The Experts below are selected from a list of 312 Experts worldwide ranked by ideXlab platform
Shachar Lovett - One of the best experts on this subject based on the ideXlab platform.
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Estimating the distance from testable affine-Invariant properties
arXiv: Computational Complexity, 2013Co-Authors: Hamed Hatami, Shachar LovettAbstract:Let $\cal{P}$ be an affine Invariant Property of functions $\mathbb{F}_p^n \to [R]$ for fixed $p$ and $R$. We show that if $\cal{P}$ is locally testable with a constant number of queries, then one can estimate the distance of a function $f$ from $\cal{P}$ with a constant number of queries. This was previously unknown even for simple properties such as cubic polynomials over $\mathbb{F}_2$. Our test is simple: take a restriction of $f$ to a constant dimensional affine subspace, and measure its distance from $\cal{P}$. We show that by choosing the dimension large enough, this approximates with high probability the global distance of $f$ from $\cP$. The analysis combines the approach of Fischer and Newman [SIAM J. Comp 2007] who established a similar result for graph properties, with recently developed tools in higher order Fourier analysis, in particular those developed in Bhattacharyya et al. [STOC 2013].
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every locally characterized affine Invariant Property is testable
Symposium on the Theory of Computing, 2013Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, Shachar LovettAbstract:Set F = Fp for any fixed prime p ≥ 2. An affine-Invariant Property is a Property of functions over Fn that is closed under taking affine transformations of the domain. We prove that all affine-Invariant properties having local characterizations are testable. In fact, we show a proximity-oblivious test for any such Property cP, meaning that given an input function f, we make a constant number of queries to f, always accept if f satisfies cP, and otherwise reject with probability larger than a positive number that depends only on the distance between f and cP. More generally, we show that any affine-Invariant 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 decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-$d$ polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on 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 low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties.
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testing low complexity affine Invariant properties
Symposium on Discrete Algorithms, 2013Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Shachar LovettAbstract:@p 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 affine-Invariant 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 affine-Invariant Property P of functions f:
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SODA - Testing low complexity affine-Invariant properties
2013Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Shachar LovettAbstract:@p 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 affine-Invariant 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 affine-Invariant Property P of functions f:
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Every locally characterized affine-Invariant Property is testable
arXiv: Computational Complexity, 2012Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, Shachar LovettAbstract:Let F = F_p for any fixed prime p >= 2. An affine-Invariant Property is a Property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-Invariant Property having local characterizations are testable. In fact, we show a proximity-oblivious test for any such Property P, meaning that there is a test that, given an input function f, makes a constant number of queries to f, always accepts if f satisfies P, and rejects with positive probability if the distance between f and P is nonzero. More generally, we show that any affine-Invariant 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 decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-d polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on 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 low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties.
Arnab Bhattacharyya - One of the best experts on this subject based on the ideXlab platform.
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Guest column: on testing affine-Invariant properties over finite fields
ACM SIGACT News, 2013Co-Authors: Arnab BhattacharyyaAbstract:An affine-Invariant Property over a finite field is a Property of functions over Fn/p that is closed under all affine transformations of the domain. This class of properties includes such well-known beasts as low-degree polynomials, polynomials that nontrivially factor, and functions of low spectral norm. The last few years has seen rapid progress in characterizing the affine-Invariant properties which are testable with a constant number of queries. We survey the current state of this project.
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every locally characterized affine Invariant Property is testable
Symposium on the Theory of Computing, 2013Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, Shachar LovettAbstract:Set F = Fp for any fixed prime p ≥ 2. An affine-Invariant Property is a Property of functions over Fn that is closed under taking affine transformations of the domain. We prove that all affine-Invariant properties having local characterizations are testable. In fact, we show a proximity-oblivious test for any such Property cP, meaning that given an input function f, we make a constant number of queries to f, always accept if f satisfies cP, and otherwise reject with probability larger than a positive number that depends only on the distance between f and cP. More generally, we show that any affine-Invariant 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 decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-$d$ polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on 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 low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties.
