Algebraic Property Testing A Survey Madhu Sudan MIT
Algebraic Property Testing: A Survey Madhu Sudan MIT April 1, 2009 Algebraic Property Testing @ DIMACS 1
Algebraic Property Testing: Personal Perspective Madhu Sudan MIT April 1, 2009 Algebraic Property Testing @ DIMACS 2
Algebraic Property Testing: Personal Perspective Madhu Sudan MIT April 1, 2009 Algebraic Property Testing @ DIMACS 3
Property Testing n n 6 g(x)] ±(f ; g) = Pr x 2 D [f (x) = Distance: ±(f ; F ) = ming 2 F f ±(f ; g)g f ¼² g if ±(f ; g) · ². Definition F is (k; ²; : ±) locally t est able if 9 a k query t est er T s. t. T f accept s w. p. ¸ 1 ¡ ² f 2 F ) T f reject s w. p. ¸ ². ±(f ; F ) ¸ ± ) n k locally t est able implies 9²; ± > 0 Notes : locally t est able implies 9 k = O(1) One sided error: Accept f 2 F w. p. 1 April 1, 2009 Algebraic Property Testing @ DIMACS 4
Brief History n n n [Blum, Luby, Rubinfeld – S’ 90] n Linearity + application to program testing [Babai, Fortnow, Lund – F’ 90] n Multilinearity + application to PCPs (MIP). [Rubinfeld+S. ] n Low-degree testing + Formal Definition [Goldreich, Goldwasser, Ron] n Graph property testing. Since then … many developments n Graph properties n Statistical properties n More algebraic properties April 1, 2009 Algebraic Property Testing @ DIMACS 5
Specific Directions in Algebraic P. T. n n More Properties n Low-degree (d < q) functions [RS] n Moderate-degree (q < d < n) functions n q=2: [AKKLR] n General q: [KR, JPRZ] n Long code/Dictator/Junta testing [PRS] n BCH codes (Trace of low-deg. poly. ) [KL] n All nicely “invariant” properties [ KS] Better Parameters (motivated by PCPs). n #queries, high-error, amortized query complexity, reduced randomness. April 1, 2009 Algebraic Property Testing @ DIMACS 6
Contrast w. Combinatorial P. T. Universe (Also usually) R is a ¯eld F Property = Linear subspace. f f : D ! Rg F F Must accept Ok t o accept Must reject w. h. p. Algebraic Property = Code! (usually) April 1, 2009 Algebraic Property Testing @ DIMACS 7
Goal of this talk n n n Implications of linearity n Constraints, Characterizations, LDPC structure n One-sided error, Non-adaptive tests [BHR] Redundancy of Constraints n Tensor Product Codes Symmetries of Code n Testing affine-invariant codes n Yields basic tests for all known algebraic codes (over small fields). April 1, 2009 Algebraic Property Testing @ DIMACS 8
Basic Implications of Linearity [BHR[ n Generic adaptive test = decision tree. f(i) 0 f(k) f(j) ² Pick pat h followed by random g 2 F. ² Query f according t o pat h. 1 0 1 ² Accept i® f on pat h consist ent wit h some h 2 F. ² Yields non adapt ive one sided error t est for linear F. April 1, 2009 Algebraic Property Testing @ DIMACS 9
Basic Implications of Linearity [BHR[ n Generic adaptive test = decision tree. f(i) 0 f(k) f(j) ² Pick pat h followed by random g 2 F. ² Query f according t o pat h. 1 0 1 ² Accept i® f on pat h consist ent wit h some h 2 F. ² Yields non adapt ive one sided error t est for linear F. April 1, 2009 Algebraic Property Testing @ DIMACS 10
