Of 38 July 29, 2011 Invariance in Property Testing: EPFL 1 Invariance in Property Testing Madhu Sudan Microsoft Research TexPoint fonts used in EMF. Read.

of 38 July 29, 2011 Invariance in Property Testing: EPFL 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

of 38 This talk Introduce Property Testing Introduce Property Testing Focus on special case of algebraic properties Focus on special case of algebraic properties (Aka Locally Testing of (algebraic) Codes) (Aka Locally Testing of (algebraic) Codes) Some general results for codes/properties with special invariance. Some general results for codes/properties with special invariance. July 29, 2011 Invariance in Property Testing: EPFL 2

of 38 Modern challenge to Algorithm Design Data = Massive; Computers = Tiny Data = Massive; Computers = Tiny How can tiny computers analyze massive data? How can tiny computers analyze massive data? Only option: Design sublinear time algorithms. Only option: Design sublinear time algorithms. Algorithms that take less time to analyze data, than it takes to read/write all the data. Algorithms that take less time to analyze data, than it takes to read/write all the data. Can such algorithms exist? Can such algorithms exist? July 29, 2011 Invariance in Property Testing: EPFL 3

of 38 Yes! 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 …) July 29, 2011 Invariance in Property Testing: EPFL 4

of 38 Recent “novel” example 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) (Not obvious) July 29, 2011 Invariance in Property Testing: EPFL 5

of 38 July 29, 2011 Invariance in Property Testing: EPFL 6 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

of 38 July 29, 2011 Invariance in Property Testing: EPFL 7 Property Testing Data = a function from D to R: Data = a function 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 (t, ε, δ)-locally testable if 9 t-query test T P is (t, ε, δ)-locally testable if 9 t-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 t(ε, δ) = O(1) for ε,δ= (1). Notes: want t(ε, δ) = O(1) for ε,δ= (1).

of 38 Locally Testable Codes Intriguing aspect of BLR test: Intriguing aspect of BLR test: Property P = {first order Reed-Muller codes} Property P = {first order Reed-Muller codes} (A Hadamard Code) Motivates “Locally Testable Code” (LTC): Motivates “Locally Testable Code” (LTC): Property P = {Error-correcting code} Property P = {Error-correcting code} t-LTC: Testable with t(n) queries. t-LTC: Testable with t(n) queries. Are there better rate LTCs than Hadamard? Are there better rate LTCs than Hadamard? Yes – example 1: RM codes. Yes – example 1: RM codes. Yes … many more sophisticated ones. Yes … many more sophisticated ones. Natural motivation: Can test massive DVD for “too many” errors Natural motivation: Can test massive DVD for “too many” errors July 29, 2011 Invariance in Property Testing: EPFL 8

of 38 July 29, 2011 Invariance in Property Testing: EPFL 9 Why is BLR special?

of 38 Why is BLR special? Impressive collection of generalizations, alternate proofs, applications (all of PCP, LTC theory, e.g.)? Impressive collection of generalizations, alternate proofs, applications (all of PCP, LTC theory, e.g.)? Why is it more interesting than just polling? Why is it more interesting than just polling? Why did the proof work? Was it a one-shot thing? Why did the proof work? Was it a one-shot thing? Most previous attempts to extend “broadly” failed … Most previous attempts to extend “broadly” failed … July 29, 2011 Invariance in Property Testing: EPFL 10

of 38 BLR Analysis Fix f s.t. Rej(f) = Pr x,y [ f(x) + f(y) ≠ f(x+y)] < ² Fix f s.t. 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. July 29, 2011 Invariance in Property Testing: EPFL 11

of 38 Key Step: 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? (Note: Prob over y,z for fixed x.) (Note: Prob over y,z for fixed x.) Proof: Proof: f(x+y) + f(z) = f(x+y+z) [w.h.p.] f(x+y) + f(z) = f(x+y+z) [w.h.p.] = f(x+z) + f(y) [w.h.p. again] = f(x+z) + f(y) [w.h.p. again] Proof from the Book. Proof from the Book. (Indisputable! Inexplicable!) (Indisputable! Inexplicable!) July 29, 2011 Invariance in Property Testing: EPFL 12

