1. of 22 August 29-30, 2011 Rabin ’80: APT 1 Invariance in Property Testing Madhu Sudan Microsoft Research TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A AAA Based on: works with/of Eli Ben-Sasson, Elena Grigorescu, Tali Kaufman, Shachar Lovett, Ghid Maatouk, Amir Shpilka.

2. of 22 Property Testing Sublinear time algorithms: Sublinear time algorithms: Algorithms running in time o(input), o(output). Algorithms running in time o(input), o(output). Probabilistic. Probabilistic. Correct on (approximation) to input. Correct on (approximation) to input. Input given by oracle, output implicit. Input given by oracle, output implicit. Crucial to modern context Crucial to modern context (Massive data, no time). (Massive data, no time). Property testing: Property testing: Restriction of sublinear time algorithms to decision problems (output = YES/NO). Restriction of sublinear time algorithms to decision problems (output = YES/NO). Amazing fact: Many non-trivial algorithms exist! Amazing fact: Many non-trivial algorithms exist! August 29-30, 2011 Rabin ’80: APT 2

3. of 22 Example 1: Polling Is the majority of the population Red/Blue Is the majority of the population Red/Blue Can find out by random sampling. Can find out by random sampling. Sample size / margin of error Sample size / margin of error Independent of size of population Independent of size of population Other similar examples: (can estimate other moments …) Other similar examples: (can estimate other moments …) August 29-30, 2011 Rabin ’80: APT 3

4. of 22 Example 2: Linearity Can test for homomorphisms: Can test for homomorphisms: Given: f: G  H (G,H finite groups), is f essentially a homomorphism? Given: f: G  H (G,H finite groups), is f essentially a homomorphism? Test: Test: Pick x,y in G uniformly, ind. at random; Pick x,y in G uniformly, ind. at random; Verify f(x) ¢ f(y) = f(x ¢ y) Verify f(x) ¢ f(y) = f(x ¢ y) Completeness: accepts homomorphisms w.p. 1 Completeness: accepts homomorphisms w.p. 1 (Obvious) (Obvious) Soundness: Rejects f w.p prob. Proportional to its “distance” (margin) from homomorphisms. Soundness: Rejects f w.p prob. Proportional to its “distance” (margin) from homomorphisms. (Not obvious, [BlumLubyRubinfeld’90]) (Not obvious, [BlumLubyRubinfeld’90]) August 29-30, 2011 Rabin ’80: APT 4

5. of 22 July 29, 2011 Invariance in Property Testing: EPFL 5 History ( slightly abbreviated ) [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 More graph properties, statistical properties, matrix properties, properties of Boolean functions … More graph properties, statistical properties, matrix properties, properties of Boolean functions … More algebraic properties More algebraic properties

6. of 22 Pictorial Summary August 29-30, 2011 Rabin ’80: APT 6 All properties Statistical Properties Linearity Low-degree Graph Properties Boolean functions Testable! Not-testable

7. of 22 Some (introspective) questions What is qualitatively novel about linearity testing relative to classical statistics? What is qualitatively novel about linearity testing relative to classical statistics? Why are the mathematical underpinnings of different themes so different? Why are the mathematical underpinnings of different themes so different? Why is there no analog of “graph property testing” (broad class of properties, totally classified wrt testability) in algebraic world? Why is there no analog of “graph property testing” (broad class of properties, totally classified wrt testability) in algebraic world? August 29-30, 2011 Rabin ’80: APT 7

8. of 22 Invariance? Property P µ {f : D  R} Property P µ {f : D  R} 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. Observation: Different property tests unified/separated by invariance class. Observation: Different property tests unified/separated by invariance class. July 29, 2011 Invariance in Property Testing: EPFL 8

9. of 22 Invariances (contd.) Some examples: Some examples: Classical statistics: Invariant under all permutations. Classical statistics: Invariant under all permutations. Graph properties: Invariant under vertex renaming. Graph properties: Invariant under vertex renaming. Boolean properties: Invariant under variable renaming. Boolean properties: Invariant under variable renaming. Matrix properties: Invariant under mult. by invertible matrix. Matrix properties: Invariant under mult. by invertible matrix. Algebraic Properties = ? Algebraic Properties = ? Goals: Goals: Possibly generalize specific results. Possibly generalize specific results. Get characterizations within each class? Get characterizations within each class? In algebraic case, get new (useful) codes? In algebraic case, get new (useful) codes? August 29-30, 2011 Rabin ’80: APT 9

10. of 22 Abstracting Linearity/Low-degree tests Affine Invariance: Affine Invariance: Domain = Big field (GF(2 n )) Domain = Big field (GF(2 n )) or vector space over small field (GF(2) n ). or vector space over small field (GF(2) n ). Property invariant under affine transformations of domain (x  A.x + b) Property invariant under affine transformations of domain (x  A.x + b) Linearity: Linearity: Range = small field (GF(2)) Range = small field (GF(2)) Property = vector space over range. Property = vector space over range. August 29-30, 2011 Rabin ’80: APT 10

11. of 22 May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 11 Testing Linear Properties 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!