Constraints, Characterizations ² Say t est queries i 1 ; : : : ; i k 6 Fk accept s hf (i 1 ); : : : ; f (i k )i 2 V = ² (i 1 ; : : : ; i k ; V ) = Const raint Every f 2 F sat is¯es it. 1 i 2 ² If every f 2 6 F reject ed w. posit ive prob. t hen F charact erized by const raint s. • Like LDPC Codes! April 1, 2009 i 1 2 in V? ik D Algebraic Property Testing @ DIMACS 11
Constraints, Characterizations ² Say t est queries i 1 ; : : : ; i k 6 Fk accept s hf (i 1 ); : : : ; f (i k )i 2 V = ² (i 1 ; : : : ; i k ; V ) = Const raint Every f 2 F sat is¯es it. 1 i 2 ² If every f 2 6 F reject ed w. posit ive prob. t hen F charact erized by const raint s. • Like LDPC Codes! April 1, 2009 i 1 2 in V? ik D Algebraic Property Testing @ DIMACS 12
Example: Linearity Testing [BLR[ • Constraints: Cx ; y = (x; y; x + y; V )jx; y 2 Fn where V = f (a; b; a + b)ja; b 2 Fg • Characterization: f is linear i® 8 x; y; Cx ; y sat is¯ed x in V? y x+y April 1, 2009 Algebraic Property Testing @ DIMACS 13
Insufficiency of local characterizations n [Ben-Sasson, Harsha, Raskhodnikova] F n n 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) April 1, 2009 Algebraic Property Testing @ DIMACS 14
Why are characterizations insufficient? n n Constraints too minimal. n Not redundant enough! n Proved formally in [Ben-Sasson, Guruswami, Kaufman, S. , Viderman] Constraints too asymmetric. n Property must show some symmetry to be testable. n Not a formal assertion … just intuitive. April 1, 2009 Algebraic Property Testing @ DIMACS 15
Redundancy? n E. g. Test: ¡ Linearity raint s on domain D (D 2 ) const n Standard LDPC analysis: n ¡ Dimension(F ) ¼ D ¡ m for m const raint s. ¡ Requires # const raint s < D. ¡ Does not allow much redundancy! What natural operations create redundant local constraints? n Tensor Products! April 1, 2009 Algebraic Property Testing @ DIMACS 16
Tensor Products of Codes! F £ G n Tensor Product: = f Mat rices such every row in F and every column in G g n Redundancy? Suppose F , G syst emat ic First ` ent ries free rest det ermined by t hem. Free F det ermined G det ermined twice, by F and G! April 1, 2009 Algebraic Property Testing @ DIMACS 17
Testability of tensor product codes? n n n Natural test: n Given Matrix M n Test if random row in F n Test if random column in G Claim: n 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. n Can we do better? Exploit local testability of F, G? April 1, 2009 Algebraic Property Testing @ DIMACS 18
Robust testability of tensors? n n Natural test (if F, G locally testable): n Given Matrix M n Run for F on random • Test. Local that. Test random row closerow to F n Run for G on randomclose column • Test. Local that. Test random column to G Suppose M close on most rows/columns to F, G. Does this imply M is close to F x G? n Generalizes test for bivariate polynomials. True for F, G = class of low-degree polynomials. [BFLS, Arora+Safra, Polishchuk+Spielman]. n General question raised by [Ben-Sasson+S. ] n [P. Valiant] Not true for every F, G ! n [Dinur, S. , Wigderson] True if F (or G) locally testable. April 1, 2009 Algebraic Property Testing @ DIMACS 19
