Algebraic Property Testing: A Survey Madhu Sudan MIT 1 1 April - - PowerPoint PPT Presentation
Algebraic Property Testing: A Survey Madhu Sudan MIT 1 1 April 1, 2009 April 1, 2009 Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS Algebraic Property Testing: Personal Perspective Madhu Sudan MIT 2 2 April
April 1, 2009 April 1, 2009 Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 1 1
Algebraic Property Testing: A Survey
Madhu Sudan MIT
April 1, 2009 April 1, 2009 Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 2 2
Algebraic Property Testing: Personal Perspective
Madhu Sudan MIT
April 1, 2009 April 1, 2009 Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 3 3
Algebraic Property Testing:
Perspective
Madhu Sudan MIT
April 1, 2009 April 1, 2009 Algebraic Property Testing @ DIMACS Algebraic Property Testing @ DIMACS 4 4
Property Testing Property Testing
Distance:
Definition:
Notes:
δ(f, g) = Prx∈D[f(x) 6= g(x)] δ(f, F) = ming∈F{δ(f, g)} f ≈² g if δ(f, g) ≤ ². F is (k, ², δ)-locally testable if ∃ a k-query tester T s.t. f ∈ F ⇒ T f accepts w.p. ≥ 1 − ² δ(f, F) ≥ δ ⇒ T f rejects w.p. ≥ ². k-locally testable implies ∃², δ > 0 locally testable implies ∃k = O(1) One-sided error: Accept f ∈ F w.p. 1
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Brief History Brief History
[ Blum,Luby,Rubinfeld Blum,Luby,Rubinfeld – – S S’ ’90] 90]
Linearity + application to program testing
[ Babai,Fortnow,Lund Babai,Fortnow,Lund – – F F’ ’90] 90]
Multilinearity + application to PCPs (MIP). + application to PCPs (MIP).
[ Rubinfeld+ S Rubinfeld+ S. .] ]
Low-
degree testing + Formal Definition Formal Definition
[ Goldreich,Goldwasser,Ron Goldreich,Goldwasser,Ron] ]
Graph property testing.
Since then … … many developments many developments
Graph properties
Statistical properties
More algebraic properties
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Specific Directions in Algebraic P.T. Specific Directions in Algebraic P.T.
More Properties
Low-
degree (d < q) functions [ RS RS] ]
Moderate-
degree (q < d < n) functions
q= 2: [ AKKLR AKKLR] ]
General q: [ KR, JPRZ KR, JPRZ] ]
Long code/ Dictator/ Junta testing [ PRS PRS] ]
BCH codes (Trace of low-
KL] ]
All nicely “ “ invariant invariant ” ” properties [ properties [ KS KS] ]
Better Parameters (motivated by PCPs).
# queries, high-
error, amortized query complexity, reduced randomness. complexity, reduced randomness.
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Contrast w . Com binatorial P.T. Contrast w . Com binatorial P.T.
Universe {f : D → R} Must accept Ok to accept Must reject w.h.p.
Algebraic Property = Code! (usually)
(Also usually) R is a field F Property = Linear subspace.
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Goal of this talk 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). codes (over small fields).
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Basic I m plications of Linearity [ BHR] Basic I m plications of Linearity [ BHR]
Generic adaptive test = decision tree. f(i) f(k) f(j) 1 1
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Basic I m plications of Linearity [ BHR] Basic I m plications of Linearity [ BHR]
Generic adaptive test = decision tree. f(i) f(k) f(j) 1 1
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Constraints, Characterizations Constraints, Characterizations
accepts hf(i1), . . . , f(ik)i ∈ V 6= Fk
1 D i1 i2 ik in V? 2
Every f ∈ F satisfies it.
then F characterized by constraints.
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Constraints, Characterizations Constraints, Characterizations
accepts hf(i1), . . . , f(ik)i ∈ V 6= Fk
1 D i1 i2 ik in V? 2
Every f ∈ F satisfies it.
then F characterized by constraints.
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Exam ple: Linearity Testing [ BLR] Exam ple: Linearity Testing [ BLR]
in V? x
y x+ y
f is linear iff ∀x, y, Cx,y satisfied
Cx,y = (x, y, x + y; V )|x, y ∈ Fn where V = {(a, b, a + b)|a, b ∈ F}
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I nsufficiency of local characterizations I nsufficiency of local characterizations
[ Ben-
Sasson, , Harsha Harsha, , Raskhodnikova Raskhodnikova] ]
There exist families characterized characterized by by k k-
local constraints constraints that are not that are not o(| D
| ) -
locally testable. .