12. of 22 Why study affine-invariance? Common abstraction of properties studied in [BLR], [RS], [ALMSS], [AKKLR], [KR], [KL], [JPRZ]. Common abstraction of properties studied in [BLR], [RS], [ALMSS], [AKKLR], [KR], [KL], [JPRZ]. (Variations on low-degree polynomials) (Variations on low-degree polynomials) Hopes Hopes Unify existing proofs Unify existing proofs Classify/characterize testability Classify/characterize testability Find new testable codes (w. novel parameters) Find new testable codes (w. novel parameters) Rest of the talk: Brief summary of findings Rest of the talk: Brief summary of findings August 29-30, 2011 Rabin ’80: APT 12

13. of 22 Basic terminology Local Constraint: Local Constraint: Example: f(1) + f(2) = f(3). Example: f(1) + f(2) = f(3). Necessary for testing Linear Properties [BHR] Necessary for testing Linear Properties [BHR] Local Characterization: Local Characterization: Example: 8 x, y, f(x) + f(y) = f(x+y), f 2 P Example: 8 x, y, f(x) + f(y) = f(x+y), f 2 P Aka: LDPC code, k-CNF property etc. Aka: LDPC code, k-CNF property etc. Necessary for affine-invariant linear properties. Necessary for affine-invariant linear properties. Single-orbit characterization: Single-orbit characterization: One linear constraint + implications by affine- invariance. One linear constraint + implications by affine- invariance. Feature in all previous algebraic properties. Feature in all previous algebraic properties. August 29-30, 2011 Rabin ’80: APT 13

14. of 22 t-local constraint t-characterized Affine-invariance & testability May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 14 t-locally testable t-S-O-C

15. of 22 State of the art in 2007 [AKKLR]: k-constraint = k’-testable, for all linear affine-invariant properties? [AKKLR]: k-constraint = k’-testable, for all linear affine-invariant properties? August 29-30, 2011 Rabin ’80: APT 15

16. of 22 t-local constraint t-characterized Affine-invariance & testability May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 16 t-locally testable t-S-O-C

17. of 22 Some results [Kaufman+S.’07]: Single-orbit ) Testable. [Kaufman+S.’07]: Single-orbit ) Testable. July 29, 2011 Invariance in Property Testing: EPFL 17

18. of 22 t-local constraint t-characterized Affine-invariance & testability May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 18 t-locally testable t-S-O-C [KS’08]

19. of 22 Some results [Kaufman+S.’07]: Single-orbit ) Testable. [Kaufman+S.’07]: Single-orbit ) Testable. Unifies known algebraic testing results. Unifies known algebraic testing results. Converts testability to purely algebraic terms. Converts testability to purely algebraic terms. Yields “Constraints = Char. = Testability” for vector spaces over small fields. Yields “Constraints = Char. = Testability” for vector spaces over small fields. Left open: Domain = Big field. Left open: Domain = Big field. 9 Many “non-polynomial” testable properties 9 Many “non-polynomial” testable properties [GKS’08]: Over big fields, Constraint ≠ Char. [GKS’08]: Over big fields, Constraint ≠ Char. [BMSS’11]: Over big fields, Char ≠ Testability. [BMSS’11]: Over big fields, Char ≠ Testability. [BGMSS’11]: Many questions/conjectures outlining a possible characterization of affine- invariant properties. [BGMSS’11]: Many questions/conjectures outlining a possible characterization of affine- invariant properties. July 29, 2011 Invariance in Property Testing: EPFL 19

20. of 22 [BS’10]k-local constraint k-characterized Affine-invariance & testability August 29-30, 2011 Rabin ’80: APT 20 k-locally testable k-S-O-C [KS’08] [GKS’08] [BMSS’11]weight-k degrees

21. of 22 Hopes Get a complete characterization of locally testable affine-invariant properties. Get a complete characterization of locally testable affine-invariant properties. Use codes of (polynomially large?) locality to build better LTCs/PCPs? Use codes of (polynomially large?) locality to build better LTCs/PCPs? In particular move from “domain = vector space” to “domain = field”. In particular move from “domain = vector space” to “domain = field”. More broadly: Apply lens of invariance more broadly to property testing. More broadly: Apply lens of invariance more broadly to property testing. August 29-30, 2011 Rabin ’80: APT 21

22. of 22 Thank You! August 29-30, 2011 Rabin ’80: APT 22

23. of 22 Pictorial Summary August 29-30, 2011 Rabin ’80: APT 23 All properties Statistical Properties Linearity Graph Properties Low-degree Boolean functions Testable! Not-testable