Tensor Products and Local Testability n Robust testability allows easy induction (essentially from [BFL, BFLS]; see also [Ben. Sasson+S. ]) ² Let F n = n fold t ensor of F. ² Given f : D n ! F Nat ural t est : Pick random axis parallel line verify f j l i ne 2 F April 1, 2009 Algebraic Property Testing @ DIMACS 20
Robust testability of tensors (contd(. n n n Unnatural test (for F x F): n Given 3 -d matrix M: n Pick random 2 -d submatrix. n Verify it is close to F x F Theorem [Ben. Sasson+S. , based on Raz+Safra]: Distance to 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 Testing @ DIMACS 21
Redundant Characterizations (contd(. n n Redundant constraints necessary for testing [BGKSV] How to get redundancy? n Tensor Products n Sufficient to get some local testability n n Invariances (Symmetries) n Sufficient? Counting (See Tali’s talk) April 1, 2009 Algebraic Property Testing @ DIMACS 22
Testing by symmetries April 1, 2009 Algebraic Property Testing @ DIMACS 23
Invariance & Property testing n n Invariances (Automorphism groups: ( For permut at ion ¼: D ! D , F is ¼ invariant if f 2 F implies f ± ¼ 2 F. Aut (F ) = f ¼ j F is ¼ invariant g Forms group under composit ion. Hope: If Automorphism group is “large” (“nice”), then property is testable. April 1, 2009 Algebraic Property Testing @ DIMACS 24
Examples n Majority: n Graph Properties: ¡ Aut. group given by renaming of vert ices ¡ Aut group = SD (full group). ¡ Easy Fact : If A’ut (F ) = SD t hen F is poly(R; 1=²) locally t est able. ¡ [AFNS, Borgs et al. ] implies regular propert ies wit h t his Aut group are t est able. n Algebraic Properties: What symmetries do they have? April 1, 2009 Algebraic Property Testing @ DIMACS 25
Algebraic Properties & Invariances n Properties: D = Fn , R = F (Linearity, Low degree, Reed Muller) Or D = K ¶ F, R = F (Dual BCH) (K ; F ¯nit e ¯elds) n Automorphism groups? Linear t ransformat ions of domain. ¼(x) = Ax where A 2 Fn £ n (Linear-Invariant) A± ne t ransformat ions of domain. ¼(x) = Ax + b where A 2 Fn £ n ; b 2 Fn n (Affine-Inv. ) Question: Are Linear/Affine-Inv. , Locally Characterized Props. Testable? ([Kaufman + S. ]) April 1, 2009 Algebraic Property Testing @ DIMACS 26
Linear-Invariance & Testability n n n 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 Testing @ DIMACS 27
Some Results [Kaufman + S[. n F µ f K n ! Fg linear, linear invariant , Theorem 1: k locally charact erized implies F is f (K ; k) locally t est able. n F µ f K n ! Fg linear, a± ne invariant , Theorem 2: has k local constraint implies F is f (K ; k) locally t est able. April 1, 2009 Algebraic Property Testing @ DIMACS 28
Examples of Linear-Invariant Families ¡ Linear funct ions from Fn t o F. ¡ Polynomials in F[x 1 ; : : : ; x n ] of degree at most d ¡ Traces of Poly in K [x 1 ; : : : ; x n ] of degree at most d ¡ (Traces of ) Homogenous polynomials of degree d ¡ F 1 + F 2 , where F 1 , F 2 are linear invariant. Polynomials support ed by degree 2; 3; 5; 7 monomials. April 1, 2009 Algebraic Property Testing @ DIMACS 30