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testing low complexity affine Invariant properties
Symposium on Discrete Algorithms, 2013Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Shachar LovettAbstract:@p 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 affine-Invariant 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 affine-Invariant Property P of functions f:
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SODA - Testing low complexity affine-Invariant properties
2013Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Shachar LovettAbstract:@p 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 affine-Invariant 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 affine-Invariant Property P of functions f:
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Every locally characterized affine-Invariant Property is testable
arXiv: Computational Complexity, 2012Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, Shachar LovettAbstract:Let F = F_p for any fixed prime p >= 2. An affine-Invariant Property is a Property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-Invariant Property having local characterizations are testable. In fact, we show a proximity-oblivious test for any such Property P, meaning that there is a test that, given an input function f, makes a constant number of queries to f, always accepts if f satisfies P, and rejects with positive probability if the distance between f and P is nonzero. More generally, we show that any affine-Invariant 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 decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-d polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on 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 low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties.
Yuichi Yoshida - One of the best experts on this subject based on the ideXlab platform.
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STOC - A characterization of locally testable affine-Invariant properties via decomposition theorems
Proceedings of the forty-sixth annual ACM symposium on Theory of computing, 2014Co-Authors: Yuichi YoshidaAbstract:Let P be a Property of function Fnp → {0, 1} for a fixed prime p. An algorithm is called a tester for P if, given a query access to the input function f, with high probability, it accepts when f satisfies P and rejects when f is "far" from satisfying P. In this paper, we give a characterization of affine-Invariant properties that are (two-sided error) testable with a constant number of queries. The characterization is stated in terms of decomposition theorems, which roughly claim that any function can be decomposed into a structured part that is a function of a constant number of polynomials, and a pseudo-random part whose Gowers norm is small. We first give an algorithm that tests whether the structured part of the input function has a specific form. Then we show that an affine-Invariant Property is testable with a constant number of queries if and only if it can be reduced to the problem of testing whether the structured part of the input function is close to one of a constant number of candidates.
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A Characterization of Locally Testable Affine-Invariant Properties via Decomposition Theorems
arXiv: Computational Complexity, 2014Co-Authors: Yuichi YoshidaAbstract:Let $\mathcal{P}$ be a Property of function $\mathbb{F}_p^n \to \{0,1\}$ for a fixed prime $p$. An algorithm is called a tester for $\mathcal{P}$ if, given a query access to the input function $f$, with high probability, it accepts when $f$ satisfies $\mathcal{P}$ and rejects when $f$ is "far" from satisfying $\mathcal{P}$. In this paper, we give a characterization of affine-Invariant properties that are (two-sided error) testable with a constant number of queries. The characterization is stated in terms of decomposition theorems, which roughly claim that any function can be decomposed into a structured part that is a function of a constant number of polynomials, and a pseudo-random part whose Gowers norm is small. We first give an algorithm that tests whether the structured part of the input function has a specific form. Then we show that an affine-Invariant Property is testable with a constant number of queries if and only if it can be reduced to the problem of testing whether the structured part of the input function is close to one of a constant number of candidates.
Eldar Fischer - One of the best experts on this subject based on the ideXlab platform.
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every locally characterized affine Invariant Property is testable
Symposium on the Theory of Computing, 2013Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, Shachar LovettAbstract:Set F = Fp for any fixed prime p ≥ 2. An affine-Invariant Property is a Property of functions over Fn that is closed under taking affine transformations of the domain. We prove that all affine-Invariant properties having local characterizations are testable. In fact, we show a proximity-oblivious test for any such Property cP, meaning that given an input function f, we make a constant number of queries to f, always accept if f satisfies cP, and otherwise reject with probability larger than a positive number that depends only on the distance between f and cP. More generally, we show that any affine-Invariant 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 decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-$d$ polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on 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 low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties.