Proof idea: Pick LDPC graph at random … … (and analyze resulting property) (and analyze resulting property)
F
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W hy are characterizations insufficient? W hy are characterizations insufficient?
Constraints too minimal.
Not redundant enough!
Proved formally in [ Ben-
Sasson, , Guruswami Guruswami, Kaufman, S., , Kaufman, S., Viderman Viderman] ]
Constraints too asymmetric.
Property must show some symmetry to be testable. testable.
Not a formal assertion … … just intuitive. just intuitive.
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Redundancy? Redundancy?
E.g. Linearity Test:
Standard LDPC analysis:
What natural operations create redundant local constraints? constraints?
Tensor Products!
− Dimension(F) ≈ D − m for m constraints. − Requires #constraints < D. − Does not allow much redundancy! − Ω(D2) constraints on domain D
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Tensor Products of Codes! Tensor Products of Codes!
Tensor Product:
Redundancy?
F × G
= { Matrices such every row in F and every column in G } Suppose F, G systematic First ` entries free rest determined by them.
Free
F determined G determined determined twice, by F and G!
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Testability of tensor product codes? Testability of tensor product codes?
Natural test:
Given Matrix M M
Test if random row in F F
Test if random column in G G
Claim:
If F, G F, G codes of constant (relative) distance; codes of constant (relative) distance; then if test accepts then if test accepts w.h.p w.h.p. then . then M M is close to is close to codeword of codeword of F x G F x G
Yields O( O( √ √n n) ) local test for codes of length local test for codes of length n n. .
Can we do better? Exploit local testability of F, F, G G? ?
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Robust testability of tensors? Robust testability of tensors?
Natural test (if F,G F,G locally testable): locally testable):
Given Matrix M M
Run Local Test for F F on random row
Run Local Test for G G on random column
Suppose M M close close on most rows/ columns to
F, G. Does this . Does this imply imply M M is is close close to to F x G F x G? ?
Generalizes test for bivariate bivariate polynomials. True for
F, G = class of low = class of low-
degree polynomials. [ BFLS, [ BFLS, Arora+ Safra Arora+ Safra, , Polishchuk+ Spielman Polishchuk+ Spielman] . ] .
General question raised by [ Ben [ Ben-
Sasson+ S.] .]
[ P. Valiant] Not true for every Not true for every F, G F, G ! !
[ Dinur Dinur, S., , S., Wigderson Wigderson] ] True if True if F F (or (or G G) locally testable. ) locally testable.
Test that random row close close to to F F
Test that random column close close to to G G
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Tensor Products and Local Testability Tensor Products and Local Testability
Robust testability allows easy induction (essentially from (essentially from [ BFL, BFLS] ; [ BFL, BFLS] ; see also see also [ Ben [ Ben-
Sasson+ S.] ) .] )
Natural test: Pick random axis-parallel line verify f|line ∈ F
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Robust testability of tensors ( contd.) Robust testability of tensors ( contd.)
Unnatural test (for F x F x F F x F x F): ):
Given 3-
d matrix M:
Pick random 2-
d submatrix submatrix. .
Verify it is close to F x F F x F
Theorem [ [ BenSasson+ S BenSasson+ S., based on ., based on Raz+ Safra Raz+ Safra] : ] : Distance to Distance to F x F x F F x F x F proportional to average proportional to average distance of random 2 distance of random 2-
d submatrix submatrix to to F x F F x F. .
[ Meir] : “ “ Linear Linear-
algebraic” ” construction of Locally construction of Locally Testable Codes (matching best known Testable Codes (matching best known parameters) using this (and many other parameters) using this (and many other ingredients). ingredients).
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Redundant Characterizations ( contd.) Redundant Characterizations ( contd.)
Redundant constraints necessary for testing [ [ BGKSV BGKSV] ]
How to get redundancy?
Tensor Products
Sufficient to get some local testability
Invariances (Symmetries) (Symmetries)
Sufficient?
Counting (See Tali Tali’ ’s s talk) talk)
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Testing by sym m etries Testing by sym m etries
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I nvariance & Property testing I nvariance & Property testing
Invariances ( ( Automorphism Automorphism groups): groups):
Hope: If Automorphism Automorphism group is group is “ “ large large” ” ( ( “ “ nice nice” ” ), ), then property is testable. then property is testable.
