A UNIFIED FRAMEWORK FOR TESTING LINEARINVARIANT PROPERTIES ARNAB
A UNIFIED FRAMEWORK FOR TESTING LINEARINVARIANT PROPERTIES ARNAB BHATTACHARYYA CSAIL, MIT (JOINT WORK WITH ELENA GRIGORESCU AND ASAF SHAPIRA)
Property Testing Does the object have a given property P or is it e-far from having P ? Input Object Queries an e-fraction of the representation of the object needs to be modified P is (one-sided) testable if the number of queries needed to always accept positive inputs and reject negative inputs with probability >90% can be made independent of size of the input.
Properties of Functions Origins of property testing in testing algebraic properties for program checking & PCP’s [Blum-Luby-Rubinfeld ‘ 93, Rubinfeld-Sudan ‘ 96] Input objects are functions on a vector space Distance of function to property P measured by smallest Hamming distance to evaluation table of a function satisfying P
Properties of Boolean Functions For this talk, focus on Boolean functions on the hypercube f: F 2 n → {0, 1} Examples of testable properties of Boolean functions: Is function f: F 2 n → F 2 linear, i. e. f(x+y)=f(x)+f(y) for all x, y? [BLR ’ 93] More generally, is it of degree at most d? [Alon-Kaufman. Krivelevich-Litsyn-Ron ‘ 03] Fourier dimensionality and sparsity [Gopalan-O’Donnell. Servedio-Shpilka-Wimmer ‘ 09] What are all the testable algebraic properties? Want the “shortest” explanation for testability.
(Dense) Graph Properties Graph properties are invariant with respect to vertex relabelings. 1 3 2 4 5 Input graph represented by its adjacency matrix Distance to property P measured by smallest Hamming distance to adjacency matrix of a graph satisfying P. Examples: bipartiteness, 3 -colorability, triangle-freeness, … [Goldreich-Goldwasser-Ron ‘ 98]
Testability of Graph Properties All hereditary graph properties are testable with one-sided error. [Alon-Shapira ‘ 05] P is hereditary if for any graph G satisfying P, every induced subgraph of G also satisfies P. “All” testable properties (with one-sided error) are hereditary! Full characterization given by [Alon-Fischer. Newman-Shapira ’ 06], [Borgs-Chayes-Lovasz-Sos -Szegedy-Vesztergombi ’ 06]
Forbidden Induced Subgraphs Given fixed collection of graphs F, a graph G is said to be F-free if G does not contain any graph in F as an induced subgraph. Bipartiteness: F is infinite A graph property is hereditary iff it is equal to Ffreeness for some collection of graphs F.
Linear Invariance [Kaufman-Sudan ‘ 07] observed that most natural properties of Boolean functions invariant under linear transformations of domain If f: F 2 n → {0, 1} in property P, then f o L also in P for every linear map L: F 2 n → F 2 n [KS ‘ 07] showed testability for linear-invariant properties if they formed a subspace and are “locally characterized” Challenge to characterize all linear-invariant testable properties [Sudan ‘ 10]
Subspace Hereditariness Linear-invariant property P is subspacehereditary if: for any function f: F 2 n → {0, 1} satisfying P, restriction of f to any linear subspace of F 2 n also satisfies P.
Our Main Conjecture All subspace-hereditary linear-invariant properties are testable.
Implied Characterization Implication: A linear-invariant property is onesided testable “iff” it is subspace-hereditary Restriction to testers whose behavior doesn’t depend on value of n “Only if” direction is a theorem [BGS 10], not conjecture. Shows importance of notion of subspacehereditariness.
Progress towards conjecture All subspace-hereditary linear-invariant properties are testable We show testability of a large subclass of subspacehereditary properties Those characterized by forbidding solutions to systems of equations of complexity 1 Technique: constructing robust arithmetic regularity lemmas Proof of full conjecture along similar lines would depend on developing arithmetic regularity lemmas with respect to higher-order Gowers norms over F 2.
