1. January 8-10, 2010 ITCS: Invariance in Property Testing 1 Invariance in Property Testing Madhu Sudan Microsoft/MIT TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA

2. January 8-10, 2010 ITCS: Invariance in Property Testing 2 Property Testing … of functions from D to R: … of functions from D to R: Property P µ {D  R} Property P µ {D  R} Distance Distance δ(f,g) = Pr x 2 D [f(x) ≠ g(x)] δ(f,g) = Pr x 2 D [f(x) ≠ g(x)] δ(f,P) = min g 2 P [δ(f,g)] δ(f,P) = min g 2 P [δ(f,g)] f is ε-close to g (f ¼ ² g) iff δ(f,g) · ε. f is ε-close to g (f ¼ ² g) iff δ(f,g) · ε. Local testability: Local testability: P is (k, ε, δ)-locally testable if 9 k-query test T P is (k, ε, δ)-locally testable if 9 k-query test T f 2 P ) T f accepts w.p. 1-ε. f 2 P ) T f accepts w.p. 1-ε. δ(f,P) > δ ) T f accepts w.p. ε. δ(f,P) > δ ) T f accepts w.p. ε. Notes: want k(ε, δ) = O(1) for ε,δ= (1). Notes: want k(ε, δ) = O(1) for ε,δ= (1).

3. January 8-10, 2010 ITCS: Invariance in Property Testing 3 Brief History [Blum,Luby,Rubinfeld – S’90] [Blum,Luby,Rubinfeld – S’90] Linearity + application to program testing Linearity + application to program testing [Babai,Fortnow,Lund – F’90] [Babai,Fortnow,Lund – F’90] Multilinearity + application to PCPs (MIP). Multilinearity + application to PCPs (MIP). [Rubinfeld+S.] [Rubinfeld+S.] Low-degree testing Low-degree testing [Goldreich,Goldwasser,Ron] [Goldreich,Goldwasser,Ron] Graph property testing Graph property testing Since then … many developments Since then … many developments Graph properties Graph properties Statistical properties Statistical properties … More algebraic properties More algebraic properties

4. January 8-10, 2010 ITCS: Invariance in Property Testing 4 Specific Directions in Algebraic P.T. More Properties More Properties Low-degree (d < q) functions [RS] Low-degree (d < q) functions [RS] Moderate-degree (q < d < n) functions Moderate-degree (q < d < n) functions q=2: [AKKLR] q=2: [AKKLR] General q: [KR, JPRZ] General q: [KR, JPRZ] Long code/Dictator/Junta testing [BGS,PRS] Long code/Dictator/Junta testing [BGS,PRS] BCH codes (Trace of low-deg. poly.) [KL] BCH codes (Trace of low-deg. poly.) [KL] Better Parameters (motivated by PCPs). Better Parameters (motivated by PCPs). #queries, high-error, amortized query complexity, reduced randomness. #queries, high-error, amortized query complexity, reduced randomness.

5. My concerns … Relatively few results … Relatively few results … Why can’t we get “rich” class of properties that are all testable? Why can’t we get “rich” class of properties that are all testable? Why are proofs so specific to property being tested? Why are proofs so specific to property being tested? What made Graph Property Testing so well- understood? What made Graph Property Testing so well- understood? What is “novel” about Property Testing, when compared to “polling”? What is “novel” about Property Testing, when compared to “polling”? January 8-10, 2010 ITCS: Invariance in Property Testing 5

6. January 8-10, 2010 ITCS: Invariance in Property Testing 6 Contrast w. Combinatorial P.T. Algebraic Property = Code! (usually) Universe Universe: {f:D  R} P Don’t care Must reject Must accept P R is a field F; P is linear!

7. Basic Implications of Linearity [BHR] If P is linear, then: If P is linear, then: Tester can be made non-adaptive. Tester can be made non-adaptive. Tester makes one-sided error Tester makes one-sided error (f 2 P ) tester always accepts). (f 2 P ) tester always accepts). Motivates: Motivates: Constraints: Constraints: k-query test => constraint of size k: k-query test => constraint of size k: value of f at ® 1,… ® k constrained to lie in subspace. value of f at ® 1,… ® k constrained to lie in subspace. Characterizations: Characterizations: If non-members of P rejected with positive probability, then P characterized by local constraints. If non-members of P rejected with positive probability, then P characterized by local constraints. functions satisfying all constraints are members of P. functions satisfying all constraints are members of P. January 8-10, 2010 ITCS: Invariance in Property Testing 7

