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22
A unified framework for testing linearinvariant properties
 In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science
, 2010
"... In the history of property testing, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, ReedMuller codes, and F ..."
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Cited by 18 (6 self)
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In the history of property testing, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, ReedMuller codes, and Fourier sparsity. In this work, we describe a framework that can lead to a unified analysis of the testability of all linearinvariant properties, drawing on techniques from additive combinatorics and from graph theory. Our main contributions here are the following: 1. We introduce a simple combinatorial condition, which we call subspaceheredity, and conjecture that any property of Boolean functions satisfying it can be efficiently tested. Verifying this conjecture will unify many individual results in this area. 2. We show that if our conjecture holds, then one can obtain a simple combinatorial characterization of properties of Boolean functions that can be efficiently tested with onesided error, thus addressing a challenge posed by Sudan recently. 3. We introduce a new technique for proving the testability of Boolean functions. Using it, we verify a special case of the conjecture. Our approach here is motivated by techniques that proved to be very successful previously in studying the testability of graph properties.
Invariance in property testing
 Electronic Colloquium on Computational Complexity (ECCC
"... Property testing considers the task of testing rapidly (in particular, with very few samples into the data), if some massive data satisfies some given property, or is far from satisfying the property. For “global properties”, i.e., properties that really depend somewhat on every piece of the data, o ..."
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Cited by 14 (2 self)
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Property testing considers the task of testing rapidly (in particular, with very few samples into the data), if some massive data satisfies some given property, or is far from satisfying the property. For “global properties”, i.e., properties that really depend somewhat on every piece of the data, one could ask how it can be tested by so few samples? We suggest that for “natural ” properties, this should happen because the property is invariant under “nice ” set of “relabellings ” of the data. We refer to this set of relabellings as the “invariance class ” of the property and advocate explicit identification of the invariance class of locally testable properties. Our hope is the explicit knowledge of the invariance class may lead to more general, broader, results. After pointing out the invariance classes associated with some the basic classes of testable properties, we focus on “algebraic properties ” which seem to be characterized by the fact that the properties are themselves vector spaces, while their domains are also vector spaces and the properties are invariant under affine transformations of the domain. We survey recent results (obtained with Tali Kaufman, Elena Grigorescu and Eli BenSasson) that give broad conditions that are sufficient for local testability among this class of properties, and some structural theorems that attempt to describe which properties exhibit the sufficient conditions. 1
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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Cited by 6 (3 self)
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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.
On the Structure of Boolean Functions with Small Spectral Norm
"... In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of f is ‖ ˆ f‖1 = ∑ α  ˆ f(α)). Specifically, we prove the following results for functions f: {0, 1} n → {0, 1} with ‖ ˆ f‖1 = A. 1. There is a subspace V of codimension at most A 2 such that ..."
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Cited by 6 (0 self)
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In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of f is ‖ ˆ f‖1 = ∑ α  ˆ f(α)). Specifically, we prove the following results for functions f: {0, 1} n → {0, 1} with ‖ ˆ f‖1 = A. 1. There is a subspace V of codimension at most A 2 such that fV is constant. 2. f can be computed by a parity decision tree of size 2A2n2A. (a parity decision tree is a decision tree whose nodes are labeled with arbitrary linear functions.) 3. If in addition f has at most s nonzero Fourier coefficients, then f can be computed by a parity decision tree of depth A 2 log s. 4. For every 0 < ɛ there is a parity decision tree of depth O(A 2 + log(1/ɛ)) and size 2 O(A2) min{1/ɛ 2, O(log(1/ɛ)) 2A} that ɛapproximates f. Furthermore, this tree can be learned, with probability 1 − δ, using poly(n, exp(A 2), 1/ɛ, log(1/δ)) membership queries. All the results above also hold (with a slight change in parameters) for functions f: Z n p →
Efficient sample extractors for juntas with applications
 Automata, Languages and Programming
, 2011
"... We develop a queryefficient sample extractor for juntas, that is, a probabilistic algorithm that can simulate random samples from the core of a kjunta f: {0, 1} n → {0, 1} given oracle access to a function f ′ : {0, 1} n → {0, 1} that is only close to f. After a preprocessing step, which takes Õ(k ..."
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Cited by 6 (1 self)
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We develop a queryefficient sample extractor for juntas, that is, a probabilistic algorithm that can simulate random samples from the core of a kjunta f: {0, 1} n → {0, 1} given oracle access to a function f ′ : {0, 1} n → {0, 1} that is only close to f. After a preprocessing step, which takes Õ(k) queries, generating each sample to the core of f takes only one query to f ′. We then plug in our sample extractor in the “testing by implicit learning ” framework of Diakonikolas et al. [DLM + 07], improving the query complexity of testers for various Boolean function classes. In particular, for some of the classes considered in [DLM + 07], such as sterm DNF formulas, sizes decision trees, sizes Boolean formulas, ssparse polynomials over F2, and sizes branching programs, the query complexity is reduced from Õ(s4 /ɛ 2) to Õ(s/ɛ2). This shows that, using the new sample extractor, testing by implicit learning can lead to testers having better query complexity than those tailored to a specific problem, such as the tester of Parnas et al. [PRS02] for the class of monotone sterm DNF formulas. In terms of techniques, we extend the tools used in [CGM11] for testing function isomorphism to juntas. Specifically, while the original analysis in [CGM11] allowed queryefficient noisy sampling
On the communication complexity of XOR functions
, 2010
"... An XOR function is a function of the form g(x, y) = f(x ⊕ y), for some boolean function f on n bits. We study the quantum and classical communication complexity of XOR functions. In the case of exact protocols, we completely characterise oneway communication complexity for all f. We also show that ..."