What Dictates Locality of Characterizations? ¡ Precise locality not yet underst ood: Depends on p ary represent at ion of degrees. Example: F support ed by monomials x pi + pj behaves like degree two polynomial ¡ For a± ne invariant family dict at ed (coarsely) by highest degree monomial in family ¡ For some linear invariant families, can be much less t han t he highest degree monomial. Example: K = F 7 ; F = F 1 + F 2 F 1 = poly of degree at most 16 F 2 = poly support ed on monomials of degree 3 mod 6. Degree(F ) = (n); Locality(F ) · 49. April 1, 2009 Algebraic Property Testing @ DIMACS 31
Property Testing from Invariances April 1, 2009 Algebraic Property Testing @ DIMACS 33
Key Notion: Formal Characterization ¡ F has single orbit charact erizat ion if 9 a single const raint C = (x 1 ; : : : ; x k ; V ) such t hat f C ± ¼g¼ 2 A ut ( F ) charact erize F. T heorem: If F has single orbit charact erizat ion by a k local const raint (wit h some rest rict ions) t hen it is k locally t est able. Rest of talk: Analysis (extending BLR) April 1, 2009 Algebraic Property Testing @ DIMACS 34
BLR Analysis: Outline ² Have f s. t. Pr x ; y [f (x) + f (y) = 6 f (x + y)] = ± < 1=20. Want t o show f close t o some g 2 F. ² De¯ne g(x) = most likely f f (x + y) ¡ f (y)g. y ² If f close t o F t hen g will be in F and close t o f. ² But if f not close? g may not even be uniquely de¯ned! ² St eps: ¡ St ep 0: Prove f close t o g ¡ St ep 1: Prove most likely is overwhelming majority. ¡ St ep 2: Prove t hat g is in F. April 1, 2009 Algebraic Property Testing @ DIMACS 36
BLR Analysis: Step 0 ² De¯ne g(x) = most likely y f f (x + y) ¡ f (y)g. = g(x)] · 2± Claim: Pr x [f (x) 6 = f (x + y) ¡ f (y)] ¸ ¡ Let B = f xj Pr y [f (x) 6 ¡ Pr x ; y [linearity t est reject s jx 2 B ] ¸ 1 g 2 1 2 ) Pr x [x 2 B ] · 2± ¡ If x 2 6 B t hen f (x) = g(x) April 1, 2009 Algebraic Property Testing @ DIMACS 37
Vot ex (y) BLR Analysis: Step 1 ² De¯ne g(x) = most likely y f f (x + y) ¡ f (y)g. ² Suppose for some x, 9 two equally likely values. Presumably, only one leads t o linear x, so which one? ² If we wish t o show g linear, t hen need t o rule out t his case. Lemma: 8 x, Pr y ; z [Vot ex (y) = 6 Vot ex (z))] · 4± April 1, 2009 Algebraic Property Testing @ DIMACS 38
Vot ex (y) BLR Analysis: Step 1 ² De¯ne g(x) = most likely y f f (x + y) ¡ f (y)g. ² Suppose for some x, 9 two equally likely values. Presumably, only one leads t o linear x, so which one? ² If we wish t o show g linear, t hen need t o rule out t his case. Lemma: 8 x, Pr y ; z [Vot ex (y) = 6 Vot ex (z))] · 4± April 1, 2009 Algebraic Property Testing @ DIMACS 39
Vot ex (y) BLR Analysis: Step 1 ² De¯ne g(x) = most likely y f f (x + y) ¡ f (y)g. = Vot ex (z))] · 4± Lemma: 8 x, Pr y ; z [Vot ex (y) 6 ? f (z) f (y) ¡ f (x + y) f (y + z) ¡ f (y + 2 z) ¡ f (x + z) ¡ f (2 y + z) f (x + 2 y + 2 z) Prob. Row/ column sum non zero · ±. April 1, 2009 Algebraic Property Testing @ DIMACS 40
Vot ex (y) BLR Analysis: Step 1 ² De¯ne g(x) = most likely y f f (x + y) ¡ f (y)g. = Vot ex (z))] · 4± Lemma: 8 x, Pr y ; z [Vot ex (y) 6 ? f (z) f (y) ¡ f (x + y) f (y + z) ¡ f (y + 2 z) ¡ f (x + z) ¡ f (2 y + z) f (x + 2 y + 2 z) Prob. Row/ column sum non zero · ±. April 1, 2009 Algebraic Property Testing @ DIMACS 41