8. f = assgm’t to left f = assgm’t to left Right = constraints Right = constraints Characterization of P: Characterization of P: P = {f sat. all constraints} P = {f sat. all constraints} Pictorially January 8-10, 2010 ITCS: Invariance in Property Testing 8 1 2 3 D {0000,1100, 0011,1111}

9. Sufficient conditions? Linearity + k-local characterization Linearity + k-local characterization ) k-local testability? ) k-local testability? [BHR] No! [BHR] No! Elegant use of expansion Elegant use of expansion Rule out obvious test; but also any test … of any “q(k)”-locality Rule out obvious test; but also any test … of any “q(k)”-locality Why is characterization insufficient? Why is characterization insufficient? Lack of symmetry? Lack of symmetry? January 8-10, 2010 ITCS: Invariance in Property Testing 9

10. Example motivating symmetry Conjecture (AKKLR ‘96): Conjecture (AKKLR ‘96): Suppose property P is a vector space over F 2 ; Suppose property P is a vector space over F 2 ; Suppose its “invariant group” is “2-transitive”. Suppose its “invariant group” is “2-transitive”. Suppose P satisfies a k-ary constraint Suppose P satisfies a k-ary constraint 8 f 2 P, f( ® 1 ) +  + f( ® k ) = 0. 8 f 2 P, f( ® 1 ) +  + f( ® k ) = 0. Then P is (q(k), ² (k,δ),δ)-locally testable. Then P is (q(k), ² (k,δ),δ)-locally testable. Inspired by “low-degree” test over F 2. Implied all previous algebraic tests (at least in weak forms). Inspired by “low-degree” test over F 2. Implied all previous algebraic tests (at least in weak forms). January 8-10, 2010 ITCS: Invariance in Property Testing 10

11. Invariances Property P invariant under permutation (function) ¼ : D  D, if Property P invariant under permutation (function) ¼ : D  D, if f 2 P ) f ο ¼ 2 P Property P invariant under group G if Property P invariant under group G if 8 ¼ 2 G, P is invariant under ¼. 8 ¼ 2 G, P is invariant under ¼. Can ask: Does invariance of P w.r.t. “nice” G leads to local testability? Can ask: Does invariance of P w.r.t. “nice” G leads to local testability? January 8-10, 2010 ITCS: Invariance in Property Testing 11

12. Invariances are the key? “Polling” works well when (because) invariant group of property is the full symmetric group. “Polling” works well when (because) invariant group of property is the full symmetric group. Modern property tests work with much smaller group of invariances. Modern property tests work with much smaller group of invariances. Graph property ~ Invariant under vertex renaming. Graph property ~ Invariant under vertex renaming. Algebraic Properties & Invariances? Algebraic Properties & Invariances? January 8-10, 2010 ITCS: Invariance in Property Testing 12

13. Abstracting Algebraic Properties [Kaufman & S.] [Kaufman & S.] Range is a field F and P is F-linear. Range is a field F and P is F-linear. Domain is a vector space over F (or some field K extending F). Domain is a vector space over F (or some field K extending F). Property is invariant under affine (sometimes only linear) transformations of domain. Property is invariant under affine (sometimes only linear) transformations of domain. “Property characterized by single constraint, and its orbit under affine (or linear) transformations.” “Property characterized by single constraint, and its orbit under affine (or linear) transformations.” January 8-10, 2010 ITCS: Invariance in Property Testing 13

14. Invariance, Orbits and Testability Single constraint implies many Single constraint implies many One for every permutation ¼ 2 Aut(P): One for every permutation ¼ 2 Aut(P): “Orbit of a constraint C” “Orbit of a constraint C” = {C ο ¼ | ¼ 2 Aut(P)} = {C ο ¼ | ¼ 2 Aut(P)} Extreme case: Extreme case: Property characterized by single constraint + its orbit: “Single orbit feature” Property characterized by single constraint + its orbit: “Single orbit feature” Most algebraic properties have this feature. Most algebraic properties have this feature. W.l.o.g. if domain = vector space over small field. W.l.o.g. if domain = vector space over small field. January 8-10, 2010 ITCS: Invariance in Property Testing 14

15. Example: Degree d polynomials Constraint: When restricted to a small dimensional affine subspace, function is polynomial of degree d (or less). Constraint: When restricted to a small dimensional affine subspace, function is polynomial of degree d (or less). #dimensions · d/(K - 1) #dimensions · d/(K - 1) Characterization: If a function satisfies above for every small dim. subspace, then it is a degree d polynomial. Characterization: If a function satisfies above for every small dim. subspace, then it is a degree d polynomial. Single orbit: Take constraint on any one subspace of dimension d/(K-1); and rotate over all affine transformations. Single orbit: Take constraint on any one subspace of dimension d/(K-1); and rotate over all affine transformations. January 8-10, 2010 ITCS: Invariance in Property Testing 15