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An XOR function is a function of the form g(x, y) = f(x ⊕ y), for some boolean function f on n bits. We study the quantum and classical communication complexity of XOR functions. In the case of exact protocols, we completely characterise oneway communication complexity for all f. We also show that, when f is monotone, g’s quantum and classical complexities are quadratically related, and that when f is a linear threshold function, g’s quantum complexity is Θ(n). More generally, we make a structural conjecture about the Fourier spectra of boolean functions which, if true, would imply that the quantum and classical exact communication complexities of all XOR functions are asymptotically equivalent. We give two randomised classical protocols for general XOR functions which are efficient for certain functions, and a third protocol for linear threshold functions with high margin. These protocols operate in the symmetric message passing model with shared randomness. 1
Testing by implicit learning: a brief survey
 Property Testing
, 2010
"... We give a highlevel survey of the “testing by implicit learning ” paradigm, and explain some of the property testing results for various Boolean function classes that have been obtained using this approach. 1 ..."
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Cited by 3 (0 self)
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We give a highlevel survey of the “testing by implicit learning ” paradigm, and explain some of the property testing results for various Boolean function classes that have been obtained using this approach. 1
Fourier sparsity, spectral norm, and the Logrank conjecture
"... Abstract—We study Boolean functions with sparse Fourier spectrum or small spectral norm, and show their applications to the Logrank Conjecture for XOR functions f(x ⊕ y) — a fairly large class of functions including well studied ones such as Equality and Hamming Distance. The rank of the communicat ..."
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Abstract—We study Boolean functions with sparse Fourier spectrum or small spectral norm, and show their applications to the Logrank Conjecture for XOR functions f(x ⊕ y) — a fairly large class of functions including well studied ones such as Equality and Hamming Distance. The rank of the communication matrix Mf for such functions is exactly the Fourier sparsity of f. Let d = deg2(f) be the F2degree of f and DCC(f ◦ ⊕) stand for the deterministic communication complexity for f(x ⊕ y). We show that 1) DCC(f ◦ ⊕) = O(2d2/2 logd−2 ‖f̂‖1). In particular, the Logrank conjecture holds for XOR functions with constant F2degree. 2) DCC(f ◦ ⊕) = O(d‖f̂‖1) = Õ( rank(Mf)). This improves the (trivial) linear bound by nearly a quadratic factor. We obtain our results through a degreereduction protocol based on a variant of polynomial rank, and actually conjecture that the communication cost of our protocol is at most logO(1) rank(Mf). The above bounds are obtained from different analysis for the number of parity queries required to reduce f ’s F2degree. Our bounds also hold for the parity decision tree complexity of f, a measure that is no less than the communication complexity. Along the way we also prove several structural results about Boolean functions with small Fourier sparsity ‖f̂‖0 or spectral norm ‖f̂‖1, which could be of independent interest. For functions f with constant F2degree, we show that: 1) f can be written as the summation of quasipolynomially many indicator functions of subspaces with ±signs, improving the previous doubly exponential upper bound by Green and Sanders; 2) being sparse in Fourier domain is polynomially equivalent to having a small parity decision tree complexity; and 3) f depends only on polylog‖f̂‖1 linear functions of input variables. For functions f with small spectral norm, we show that: 1) there is an affine subspace of codimension O(‖f̂‖1) on which f(x) is a constant, and 2) there is a parity decision tree of depth O(‖f̂‖1 log ‖f̂‖0) for computing f.
Testing linearinvariant function isomorphism
 In ICALP
, 2013
"... Abstract. A function f: F n 2 → {−1, 1} is called linearisomorphic to g if f = g ◦ A for some nonsingular matrix A. In the gisomorphism problem, we want a randomized algorithm that distinguishes whether an input function f is linearisomorphic to g or far from being so. We show that the query com ..."
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Abstract. A function f: F n 2 → {−1, 1} is called linearisomorphic to g if f = g ◦ A for some nonsingular matrix A. In the gisomorphism problem, we want a randomized algorithm that distinguishes whether an input function f is linearisomorphic to g or far from being so. We show that the query complexity to test gisomorphism is essentially determined by the spectral norm of g. That is, if g is close to having spectral norm s, then we can test gisomorphism with poly(s) queries, and if g is far from having spectral norm s, then we cannot test gisomorphism with o(log s) queries. The upper bound is almost tight since there is indeed a function g close to having spectral norm s whereas testing gisomorphism requires Ω(s) queries. As far as we know, our result is the first characterization of this type for functions. Our upper bound is essentially the KushilevitzMansour learning algorithm, modified for use in the implicit setting. Exploiting our upper bound, we show that any property is testable if it can be wellapproximated by functions with small spectral norm. We also extend our algorithm to the setting where A is allowed to be singular. 1
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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