of 38 Extensions [Rubinfeld + S. 92-96]: Low degree tests [Rubinfeld + S. 92-96]: Low degree tests [Rubinfeld 94]: Functional equations [Rubinfeld 94]: Functional equations [ALMSS, etc. ]: PCP theory [ALMSS, etc. ]: PCP theory [AKKLR 02]: Reed-Muller tests [AKKLR 02]: Reed-Muller tests [KaufmanRon, JPRZ]: Generalized RM tests. [KaufmanRon, JPRZ]: Generalized RM tests. … each time a new proof of key step. … each time a new proof of key step. July 29, 2011 Invariance in Property Testing: EPFL 13

of 38 Abstraction of BLR (in special case) Restrict to G = F n and H = F Restrict to G = F n and H = F (F = finite field; with q elements) (F = finite field; with q elements) Property: Property: Linear: (sum of linear functions is linear) Linear: (sum of linear functions is linear) Locally characterized: 8 x,y f(x) + f(y) = f(x+y) Locally characterized: 8 x,y f(x) + f(y) = f(x+y) Linear-invariant: Linear function remains linear after linear transformation of domain. Linear-invariant: Linear function remains linear after linear transformation of domain. Single-orbit: Constraints above given by one constraint and implication of linear-invariance. Single-orbit: Constraints above given by one constraint and implication of linear-invariance. Our hope: Such abstractions explain, extend and unify algebraic property testing. Our hope: Such abstractions explain, extend and unify algebraic property testing. July 29, 2011 Invariance in Property Testing: EPFL 14

of 38 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? July 29, 2011 Invariance in Property Testing: EPFL 15

of 38 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? July 29, 2011 Invariance in Property Testing: EPFL 16

of 38 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 t-ary constraint Suppose P satisfies a t-ary constraint 8 f 2 P, f( ® 1 ) +  + f( ® t ) = 0. 8 f 2 P, f( ® 1 ) +  + f( ® t ) = 0. (dual(P) has distance ≤ t) (dual(P) has distance ≤ t) Then P is (q(t), ² (t,δ),δ)-locally testable. Then P is (q(t), ² (t,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). July 29, 2011 Invariance in Property Testing: EPFL 17

of 38 t-local constraint Affine-invariance & testability May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 18 t-locally testable =?

of 38 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.” July 29, 2011 Invariance in Property Testing: EPFL 19

of 38 Terminology t-Constraint: Sequence of t elements of domain, and set of forbidden values for this sequence. t-Constraint: Sequence of t elements of domain, and set of forbidden values for this sequence. e.g. f(a) + f(b) = f(a+b) e.g. f(a) + f(b) = f(a+b) t-characterization: Collection of t-constraints, satisfaction of which is necessary and sufficient criterion for satisfying property t-characterization: Collection of t-constraints, satisfaction of which is necessary and sufficient criterion for satisfying property e.g. f(a) + f(b) = f(a+b), f(c) + f(d) = f(c+d) … e.g. f(a) + f(b) = f(a+b), f(c) + f(d) = f(c+d) … [t-LDPC] t-single-orbit characterization: One k-constraint such that its translations under affine group yields k-characterization. t-single-orbit characterization: One k-constraint such that its translations under affine group yields k-characterization. f(L(a)) + f(L(b)) = f(L(a+b)) ; a,b fixed, all linear L. f(L(a)) + f(L(b)) = f(L(a+b)) ; a,b fixed, all linear L. July 29, 2011 Invariance in Property Testing: EPFL 20

of 38 t-local constraint t-characterized Affine-invariance & testability May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 21 t-locally testable t-S-O-C

of 38 July 29, 2011 Invariance in Property Testing: EPFL 22 Main Results

of 38 Some results If P is affine-invariant and has t-single orbit characterization then it is (t, δ/t 3, δ)-locally testable. If P is affine-invariant and has t-single orbit characterization then it is (t, δ/t 3, δ)-locally testable. Unifies previous algebraic tests (in basic form) with single proof. Unifies previous algebraic tests (in basic form) with single proof. July 29, 2011 Invariance in Property Testing: EPFL 23

