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Interactive proofs and the hardness of approximating cliques
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Property testing and its connection to learning and approximation
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Robust Characterizations of Polynomials withApplications to Program Testing
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Testing Low-Degree Polynomials over Prime Fields
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A combinatorial characterization of the testable graph properties: it's all about regularity
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Graph limits and parameter testing
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Testing Polynomials over General Fields
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Testing versus Estimation of Graph Properties
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A Characterization of the (Natural) Graph Properties Testable with One-Sided Error
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Algebraic property testing: the role of invariance
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Inverse conjecture for the gowers norm is false
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Every Monotone Graph Property Is Testable
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Green's conjecture and testing linear-invariant properties
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Testing Fourier Dimensionality and Sparsity
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Algorithmic and Analysis Techniques in Property Testing
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Correlation testing for affine invariant properties on Fpn in the high error regime
Proceedings of the forty-third annual ACM symposium on Theory of computing
On Proximity-Oblivious Testing
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IEEE Transactions on Information Theory
Testing linear-invariant function isomorphism
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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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.