Forbidden Linear System Given m-by-k matrix M over F 2, say subset S of F 2 n is M-free if there is no x = (x 1, …, xk) with each xi in S such that Mx = 0. Always a monotone property Example: If M=[1 1 1], then M-freeness is property of having no x, y, x+y all in the set
Forbidden “Induced” Linear System Given m-by-k matrix M over F 2 and a binary string s in {0, 1}k, say function f: F 2 n → {0, 1} is (M, s)-free if there is no x = (x 1, …, xk) with each xi in F 2 n and Mx = 0, such that: f(xi) = si for all i in [k] Example: With m=1, k=3, M=[1 1 1] and s=001, (M, s)-freeness is property of having no x, y with f(x)=f(y)=0 and f(x+y)=1.
Forbidden Family of Linear Systems Given fixed collection F = {(M 1, s 1), (M 2, s 2), …}, a function f: F 2 n → {0, 1} is F-free if it is (Mi, si)free for every i. Example: If M=[1 1 1], s 1=111 and s 2=001 and F={(M, s 1), (M, s 2)}, then F-freeness is linearity No x, y with f(x) + f(y) + f(x+y) = 1 Similarly for Reed-Muller codes
Forbidden Family of Linear Systems Given fixed (possibly infinite) collection F = {(M 1, s 1), (M 2, s 2), …}, a function f: F 2 n → {0, 1} is F-free if it is (Mi, si)-free for every i. Property may no longer be “locally characterized”, a requirement in [Kaufman-Sudan ‘ 07] Example: ODD-CYCLE-FREENESS (to be discussed tomorrow by Asaf)
Why forbidden linear systems? Fact: Property P is characterized by F-freeness for some collection F iff it is a subspacehereditary linear-invariant property
Why forbidden linear systems? Fact: Property P is characterized by F-freeness iff it is a subspace-hereditary linear-invariant property Property being subspace-hereditary means certain restrictions to subspaces are forbidden. Linear systems encode these subspaces, pattern strings encode the forbidden restrictions on them
Our Main Conjecture F -freeness is testable, for any fixed collection F.
Our Main Result F-freeness is testable, where F= {(M 1, s 1), (M 2, s 2), …} is possibly infinite, each si is arbitrary, and each Mi is of complexity 1.
Complexity of Linear Systems Introduced by [Green-Tao ‘ 06]. Also called “Cauchy-Schwarz complexity” [Gowers-Wolf ‘ 07]. Every system of equations assigned a complexity. Exact definition unimportant for purposes of this talk. Any system of rank at most 2 is of complexity 1 Linear systems used to define RM codes of order d have complexity d
Our Main Result F-freeness is testable, where F= {(M 1, s 1), (M 2, s 2), …} is possibly infinite, each si is not necessarily all-ones, and each Mi is of complexity 1. Linearity is testable…once again Price of generality: bound on the query complexity is extremely weak in terms of distance parameter (tower of exponentials)
Previous Work Testability results: ‘ 05]: (M, s)-freeness for M with rank 1 and s is all-ones. [B. -Chen-Sudan-Xie ‘ 09]: (M, s)-freeness for M of complexity 1 and s is all-ones [Kràl’-Serra-Vena ‘ 09, Shapira ‘ 09]: F-freeness where F is finite collection, each M of arbitrary complexity but each s still all-ones [Green
Regularity Partitioning Restriction not “pseudorandom” F 2 Restriction “pseudorandom” n H [G ‘ 05]: Can choose H such that very few shifts are red, and # of cosets independent of n. Say f is “pseudorandom” if it does not correlate well with any nonzero linear function.
Green’s Regularity Lemma For every e, given function f: F 2 n → {0, 1}, there is a subspace H of codimension at most T(e) such that f. H+g is not e-regular for < e 2 n many shifts g. e-regular: correlation with every nonzero linear function at most e.
Regularity Lemma: Functional version Actual statement used in the proof more complicated
One-sided testers and hereditariness A tester T is oblivious if it inspects a uniformly chosen random subspace and then acts the same independent of the value of n First Theorem: Any linear-invariant property that is one-sided testable by an oblivious tester is <e semi-subspace-hereditary. } condition is without loss of generality Semi-subspace-hereditary property Subspace-hereditary property
Other Open Questions Testability over other fields? Testability of non-Boolean functions? Are there better query complexity upper bounds, even for Green’s problem? Best lower bound only poly(1/e) [B. -Xie ’ 10] Characterization with respect to other invariance groups?
Thanks!
- Slides: 31