A unified framework for testing linear‐invariant properties

Bhattacharyya, Arnab; Grigorescu, Elena; Shapira, A.

doi:10.1002/rsa.20507

Cited by 25 publications

(18 citation statements)

References 63 publications

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“…For example, given a finite field F , for properties of functions over a linear space over F that are known to be invariant under linear transformations, a canonical testing scheme would consist of querying the function over an entire small dimensional subspace picked uniformly at random [2].…”

Section: Introductionmentioning

confidence: 99%

Trading Query Complexity for Sample-Based Testing and Multi-testing Scalability

Fischer

Lachish

Vasudev

2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

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We show here that every non-adaptive property testing algorithm making a constant number of queries, over a fixed alphabet, can be converted to a sample-based (as per [Goldreich and Ron, 2015]) testing algorithm whose average number of queries is a fixed, smaller than 1, power of n. Since the query distribution of the sample-based algorithm is not dependent at all on the property, or the original algorithm, this has many implications in scenarios where there are many properties that need to be tested for concurrently, such as testing (relatively large) unions of properties, or converting a Merlin-Arthur Proximity proof (as per [Gur and Rothblum, 2013]) to a proper testing algorithm.The proof method involves preparing the original testing algorithm for a combinatorial analysis, which in turn involves a new result about the existence of combinatorial structures (essentially generalized sunflowers) that allow the sample-based tester to replace the original constant query complexity tester.

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Section: Introductionmentioning

confidence: 99%

Trading Query Complexity for Sample-Based Testing and Multi-testing Scalability

Fischer

Lachish

Vasudev

2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

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“…An ambitious open problem is obtaining a similar characterization for properties of Boolean functions. Recently there has been a lot of progress on the restriction of this question to properties that are closed under linear or affine transformations [6,23]. More generally, one might hope to settle this open problem for all properties of Boolean functions that are closed under relabeling of the input variables.…”

Section: Definition 1 ([27]mentioning

confidence: 99%

Partially Symmetric Functions Are Efficiently Isomorphism-Testable

Blais

Weinstein

Yoshida

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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Given a function f : {0, 1} n → {0, 1}, the f -isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to f up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions f we can test f -isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas.We unify and extend these results by showing that all partially symmetric functionsfunctions invariant to the reordering of all but a constant number of their variables-are efficiently isomorphism-testable. This class of functions, first introduced by Shannon, includes symmetric functions, juntas, and many other functions as well. We conjecture that these functions are essentially the only functions efficiently isomorphism-testable.To prove our main result, we also show that partial symmetry is efficiently testable. In turn, to prove this result we had to revisit the junta testing problem. We provide a new proof of correctness of the nearly-optimal junta tester. Our new proof replaces the Fourier machinery of the original proof with a purely combinatorial argument that exploits the connection between sets of variables with low influence and intersecting families.Another important ingredient in our proofs is a new notion of symmetric influence. We use this measure of influence to prove that partial symmetry is efficiently testable and also to construct an efficient sample extractor for partially symmetric functions. We then combine the sample extractor with the testing-by-implicit-learning approach to complete the proof that partially symmetric functions are efficiently isomorphism-testable.

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“…An ambitious open problem is obtaining a similar characterization for properties of Boolean functions. Recently there has been a lot of progress on the restriction of this question to linear-invariant properties [7][8][9]. A property P of Boolean functions F n 2 → {−1, 1} is called linear-invariant if a function f satisfies P , then f • A is also in P for any square matrix A, where (f • A)(x) := f (Ax) for any x ∈ F n 2 .…”

Section: Introductionmentioning

confidence: 99%

Testing Linear-Invariant Function Isomorphism

Wimmer

Yoshida

2013

Automata, Languages, and Programming

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Abstract. A function f : F n 2 → {−1, 1} is called linear-isomorphic to g if f = g • A for some non-singular matrix A. In the g-isomorphism problem, we want a randomized algorithm that distinguishes whether an input function f is linear-isomorphic to g or far from being so. We show that the query complexity to test g-isomorphism is essentially determined by the spectral norm of g. That is, if g is close to having spectral norm s, then we can test g-isomorphism with poly(s) queries, and if g is far from having spectral norm s, then we cannot test g-isomorphism 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 Kushilevitz-Mansour learning algorithm, modified for use in the implicit setting. Exploiting our upper bound, we show that any property is testable if it can be well-approximated by functions with small spectral norm. We also extend our algorithm to the setting where A is allowed to be singular.

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A unified framework for testing linear‐invariant properties

Cited by 25 publications

References 63 publications

Trading Query Complexity for Sample-Based Testing and Multi-testing Scalability

Trading Query Complexity for Sample-Based Testing and Multi-testing Scalability

Partially Symmetric Functions Are Efficiently Isomorphism-Testable

Testing Linear-Invariant Function Isomorphism

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