BLR Analysis: Step 2 (Similar( Lemma: If ± < g(x) f (z) 1 , 20 t hen 8 x; y, g(x) + g(y) = g(x + y) g(y) ¡ g(x + y) Prob. Row/ column sum non zero · 4±. f (y + z) ¡ f (y + 2 z) ¡ f (x + z) ¡ f (2 y + z) f (x + 2 y + 2 z) April 1, 2009 Algebraic Property Testing @ DIMACS 42
Our Analysis: Outline ² f s. t. Pr L [hf (L (x 1 ); : : : ; f (L (x k ))i 2 V ] = ± ¿ 1. ² De¯ne g(x) = ® t hat maximizes Pr f L j L ( x ) = x g [h®; f (L (x 2 )); : : : ; f (L (x k ))i 2 V ] 1 ² St eps: ¡ St ep 0: Prove f close t o g ¡ Step 1: Prove “most likely” is overwhelming majority. ¡ St ep 2: Prove t hat g is in F. April 1, 2009 Algebraic Property Testing @ DIMACS 43
Our Analysis: Outline ² f s. t. Pr L [hf (L (x 1 ); : : : ; f (L (x k ))i 2 V ] = ± ¿ 1. ² De¯ne g(x) = ® t hat maximizes Pr f L j L ( x ) = x g [h®; f (L (x 2 )); : : : ; f (L (x k ))i 2 V ] 1 Same as before ² St eps: ¡ St ep 0: Prove f close t o g ¡ Step 1: Prove “most likely” is overwhelming majority. ¡ St ep 2: Prove t hat g is in F. April 1, 2009 Algebraic Property Testing @ DIMACS 44
Vot ex (L ) Matrix ? ² De¯ne Magic g(x) = ® t hat maximizes Pr f L j L ( x = 1) xg [h®; f (L (x 2 )); : : : ; f (L (x k ))i 2 V ] 6 Vot ex (K ))] · 2(k ¡ 1)± Lemma: 8 x, Pr L ; K [Vot ex (L ) = x K (x 2 ). . . L (x 2 ) ? ¢¢¢ L (x k ) K (x k ) April 1, 2009 Algebraic Property Testing @ DIMACS 45
Matrix Magic? x K (x 2 ). . . L (x 2 ) ? ¢¢¢ L (x k ) K (x k ) ² Want marked rows t o be random const raint s. ² Suppose x 1 ; : : : ; x ` linearly independent ; and rest dependent on t hem. 46
n Matrix Magic? n ` x ` L (x 2 ) n ¢¢¢ Fill with random entries Fill so as to form constraints Tensor magic implies final columns are also constraints. L (x k ) K (x 2 ). . . K (x k ) ² Suppose x 1 ; : : : ; x ` linearly independent ; and rest dependent on t hem. April 1, 2009 47
n Matrix Magic? n ` x ` L (x 2 ) n ¢¢¢ Fill with random entries Fill so as to form constraints Tensor magic implies final columns are also constraints! L (x k ) K (x 2 ). . . K (x k ) ² Suppose x 1 ; : : : ; x ` linearly independent ; and rest dependent on t hem. April 1, 2009 48
Summarizing n n 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 Testing @ DIMACS 49
Concluding thoughts - 1 n n Didn’t get to talk about n PCPs, LTCs (though we did implicitly) n Optimizing parameters n Parameters In general n Broad reasons why property testing works worth examining. n Tensoring explains a few algebraic examples. n Invariance explains many other algebraic ones. (More about invariances in [Grigorescu, Kaufman, S. ’ 08], [GKS’ 09]) April 1, 2009 Algebraic Property Testing @ DIMACS 50
Concluding thoughts - 2 n Invariance: n Seems to be a nice lens to view all property testing results (combinatorial, statistical, algebraic). n Many open questions: n What groups of symmetries aid testing? n What additional properties needed? n n n Local constraints? Linearity? Does sufficient symmetry imply testability? n April 1, 2009 Give an example of a non-testable property with a ksingle orbit characterization. Algebraic Property Testing @ DIMACS 51
Thank You! April 1, 2009 Algebraic Property Testing @ DIMACS 52
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