Aut(F) = {π | F is π-invariant} Forms group under composition. For permutation π : D → D, F is π-invariant if f ∈ F implies f ◦ π ∈ F.
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Exam ples Exam ples
Majority:
Graph Properties:
Algebraic Properties: What symmetries do they have? have?
− Easy Fact: If A’ ut(F) = SD then F is poly(R, 1/²)-locally testable. − Aut group = SD (full group). − Aut. group given by renaming of vertices − [AFNS, Borgs et al.] implies regular properties with this Aut group are testable.
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Algebraic Properties & Algebraic Properties & I nvariances I nvariances
Properties:
Automorphism groups? groups?
Question: Are Linear/ Affine-
Inv., Locally Characterized Props. Testable? ( Characterized Props. Testable? ( [ Kaufman + S.] ) [ Kaufman + S.] ) (Linear-Invariant)
D = Fn, R = F (Linearity, Low-degree, Reed-Muller) Or D = K ⊇ F, R = F (Dual-BCH) (K, F finite fields) Linear transformations of domain. π(x) = Ax where A ∈ Fn×n Affine transformations of domain. π(x) = Ax + b where A ∈ Fn×n, b ∈ Fn
(Affine-Inv.)
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Linear Linear-
I nvariance & Testability
Unifies previous studies on Alg. Prop. Testing. (And captures some new properties) (And captures some new properties)
Nice family of 2 2-
transitive group of symmetries. .
Conjecture [
[ Alon Alon, Kaufman, , Kaufman, Krivelevich Krivelevich, , Litsyn Litsyn, Ron] , Ron] :
: Linear code with Linear code with k k-
local constraint and and 2 2-
transitive group of symmetries must be must be testable testable. .
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Som e Results [ Kaufm an + S.] Som e Results [ Kaufm an + S.]
Theorem 1:
Theorem 2:
F ⊆ {Kn → F} linear, linear-invariant, k-locally characterized implies F is f(K, k)-locally testable. F ⊆ {Kn → F} linear, affine-invariant, has k-local constraint implies F is f(K, k)-locally testable.
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Exam ples of Linear Exam ples of Linear-
I nvariant Fam ilies
− Polynomials in F[x1, . . . , xn] of degree at most d − Traces of Poly in K[x1, . . . , xn] of degree at most d − F1 + F2, where F1, F2 are linear-invariant. Polynomials supported by degree 2, 3, 5, 7 monomials. − (Traces of) Homogenous polynomials of degree d − Linear functions from Fn to F.
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W hat Dictates Locality of Characterizations? W hat Dictates Locality of Characterizations?
− For affine-invariant family dictated (coarsely) by highest degree monomial in family − For some linear-invariant families, can be much less than the highest degree monomial. − Precise locality not yet understood: Depends on p-ary representation of degrees. Example: F supported by monomials xpi+pj behaves like degree two polynomial Example: K = F = F7; F = F1 + F2 F1 = poly of degree at most 16 F2 = poly supported on monomials of degree 3 mod 6. Degree(F) = Ω(n); Locality(F) ≤ 49.
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Property Testing from Property Testing from I nvariances I nvariances
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Key Notion: Form al Characterization Key Notion: Form al Characterization
− F has single-orbit characterization if ∃ a single constraint C = (x1, . . . , xk; V ) such that {C ◦ π}π∈Aut(F) characterize F. Theorem: If F has single-orbit characterization by a k-local constraint (with some restrictions) then it is k-locally testable.
Rest of talk: Analysis (extending BLR)
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BLR Analysis: Outline BLR Analysis: Outline
− Step 0: Prove f close to g − Step 2: Prove that g is in F.
− Step 1: Prove most likely is overwhelming majority.
Want to show f close to some g ∈ F.