of 38 t-local constraint t-characterized Affine-invariance & testability May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 24 t-locally testable t-S-O-C [KS’08]

of 38 Analysis of Invariance-based test Property P given by ® 1,…, ® t ; V µ F k Property P given by ® 1,…, ® t ; V µ F k P = {f | (f(A( ® 1 )), …, f(A( ® t ))) 2 V, P = {f | (f(A( ® 1 )), …, f(A( ® t ))) 2 V, 8 affine A:K n K n } 8 affine A:K n K n } Rej(f) = Prob A [ (f(A( ® 1 )), …, f(A( ® t )))  V ] Rej(f) = Prob A [ (f(A( ® 1 )), …, f(A( ® t )))  V ] Wish to show: If Rej(f) < 1/t 3, Wish to show: If Rej(f) < 1/t 3, then δ(f,P) = O(Rej(f)). then δ(f,P) = O(Rej(f)). July 29, 2011 Invariance in Property Testing: EPFL 25

of 38 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. July 29, 2011 Invariance in Property Testing: EPFL 26

of 38 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( ® t ))) 2 V] Pr A [( ¯,f(A( ® 2 ),…,f(A( ® t ))) 2 V] Step 0: δ(f,g) small. Step 0: δ(f,g) small. Vote x (A) = ¯ s.t. ( ¯, f(A( ® 2 ))…f(A( ® t ))) 2 V Vote x (A) = ¯ s.t. ( ¯, f(A( ® 2 ))…f(A( ® t ))) 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. July 29, 2011 Invariance in Property Testing: EPFL 27

of 38 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? July 29, 2011 Invariance in Property Testing: EPFL 28 - f(x+z) f(y) - f(x+y) f(z) -f(y) f(x+y+z) -f(z) 0 ?

of 38 Matrix Magic? July 29, 2011 Invariance in Property Testing: EPFL 29 A( ® 2 ) B( ® t ) B( ® 2 ) A( ® t ) x s Say A( ® 1 ) … A( ® s ) independent; rest dependent s Random No Choice Doesn’t Matter!

of 38 Results (contd.) Thm 2: If P is affine-invariant over K and has a single t-local constraint, then it is has a q-single orbit feature (for some q = q(K,t)) Thm 2: If P is affine-invariant over K and has a single t-local constraint, then it is has a q-single orbit feature (for some q = q(K,t)) Proof ingredients: Proof ingredients: Analysis of all affine invariant properties. Analysis of all affine invariant properties. Characterization of all affine invariant properties in terms of degrees of monomials in support of polynomials in family Characterization of all affine invariant properties in terms of degrees of monomials in support of polynomials in family Rough characterization of locality of constraints, in terms of degrees. Rough characterization of locality of constraints, in terms of degrees. Infinitely many (new) properties … Infinitely many (new) properties … July 29, 2011 Invariance in Property Testing: EPFL 30

of 38 Results from [KS ‘08] Thm 1: If P is affine-invariant and has t-single orbit feature then it is (t, δ/t 3, δ)-locally testable. Thm 1: If P is affine-invariant and has t-single orbit feature then it is (t, δ/t 3, δ)-locally testable. Unifies previous algebraic tests with single proof. Unifies previous algebraic tests with single proof. Thm 2: If P is affine-invariant over K and has a single t-local constraint, then it is has a q-single orbit feature (for some q = q(K,t)) Thm 2: If P is affine-invariant over K and has a single t-local constraint, then it is has a q-single orbit feature (for some q = q(K,t)) (explains the AKKLR optimism) (explains the AKKLR optimism) Completely characterizes local testability of affine-invariant properties over vector spaces over small fields. Completely characterizes local testability of affine-invariant properties over vector spaces over small fields. July 29, 2011 Invariance in Property Testing: EPFL 31

