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18
Testing low complexity affineinvariant properties
 In Khanna [Kha13
"... Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affineinvariant property of multivariate functions over finite fields is testable with a constant n ..."
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Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affineinvariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, for instance, that the ReedMuller code over Fp of degree d < p is testable, with an argument that uses no detailed algebraic information about polynomials, except that having low degree is preserved by composition with affine maps. The complexity of an affineinvariant property P refers to the maximum complexity, as defined by Green and Tao (Ann. Math. 2008), of the sets of linear forms used to characterize P. A more precise statement of our main result is that for any fixed prime p ≥ 2 and fixed integer R ≥ 2, any affineinvariant property P of functions f: F n p → [R] is testable, if the complexity of the property is less than p. Our proof involves developing analogs of graphtheoretic techniques in an algebraic setting, using tools from higherorder Fourier analysis. 1
On Sums of Locally Testable Affine Invariant Properties
"... Affineinvariant properties are an abstract class of properties that generalize some central algebraic ones, such as linearity and lowdegreeness, that have been studied extensively in the context of property testing. Affine invariant properties consider functions mapping a big field Fqn to the sub ..."
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Cited by 8 (6 self)
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Affineinvariant properties are an abstract class of properties that generalize some central algebraic ones, such as linearity and lowdegreeness, that have been studied extensively in the context of property testing. Affine invariant properties consider functions mapping a big field Fqn to the subfield Fq and include all properties that form an Fqvector space and are invariant under affine transformations of the domain. Almost all the known locally testable affineinvariant properties have socalled “singleorbit characterizations ” — namely they are specified by a single local constraint on the property, and the “orbit ” of this constraint, i.e., translations of this constraint induced by affineinvariance. Singleorbit characterizations by a local constraint are also known to imply local testability. Despite this prominent role in local testing for affineinvariant properties, singleorbit characterizations are not wellunderstood. In this work we show that properties with singleorbit characterizations are closed under “summation”. Such a closure does not follow easily from definitions, and our proof uses some of the rich developing theory of affineinvariant properties. To complement this result, we also show that the property of being an nvariate lowdegree polynomial over Fq has a singleorbit
Every locally characterized affineinvariant property is testable
, 2013
"... Let F = Fp for any fixed prime p> 2. An affineinvariant property is a property of functions on Fn that is closed under taking affine transformations of the domain. We prove that all affineinvariant properties that have local characterizations are testable. In fact, we give a proximityoblivious ..."
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Let F = Fp for any fixed prime p> 2. An affineinvariant property is a property of functions on Fn that is closed under taking affine transformations of the domain. We prove that all affineinvariant properties that have local characterizations are testable. In fact, we give a proximityoblivious test for any such property P, meaning that given an input function f, we make a constant number of queries to f, always accept if f satisfies P, and otherwise reject with probability larger than a positive number that depends only on the distance between f and P. More generally, we show that any affineinvariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable. We also prove that any property that can be described as the property of being decomposable into a known structure of lowdegree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degreed polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degreestructural properties and are therefore locally characterized. Our results use a new Gowers inverse theorem by Tao and Ziegler for low characteristic fields that decomposes any polynomial with large Gowers norm into a function of a small number of lowdegree nonclassical polynomials. We establish a new equidistribution result for high rank nonclassical polynomials that drives the proofs of both the testability results and the local characterization of degreestructural properties.
Improved Lower Bounds for Testing Trianglefreeness in Boolean Functions via Fast Matrix Multiplication
, 2013
"... Understanding the query complexity for testing linearinvariant properties has been a central open problem in the study of algebraic property testing. Trianglefreeness in Boolean functions is a simple property whose testing complexity is unknown. Three Boolean functions f1, f2 and f3: Fk2 → {0, 1} ..."