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BLR Analysis: Step 0 BLR Analysis: Step 0
− Prx,y[linearity test rejects |x ∈ B] ≥ 1
2
− If x 6∈ B then f(x) = g(x) ⇒ Prx[x ∈ B] ≤ 2δ Claim: Prx[f(x) 6= g(x)] ≤ 2δ − Let B = {x| Pry[f(x) 6= f(x + y) − f(y)] ≥ 1
2}
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BLR Analysis: Step 1 BLR Analysis: Step 1
Votex(y)
Presumably, only one leads to linear x, so which one?
then need to rule out this case. Lemma: ∀ x, Pry,z[Votex(y) 6= Votex(z))] ≤ 4δ
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BLR Analysis: Step 1 BLR Analysis: Step 1
Votex(y)
Presumably, only one leads to linear x, so which one?
then need to rule out this case. Lemma: ∀ x, Pry,z[Votex(y) 6= Votex(z))] ≤ 4δ
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BLR Analysis: Step 1 BLR Analysis: Step 1
? Lemma: ∀ x, Pry,z[Votex(y) 6= Votex(z))] ≤ 4δ Votex(y) f(y) −f(x + y) f(z) f(y + z) −f(y + 2z) −f(x + z) −f(2y + z) f(x + 2y + 2z)
sum non-zero ≤ δ.
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BLR Analysis: Step 1 BLR Analysis: Step 1
? Lemma: ∀ x, Pry,z[Votex(y) 6= Votex(z))] ≤ 4δ Votex(y) f(y) −f(x + y) f(z) f(y + z) −f(y + 2z) −f(x + z) −f(2y + z) f(x + 2y + 2z)
sum non-zero ≤ δ.
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BLR Analysis: Step 2 ( Sim ilar) BLR Analysis: Step 2 ( Sim ilar)
Lemma: If δ <
1 20, then ∀ x, y, g(x) + g(y) = g(x + y)
sum non-zero ≤ 4δ. g(x) g(y) −g(x + y) f(z) f(y + z) −f(y + 2z) −f(x + z) −f(2y + z) f(x + 2y + 2z)
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Our Analysis: Outline Our Analysis: Outline
Step 1: Prove Step 1: Prove “ “ most likely most likely” ” is overwhelming majority. is overwhelming majority.
−
Pr{L|L(x1)=x}[hα, f(L(x2)), . . . , f(L(xk))i ∈ V ] − Step 0: Prove f close to g − Step 2: Prove that g is in F.
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Our Analysis: Outline Our Analysis: Outline
Step 1: Prove Step 1: Prove “ “ most likely most likely” ” is overwhelming majority. is overwhelming majority.
− Same as before
Pr{L|L(x1)=x}[hα, f(L(x2)), . . . , f(L(xk))i ∈ V ]
− Step 0: Prove f close to g − Step 2: Prove that g is in F.
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Matrix Magic? Matrix Magic?
· · · Votex(L) Lemma: ∀ x, PrL,K[Votex(L) 6= Votex(K))] ≤ 2(k − 1)δ
Pr{L|L(x1)=x}[hα, f(L(x2)), . . . , f(L(xk))i ∈ V ] K(x2) . . . x L(x2) L(xk) K(xk)
43 43
Matrix Magic? Matrix Magic?
· · ·
K(x2) K(xk) . . . x L(x2) L(xk)
and rest dependent on them.
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Matrix Magic? Matrix Magic?
Fill with random entries
Fill so as to form constraints
Tensor magic implies final columns are also constraints. columns are also constraints.
and rest dependent on them. K(xk) K(x2) x L(x2) L( xk) ` ` · · · . . .
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Matrix Magic? Matrix Magic?
Fill with random entries
Fill so as to form constraints
Tensor magic implies final columns are also constraints! columns are also constraints!
and rest dependent on them. K(xk) K(x2) x L(x2) L(xk) ` ` · · · . . .
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Sum m arizing Sum m arizing
Affine invariance + single-
imply testing. imply testing.
Unifies analysis of linearity test, basic low-
degree tests, moderate tests, moderate-
degree test (all A.P.T. except dual dual-
BCH?)
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Concluding thoughts Concluding thoughts -
1
Didn’ ’t get to talk about t get to talk about
PCPs, LTCs LTCs (though we did implicitly) (though we did implicitly)
Optimizing parameters
Parameters
In general
Broad reasons why property testing works worth examining. worth examining.
Tensoring explains a few algebraic examples. explains a few algebraic examples.
Invariance explains many other algebraic ones. (More about (More about invariances invariances in in [ [ Grigorescu,Kaufman,S Grigorescu,Kaufman,S. . ’ ’08] , [ GKS 08] , [ GKS’ ’09] ) 09] )
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Concluding thoughts Concluding thoughts -
2
Invariance:
Seems to be a nice lens to view all property testing results (combinatorial, statistical, testing results (combinatorial, statistical, algebraic). 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.
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