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testing low complexity affine Invariant properties
Symposium on Discrete Algorithms, 2013Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Shachar LovettAbstract:@p 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 affine-Invariant 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 affine-Invariant Property P of functions f:
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SODA - Testing low complexity affine-Invariant properties
2013Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Shachar LovettAbstract:@p 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 affine-Invariant 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 affine-Invariant Property P of functions f:
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Every locally characterized affine-Invariant Property is testable
arXiv: Computational Complexity, 2012Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, Shachar LovettAbstract:Let F = F_p for any fixed prime p >= 2. An affine-Invariant Property is a Property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-Invariant Property having local characterizations are testable. In fact, we show a proximity-oblivious test for any such Property P, meaning that there is a test that, given an input function f, makes a constant number of queries to f, always accepts if f satisfies P, and rejects with positive probability if the distance between f and P is nonzero. More generally, we show that any affine-Invariant 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 decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-d polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on 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 low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties.
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testing low complexity affine Invariant properties
arXiv: Computational Complexity, 2012Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Shachar LovettAbstract: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 affine-Invariant Property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, for instance, that the Reed-Muller code over F_p of degree d < p is testable, with an argument that uses no detailed algebraic information about polynomials except that low degree is preserved by composition with affine maps. The complexity of an affine-Invariant 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 affine-Invariant Property P of functions f: F_p^n -> [R] is testable, assuming the complexity of the Property is less than p. Our proof involves developing analogs of graph-theoretic techniques in an algebraic setting, using tools from higher-order Fourier analysis.
Hamed Hatami - One of the best experts on this subject based on the ideXlab platform.
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Estimating the distance from testable affine-Invariant properties
arXiv: Computational Complexity, 2013Co-Authors: Hamed Hatami, Shachar LovettAbstract:Let $\cal{P}$ be an affine Invariant Property of functions $\mathbb{F}_p^n \to [R]$ for fixed $p$ and $R$. We show that if $\cal{P}$ is locally testable with a constant number of queries, then one can estimate the distance of a function $f$ from $\cal{P}$ with a constant number of queries. This was previously unknown even for simple properties such as cubic polynomials over $\mathbb{F}_2$. Our test is simple: take a restriction of $f$ to a constant dimensional affine subspace, and measure its distance from $\cal{P}$. We show that by choosing the dimension large enough, this approximates with high probability the global distance of $f$ from $\cP$. The analysis combines the approach of Fischer and Newman [SIAM J. Comp 2007] who established a similar result for graph properties, with recently developed tools in higher order Fourier analysis, in particular those developed in Bhattacharyya et al. [STOC 2013].
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every locally characterized affine Invariant Property is testable
Symposium on the Theory of Computing, 2013Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, Shachar LovettAbstract:Set F = Fp for any fixed prime p ≥ 2. An affine-Invariant Property is a Property of functions over Fn that is closed under taking affine transformations of the domain. We prove that all affine-Invariant properties having local characterizations are testable. In fact, we show a proximity-oblivious test for any such Property cP, meaning that given an input function f, we make a constant number of queries to f, always accept if f satisfies cP, and otherwise reject with probability larger than a positive number that depends only on the distance between f and cP. More generally, we show that any affine-Invariant 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 decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-$d$ polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on 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 low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties.
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Every locally characterized affine-Invariant Property is testable
arXiv: Computational Complexity, 2012Co-Authors: Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, Shachar LovettAbstract:Let F = F_p for any fixed prime p >= 2. An affine-Invariant Property is a Property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-Invariant Property having local characterizations are testable. In fact, we show a proximity-oblivious test for any such Property P, meaning that there is a test that, given an input function f, makes a constant number of queries to f, always accepts if f satisfies P, and rejects with positive probability if the distance between f and P is nonzero. More generally, we show that any affine-Invariant 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 decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-d polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on 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 low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties.