic P.T.
More Properties
Low-degree (d < q) functions [RS]
Moderate-degree (q < d < n) functions
q=2: [AKKLR]
General q: [KR, JPRZ]
Long code/Dictator/Junta testing [PRS]
BCH codes (Trace of low-deg. poly.) [KL]
All nicely “invariant” properties [KS]
Better Parameters (motivated by PCPs).
#queries, high-error, amortized query complexity, reduced randomness.
April 1, 2009
Algebraic Property DIMACS

7. Contrast w. Combinatorial P.T.U n i v e r s f : D ! R g ( A l s o u a y ) R i e d F P r p t = L n b c . M u s t a c e p O k o r j w . h F F
Algebraic Property = Code! (usually)
April 1, 2009
Algebraic Property DIMACS

8. Algebraic Property Testing @ DIMACS
Goal of this talk
Implications of linearity
Constraints, Characterizations, LDPC structure
One-sided error, Non-adaptive tests [BHR]
Redundancy of Constraints
Tensor Product Codes
Symmetries of Code
Testing affine-invariant codes
Yields basic tests for all known algebraic codes (over small fields).
April 1, 2009
Algebraic Property DIMACS

9. Basic Implications of Linearity [BHR]
Generic adaptive test = decision tree.
f(i) 1
f(j)
f(k) P i c k p a t h f o l w e d b y r n m g 2 F . 1 Q u e r y f a c o d i n g t p h . A c e p t i f o n a h s w m 2 F . Y i e l d s n o - a p t v r f F .
April 1, 2009
Algebraic Property DIMACS

10. Basic Implications of Linearity [BHR]
Generic adaptive test = decision tree.
f(i) 1
f(j)
f(k) P i c k p a t h f o l w e d b y r n m g 2 F . 1 Q u e r y f a c o d i n g t p h . A c e p t i f o n a h s w m 2 F . Y i e l d s n o - a p t v r f F .
April 1, 2009
Algebraic Property DIMACS

11. Constraints, CharacterizationsS a y t e s q u r i 1 ; : k c p h f ( ) 2 V 6 = F ( i 1 ; : k V ) = C o n s t r a E v e y f 2 F . 1 D i1 i2 ik
in V? 2 I f e v r y 6 2 F j c t d w . p o s i b h n a z
Like LDPC Codes!
April 1, 2009
Algebraic Property DIMACS

12. Constraints, CharacterizationsS a y t e s q u r i 1 ; : k c p h f ( ) 2 V 6 = F ( i 1 ; : k V ) = C o n s t r a E v e y f 2 F . 1 D i1 i2 ik
in V? 2 I f e v r y 6 2 F j c t d w . p o s i b h n a z
Like LDPC Codes!
April 1, 2009
Algebraic Property DIMACS

13. Example: Linearity Testing [BLR]
Constraints:
Characterization: C x ; y = ( + V ) j 2 F n w h e r f a b g x
in V? f i s l n e a r 8 x ; y C t d y
x+y
April 1, 2009
Algebraic Property DIMACS

14. Insufficiency of local characterizations
[Ben-Sasson, Harsha, Raskhodnikova]
There exist families characterized by k-local constraints that are not o(|D|)-locally testable.
Proof idea: Pick LDPC graph at random …
(and analyze resulting property) F
April 1, 2009
Algebraic Property DIMACS

15. Why are characterizations insufficient?
Constraints too minimal.
Not redundant enough!
Proved formally in [Ben-Sasson, Guruswami, Kaufman, S., Viderman]
Constraints too asymmetric.
Property must show some symmetry to be testable.
Not a formal assertion … just intuitive.
April 1, 2009
Algebraic Property DIMACS

16. Algebraic Property Testing @ DIMACS
Redundancy?
E.g. Linearity Test:
Standard LDPC analysis:
What natural operations create redundant local constraints?
Tensor Products! ( D 2 ) c o n s t r a i d m D i m e n s o ( F ) f r c t a . R q u #
< l w h d y !
April 1, 2009
Algebraic Property DIMACS

17. Tensor Products of Codes!
Redundancy? F G = f M a t r i c e s u h v y o w n F d l m G g S u p o s e F , G y t m a i c F i r s t ` e n f d m b y h .
Free F d e t r m i n G d e t r m i n d e t r m i n w c , b y F a G !
April 1, 2009
Algebraic Property DIMACS

18. Testability of tensor product codes?
Natural test:
Given Matrix M
Test if random row in F
Test if random column in G
Claim:
If F, G codes of constant (relative) distance; then if test accepts w.h.p. then M is close to codeword of F x G
Yields O(√n) local test for codes of length n.
Can we do better? Exploit local testability of F, G?
April 1, 2009
Algebraic Property DIMACS

19. Robust testability of tensors?
Natural test (if F,G locally testable):
Given Matrix M
Run Local Test for F on random row
Run Local Test for G on random column
Suppose M close on most rows/columns to F, G. Does this imply M is close to F x G?
Generalizes test for bivariate polynomials. True for F, G = class of low-degree polynomials. [BFLS, Arora+Safra, Polishchuk+Spielman].
General question raised by [Ben-Sasson+S.]
[P. Valiant] Not true for every F, G !
[Dinur, S., Wigderson] True if F (or G) locally testable.
Test that random row close to F
Test that random column close to G
April 1, 2009
Algebraic Property DIMACS

20. Tensor Products and Local Testability
Robust testability allows easy induction (essentially from [BFL, BFLS]; see also [Ben-Sasson+S.]) L e t F n = - f o l d s r . G i v e n f : D ! F N a t u r l s P c k d o m x - p y j 2
April 1, 2009
Algebraic Property DIMACS

21. Robust testability of tensors (contd.)
Unnatural test (for F x F x F):
Given 3-d matrix M:
Pick random 2-d submatrix.
Verify it is close to F x F
Theorem [BenSasson+S., based on Raz+Safra]: Distance to F x F x F proportional to average distance of random 2-d submatrix to F x F.
[Meir]: “Linear-algebraic” construction of Locally Testable Codes (matching best known parameters) using this (and many other ingredients).
April 1, 2009
Algebraic Property DIMACS

22. Redundant Characterizations (contd.)
Redundant constraints necessary for testing [BGKSV]
How to get redundancy?
Tensor Products
Sufficient to get some local testability
Invariances (Symmetries)
Sufficient?
Counting (See Tali’s talk) 
April 1, 2009
Algebraic Property DIMACS

23. Algebraic Property Testing @ DIMACS
Testing by symmetries
April 1, 2009
Algebraic Property DIMACS

24. 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 .
April 1, 2009
Algebraic Property DIMACS

25. Algebraic Property Testing @ DIMACS
Examples
Majority:
Graph Properties:
Algebraic Properties: What symmetries do they have? 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 ] w h
April 1, 2009
Algebraic Property DIMACS

26. Algebraic Properties & Invariances
Automorphism groups?
Question: Are Linear/Affine-Inv., Locally Characterized Props. Testable? ([Kaufman + S.]) 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) A n e t r a s f o m i d . ( x ) = + b w h 2 F ;
(Affine-Inv.)
April 1, 2009
Algebraic Property DIMACS

27. Linear-Invariance & Testability
Unifies previous studies on Alg. Prop. Testing.
(And captures some new properties)
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.
April 1, 2009
Algebraic Property DIMACS

28. Some Results [Kaufman + S.]
Theorem 1:
Theorem 2: 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 .
April 1, 2009
Algebraic Property DIMACS

29. Examples of Linear-Invariant FamiliesL i n e a r f u c t o s m F . 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
April 1, 2009
Algebraic Property DIMACS

30. 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
April 1, 2009
Algebraic Property DIMACS

31. Property Testing from Invariances
April 1, 2009
Algebraic Property DIMACS

32. Key Notion: Formal CharacterizationF h a s i n g l e - o r b t c z f 9 C = ( x 1 ; : k V ) u 2 A . T h e o r m : I f F a s i n g l - b t c z y k ( w ) .
Rest of talk: Analysis (extending BLR)
April 1, 2009
Algebraic Property DIMACS

33. Algebraic Property Testing @ DIMACS
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 .
April 1, 2009
Algebraic Property DIMACS

34. Algebraic Property Testing @ DIMACS
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
April 1, 2009
Algebraic Property DIMACS

35. Algebraic Property Testing @ DIMACSV o t e x ( 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
April 1, 2009
Algebraic Property DIMACS

36. Algebraic Property Testing @ DIMACSV o t e x ( 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
April 1, 2009
Algebraic Property DIMACS

37. Algebraic Property Testing @ DIMACS
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
April 1, 2009
Algebraic Property DIMACS

38. Algebraic Property Testing @ DIMACS
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
April 1, 2009
Algebraic Property DIMACS

39. 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 )
April 1, 2009
Algebraic Property DIMACS

40. 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 .
April 1, 2009
Algebraic Property DIMACS

41. 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 .
April 1, 2009
Algebraic Property DIMACS

42. Algebraic Property Testing @ DIMACSV o t e x ( 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 )
April 1, 2009
Algebraic Property DIMACS

43. ? 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 .

44. 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
Tensor magic 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 .
April 1, 2009

45. 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
Tensor magic 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 .
April 1, 2009

46. Algebraic Property Testing @ DIMACS
Summarizing
Affine invariance + single-orbit characterizations imply testing.
Unifies analysis of linearity test, basic low-degree tests, moderate-degree test (all A.P.T. except dual-BCH?)
April 1, 2009
Algebraic Property DIMACS

47. Algebraic Property Testing @ DIMACS
Concluding thoughts - 1
Didn’t get to talk about
PCPs, LTCs (though we did implicitly)
Optimizing parameters
Parameters
In general
Broad reasons why property testing works worth examining.
Tensoring explains a few algebraic examples.
Invariance explains many other algebraic ones.
(More about invariances in [Grigorescu,Kaufman,S. ’08], [GKS’09])
April 1, 2009
Algebraic Property DIMACS

48. Algebraic Property Testing @ DIMACS
Concluding thoughts - 2
Invariance:
Seems to be a nice lens to view all property testing results (combinatorial, statistical, algebraic).
Many open questions:
What groups of symmetries aid testing?
What additional properties needed?
Local constraints?
Linearity?
Does sufficient symmetry imply testability?
Give an example of a non-testable property with a k-single orbit characterization.
April 1, 2009
Algebraic Property DIMACS

49. Algebraic Property Testing @ DIMACS
Thank You!
April 1, 2009
Algebraic Property DIMACS