16. Some results If P is affine-invariant and has k-single orbit feature (characterized by orbit of single k-local constraint); then it is (k, δ/k 3, δ)-locally testable. If P is affine-invariant and has k-single orbit feature (characterized by orbit of single k-local constraint); then it is (k, δ/k 3, δ)-locally testable. Unifies previous algebraic tests (in weak form) with single proof. Unifies previous algebraic tests (in weak form) with single proof. January 8-10, 2010 ITCS: Invariance in Property Testing 16

17. Analysis of Invariance-based test Property P given by ® 1,…, ® k ; V 2 F k Property P given by ® 1,…, ® k ; V 2 F k P = {f | f(A( ® 1 )) … f(A( ® k )) 2 V, 8 affine A:K n K n } P = {f | f(A( ® 1 )) … f(A( ® k )) 2 V, 8 affine A:K n K n } Rej(f) = Prob A [ f(A( ® 1 )) … f(A( ® k )) not in V ] Rej(f) = Prob A [ f(A( ® 1 )) … f(A( ® k )) not in V ] Wish to show: If Rej(f) < 1/k 3, Wish to show: If Rej(f) < 1/k 3, then δ(f,P) = O(Rej(f)). then δ(f,P) = O(Rej(f)). January 8-10, 2010 ITCS: Invariance in Property Testing 17

18. BLR Analog Rej(f) = Pr x,y [ f(x) + f(y) ≠ f(x+y)] < ² Rej(f) = Pr x,y [ f(x) + f(y) ≠ f(x+y)] < ² Define g(x) = majority y {Vote x (y)}, Define g(x) = majority y {Vote x (y)}, where Vote x (y) = f(x+y) – f(y). where Vote x (y) = f(x+y) – f(y). Step 0: Show δ(f,g) small Step 0: Show δ(f,g) small Step 1: 8 x, Pr y,z [Vote x (y) ≠ Vote x (z)] small. Step 1: 8 x, Pr y,z [Vote x (y) ≠ Vote x (z)] small. Step 2: Use above to show g is well-defined and a homomorphism. Step 2: Use above to show g is well-defined and a homomorphism. January 8-10, 2010 ITCS: Invariance in Property Testing 18

19. BLR Analysis of Step 1 Why is f(x+y) – f(y) = f(x+z) – f(z), usually? Why is f(x+y) – f(y) = f(x+z) – f(z), usually? January 8-10, 2010 ITCS: Invariance in Property Testing 19 - f(x+z) f(y) - f(x+y) f(z) -f(y) f(x+y+z) -f(z) 0 ?

20. Generalization g(x) = ¯ that maximizes, over A s.t. A( ® 1 ) = x, g(x) = ¯ that maximizes, over A s.t. A( ® 1 ) = x, Pr A [ ¯,f(A( ® 2 ),…,f(A( ® k )) 2 V] Pr A [ ¯,f(A( ® 2 ),…,f(A( ® k )) 2 V] Step 0: δ(f,g) small. Step 0: δ(f,g) small. Vote x (A) = ¯ s.t. ¯, f(A( ® 2 ))…f(A( ® k )) 2 V Vote x (A) = ¯ s.t. ¯, f(A( ® 2 ))…f(A( ® k )) 2 V (if such ¯ exists) (if such ¯ exists) Step 1 (key): 8 x, whp Vote x (A) = Vote x (B). Step 1 (key): 8 x, whp Vote x (A) = Vote x (B). Step 2: Use above to show g 2 P. Step 2: Use above to show g 2 P. January 8-10, 2010 ITCS: Invariance in Property Testing 20

21. Matrix Magic? January 8-10, 2010 ITCS: Invariance in Property Testing 21 A( ® 2 ) B( ® k ) B( ® 2 ) A( ® k ) x t Say A( ® 1 ) … A( ® t ) independent; rest dependent t Random No Choice Doesn’t Matter!