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Cited by 2 (0 self)
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Understanding the query complexity for testing linearinvariant properties has been a central open problem in the study of algebraic property testing. Trianglefreeness in Boolean functions is a simple property whose testing complexity is unknown. Three Boolean functions f1, f2 and f3: Fk2 → {0, 1} are said to be triangle free if there is no x, y ∈ Fk2 such that f1(x) = f2(y) = f3(x + y) = 1. This property is known to be strongly testable (Green, 2005), but the number of queries needed is upperbounded only by a tower of twos whose height is polynomial in 1/, where is the distance between the tested function triple and trianglefreeness, i.e., the minimum fraction of function values that need to be modified to make the triple triangle free. A lower bound of ( 1 ) 2.423 for any onesided tester was given by Bhattacharyya and Xie (2010). In this work we improve this bound to ( 1 ) 6.619. Interestingly, we prove this by way of a combinatorial construction called uniquely solvable puzzles that was at the heart of Coppersmith and Winograd (1990)’s renowned matrix multiplication algorithm. 1
The Complexity of Linear Dependence Problems in Vector Spaces
"... We study extensions of the natural kSUM problem to vector spaces over finite fields. Given a subset S ⊆ F n q of size r ≤ q n, an integer k, 2 ≤ k ≤ n, and a vector v ∈ (F k q \ {0}) k, we define the TargetSum problem to be the problem of finding k elements xi1,..., xik ∈ S for which ∑k vjxij j=1 = ..."
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We study extensions of the natural kSUM problem to vector spaces over finite fields. Given a subset S ⊆ F n q of size r ≤ q n, an integer k, 2 ≤ k ≤ n, and a vector v ∈ (F k q \ {0}) k, we define the TargetSum problem to be the problem of finding k elements xi1,..., xik ∈ S for which ∑k vjxij j=1 = z, where z may either be an input or a fixed vector. We also study a variant of this, where instead of finding xi1,..., xik ∈ S for which ∑k vjxij = z, we require that z be in j=1 span(xi1,..., xik), which we call the (k, r)LinDependenceq problem. These problems are natural generalizations of wellstudied problems that occur in coding theory and property testing. Indeed, the (k, r)LinDependenceq problem is just the Maximum Likelihood Decoding problem. Also, in the TargetSum problem, if instead of general z we require z = 0n, then this is the Weight Distribution problem. In property testing, these problems have been implicitly studied in the context of testing linearinvariant properties. We give nearly optimal bounds for TargetSum and (k, r)LinDependenceq for every r, k, and constant q. Namely, assuming 3SAT requires exponential time, we show that any algorithm for these problems must run in min(2Θ(n) , rΘ(k) ) time. Moreover, we give deterministic upper bounds that match this complexity, up to small factors. Our lower bound strengthens and simplifies an earlier min(2Θ(n) , rΩ(k1/4)) lower bound for both the Maximum Likelihood Decoding and Weight Distribution problems. We also give upper and lower bounds for variants of these problems, e.g., for the problem for which we must find xi1,..., xik for which z ∈ span(xi1,..., xik) but z is not spanned by any proper subset of these vectors, and for the counting version of this problem. Part of this work was done while the author was an intern at IBM Almaden.
A characterization of locally testable affineinvariant properties via decomposition theorems
 In Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC
, 2014
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Testing OddCycleFreeness in Boolean Functions
, 2012
"... A function f: Fn 2 → {0, 1} is oddcyclefree if there are no x1,..., xk ∈ Fn 2 with k an odd integer such that f(x1) = · · · = f(xk) = 1 and x1 + · · · + xk = 0. We show that one can distinguish oddcyclefree functions from those ɛfar from being oddcyclefree by making poly(1/ɛ) queries ..."