of 38 Vector spaces over big fields? Most general case: Most general case: f : K  F m f : K  F m Most interesting cases Most interesting cases K = huge field; F, m small. K = huge field; F, m small. Reasons to study: Reasons to study: Broader class: Potential counterexamples to intuitive beliefs. Broader class: Potential counterexamples to intuitive beliefs. Include starting point for all LTCs (so far). Include starting point for all LTCs (so far). July 29, 2011 Invariance in Property Testing: EPFL 32

of 38 Subsequent results [GrigorescuKaufmanS’08]: 1 st Counterexample to AKKLR Conjecture (t-local constraint ≠ t-LDPC.) [GrigorescuKaufmanS’08]: 1 st Counterexample to AKKLR Conjecture (t-local constraint ≠ t-LDPC.) [GrigorescuKaufmanS.’09]: Single orbit characterization of some BCH (and other) codes. [GrigorescuKaufmanS.’09]: Single orbit characterization of some BCH (and other) codes. [Ben-SassonS.’11]: Limitations on rate of (O(1)- locally testable) affine-invariant codes. [Ben-SassonS.’11]: Limitations on rate of (O(1)- locally testable) affine-invariant codes. [Ben-SassonMaatoukShpilkaS.’11]: 2 nd counterexample to AKKLR (t-LDPC ≠ t-testable) [Ben-SassonMaatoukShpilkaS.’11]: 2 nd counterexample to AKKLR (t-LDPC ≠ t-testable) [above+Grigorescu’11]: Sums of SOC are SOC. [above+Grigorescu’11]: Sums of SOC are SOC. [KaufmanWigderson]: LDPC codes with invariance (not affine-invariant) [KaufmanWigderson]: LDPC codes with invariance (not affine-invariant) [Bhattacharyya et al.]: Affine-invariant non-linear properties. [Bhattacharyya et al.]: Affine-invariant non-linear properties. July 29, 2011 Invariance in Property Testing: EPFL 33

of 38 [BS’10]t-local constraint t-characterized Affine-invariance & testability May 23-28, 2011 Bertinoro: Testing Affine-Invariant Properties 34 t-locally testable t-S-O-C [KS’08] [GKS’08] [BMSS’11]weight-t degrees

of 38 Technical nature of questions Given: t points ® 1, …, ® t from K; Given: t points ® 1, …, ® t from K; and set of positive integers D, and set of positive integers D, When is the t x |D| generalized Vandermonde matrix with columns indexed by [t] and rows by D, with (i,d)th entry being ® i d, of full column rank? When is the t x |D| generalized Vandermonde matrix with columns indexed by [t] and rows by D, with (i,d)th entry being ® i d, of full column rank? Nice connections to symmetric polynomials, and we have new results (we think). Nice connections to symmetric polynomials, and we have new results (we think). July 29, 2011 Invariance in Property Testing: EPFL 35

of 38 Other Invariances [KaufmanWigderson]: LDPC codes with invariance (not affine-invariant; probably not LTC). [KaufmanWigderson]: LDPC codes with invariance (not affine-invariant; probably not LTC). [Bhattacharyya et al. ‘09…’11]: Linear-invariant non-linear properties. [Bhattacharyya et al. ‘09…’11]: Linear-invariant non-linear properties. July 29, 2011 Invariance in Property Testing: EPFL 36

of 38 Broad directions to consider What groups of invariances lead to testability? What groups of invariances lead to testability? Is there a subclass of affine-invariant codes that will lead to linear-rate LTCs? (n o(1) -locally testable with linear rate?) Is there a subclass of affine-invariant codes that will lead to linear-rate LTCs? (n o(1) -locally testable with linear rate?) (General program): (General program): To understand structure. To understand structure. To understand locality vs. structure. To understand locality vs. structure. To get new performance parameters. To get new performance parameters. In general … seek invariances In general … seek invariances July 29, 2011 Invariance in Property Testing: EPFL 37

of 38 July 29, 2011 Invariance in Property Testing: EPFL 38 Thanks

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