22. Some results If P is affine-invariant and has k-single orbit feature (characterized by orbit of single k-local constraint); then it is (k, δ/k 3, δ)-locally testable. If P is affine-invariant and has k-single orbit feature (characterized by orbit of single k-local constraint); then it is (k, δ/k 3, δ)-locally testable. Unifies previous algebraic tests with single proof. Unifies previous algebraic tests with single proof. If P is affine-invariant over K and has a single k- local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) If P is affine-invariant over K and has a single k- local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) (explains the AKKLR optimism) (explains the AKKLR optimism) January 8-10, 2010 ITCS: Invariance in Property Testing 22

23. Results (contd.) If P is affine-invariant over K and has a single k- local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) If P is affine-invariant over K and has a single k- local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) Proof Ingredients: Proof Ingredients: Analysis of all affine invariant properties. Analysis of all affine invariant properties. Rough characterization of locality of constraints, in terms of degrees of polynomials in the family. Rough characterization of locality of constraints, in terms of degrees of polynomials in the family. Infinitely many (new) properties … Infinitely many (new) properties … January 8-10, 2010 ITCS: Invariance in Property Testing 23

24. More details Understanding invariant properties: Understanding invariant properties: Recall: all functions from K n to F are Traces of polynomials Recall: all functions from K n to F are Traces of polynomials ( Trace(x) = x + x p + x p 2 + … + x q/p ( Trace(x) = x + x p + x p 2 + … + x q/p where K = F q and F = F p ) where K = F q and F = F p ) If P contains Tr(3x 5 + 4x 2 + 2); then P contains Tr(4x 2 ) … If P contains Tr(3x 5 + 4x 2 + 2); then P contains Tr(4x 2 ) … So affine invariant properties characterized by degree of monomials in family. So affine invariant properties characterized by degree of monomials in family. Most of the study … relate degrees to upper and lower bounds on locality of constraints. Most of the study … relate degrees to upper and lower bounds on locality of constraints. January 8-10, 2010 ITCS: Invariance in Property Testing 24

25. Some results If P is affine-invariant over K and has a single k- local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) If P is affine-invariant over K and has a single k- local constraint, then it is has a q-single orbit feature (for some q = q(K,k)) (explains the AKKLR optimism) (explains the AKKLR optimism) Unfortunately, q depends inherently on K, not just F … giving counterexample to AKKLR conjecture [joint with Grigorescu & Kaufman] Unfortunately, q depends inherently on K, not just F … giving counterexample to AKKLR conjecture [joint with Grigorescu & Kaufman] Linear invariance when P is not F-linear: Linear invariance when P is not F-linear: Abstraction of some aspects of Green’s regularity lemma … [ Bhattacharyya, Chen, S., Xie ] Abstraction of some aspects of Green’s regularity lemma … [ Bhattacharyya, Chen, S., Xie ] Nice results due to [Shapira] Nice results due to [Shapira] January 8-10, 2010 ITCS: Invariance in Property Testing 25

26. More results Invariance of some standard codes Invariance of some standard codes E.g. “dual-BCH”: Have k-single orbit feature! So are “more uniformly” testable. E.g. “dual-BCH”: Have k-single orbit feature! So are “more uniformly” testable. [Grigorescu, Kaufman, S.] [Grigorescu, Kaufman, S.] Side effect: New (essentially tight) relationships between Rej AKKLR (f) and δ(f,Degree-d) over F 2 [with Bhattacharyya, Kopparty, Schoenebeck, Zuckerman] Side effect: New (essentially tight) relationships between Rej AKKLR (f) and δ(f,Degree-d) over F 2 [with Bhattacharyya, Kopparty, Schoenebeck, Zuckerman] January 8-10, 2010 ITCS: Invariance in Property Testing 26

27. More results (contd.) Invariance of some standard codes Invariance of some standard codes Side effect: New (essentially tight) relationships between Rej AKKLR (f) and δ(f,Degree-d) over F 2 Side effect: New (essentially tight) relationships between Rej AKKLR (f) and δ(f,Degree-d) over F 2 One hope: Could lead to “simple, good locally testable code”? One hope: Could lead to “simple, good locally testable code”? (Sadly, not with affine-inv. [Ben-Sasson, S.]) (Sadly, not with affine-inv. [Ben-Sasson, S.]) Still … other groups could be used? [Kaufman+Wigderson] Still … other groups could be used? [Kaufman+Wigderson] January 8-10, 2010 ITCS: Invariance in Property Testing 27

28. Conclusions Invariance seems to be a nice perspective on “property testing” … Invariance seems to be a nice perspective on “property testing” … Certainly helps unify many algebraic property tests. Certainly helps unify many algebraic property tests. But should be a general lens in sublinear time algorithmics. But should be a general lens in sublinear time algorithmics. January 8-10, 2010 ITCS: Invariance in Property Testing 28

29. January 8-10, 2010 ITCS: Invariance in Property Testing 29 Thanks