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A function f: Fn 2 → {0, 1} is oddcyclefree if there are no x1,..., xk ∈ Fn 2 with k an odd integer such that f(x1) = · · · = f(xk) = 1 and x1 + · · · + xk = 0. We show that one can distinguish oddcyclefree functions from those ɛfar from being oddcyclefree by making poly(1/ɛ) queries to an evaluation oracle. We give two proofs of this result, each shedding light on a different connection between testability of properties of Boolean functions and of dense graphs. The first issue we study is directly reducing testing of linearinvariant properties of Boolean functions to testing associated graph properties. We show a blackbox reduction from testing oddcyclefreeness to testing bipartiteness of graphs. Such reductions have been shown previously (KrálSerraVena, Israel J. Math 2011; Shapira, STOC 2009) for monotone linearinvariant properties defined by forbidding solutions to a finite number of equations. But for oddcyclefreeness whose description involves an infinite number of forbidden equations, a reduction to graph property testing was not previously known. If one could show such a reduction more generally for any linearinvariant property closed under restrictions to subspaces, then it would likely lead to a characterization of the onesided testable linearinvariant properties, an open problem raised by Sudan. The second issue we study is whether there is an efficient canonical tester for linearinvariant properties of Boolean functions. A canonical tester for linearinvariant properties operates by picking a random linear subspace and then checking if the restriction of the input function to the subspace satisfies a fixed property or not. The question is whether for every linearinvariant property, there is a canonical tester for which there is only a polynomial blowup from the optimal query complexity. We answer the question affirmatively for oddcyclefreeness. The general question still remains open.
Property Testing via SetTheoretic Operations
"... Given two testable properties P1 and P2, under what conditions are the union, intersection or setdifference of these two properties also testable? We initiate a systematic study of these basic settheoretic operations in the context of property testing. As an application, we give a conceptually dif ..."
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Given two testable properties P1 and P2, under what conditions are the union, intersection or setdifference of these two properties also testable? We initiate a systematic study of these basic settheoretic operations in the context of property testing. As an application, we give a conceptually different proof that linearity is testable, albeit with much worse query complexity. Furthermore, for the problem of testing disjunction of linear functions, which was previously known to be onesided testable with a superpolynomial query complexity, we give an improved analysis and show it has query complexity O(1/ɛ 2), where ɛ is the distance parameter.
Sunflowers and Testing TriangleFreeness of Functions
"... A function f: Fn2 → {0, 1} is trianglefree if there are no x1, x2, x3 ∈ Fn2 satisfying x1 + x2 + x3 = 0 and f (x1) = f (x2) = f (x3) = 1. In testing trianglefreeness, the goal is to distinguish with high probability trianglefree functions from those that are εfar from being trianglefree. It ..."
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A function f: Fn2 → {0, 1} is trianglefree if there are no x1, x2, x3 ∈ Fn2 satisfying x1 + x2 + x3 = 0 and f (x1) = f (x2) = f (x3) = 1. In testing trianglefreeness, the goal is to distinguish with high probability trianglefree functions from those that are εfar from being trianglefree. It was shown by Green that the query complexity of the canonical tester for the problem is upper bounded by a function that depends only on ε (GAFA, 2005), however the best known upper bound is a tower type function of 1/ε. The best known lower bound on the query complexity of the canonical tester is 1/ε13.239 (Fu and Kleinberg, RANDOM, 2014). In this work we introduce a new approach to proving lower bounds on the query complexity of trianglefreeness. We relate the problem to combinatorial questions on collections of vectors in ZnD and to sunflower conjectures studied by Alon, Shpilka, and Umans (Comput. Complex., 2013). The relations yield that a refutation of the Weak Sunflower Conjecture over Z4 implies a superpolynomial lower bound on the query complexity of the canonical tester for trianglefreeness. Our results are extended to testing kcyclefreeness of functions with domain Fnp for every k ≥ 3 and a prime p. In addition, we generalize the lower bound of Fu and Kleinberg to kcyclefreeness for k ≥ 4 by generalizing the construction of uniquely solvable puzzles due to Coppersmith and Winograd (J. Symbolic Comput., 1990). 1
Source: Question attributed to Tomaszewski in [Guy89]
, 1204
"... Statement: Find an explicit (i.e., in NP) function f: � n 2 → �2 such that we have the correlation bound E[(−1)〈f (x),p(x)〉]≤1/n for every �2polynomial p: � n 2 → �2 of degree at most log2 n. ..."
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Statement: Find an explicit (i.e., in NP) function f: � n 2 → �2 such that we have the correlation bound E[(−1)〈f (x),p(x)〉]≤1/n for every �2polynomial p: � n 2 → �2 of degree at most log2 n.