TY - JOUR

T1 - Non-Abelian homomorphism testing, and distributions close to their self-convolutions

AU - Ben-or, Michael

AU - Coppersmith, Don

AU - Luby, Mike

AU - Rubinfeld, Ronitt

PY - 2008/1

Y1 - 2008/1

N2 - In this paper, we study two questions related to the problem of testing whether a function is close to a homomorphism. For two finite groups G, H (not necessarily Abelian), an arbitrary map f : G → H, and a parameter 0 < ε < 1, say that f is ε-close to a homomorphism if there is some homomorphism g such that g and/differ on at most ε|G| elements of G, and say that/is ε-far otherwise. For a given f and ε, a homomorphism tester should distinguish whether f is a homomorphism, or if f is ε-far from a homomorphism. When G is Abelian, it was known that the test which picks 0(1/ε) random pairs x,y and tests that f(x) + f(y) = f(x + y) gives a homomorphism tester. Our first result shows that such a test works for all groups G. Next, we consider functions that are close to their self-convolutions. Let A = {ag|g ε G) be a distribution on G. The self-convolution of A, A′ = {a′g|g ε G], is defined by (Equation Presented) It is known that A = A′ exactly when A is the uniform distribution over a subgroup of G. We show that there is a sense in which this characterization is robust - that is, if A is close in statistical distance to A′, then A must be close to uniform over some subgroup of G. Finally, we show a relationship between the question of testing whether a function is close to a homomorphism via the above test and the question of characterizing functions that are close to their self-convolutions.

AB - In this paper, we study two questions related to the problem of testing whether a function is close to a homomorphism. For two finite groups G, H (not necessarily Abelian), an arbitrary map f : G → H, and a parameter 0 < ε < 1, say that f is ε-close to a homomorphism if there is some homomorphism g such that g and/differ on at most ε|G| elements of G, and say that/is ε-far otherwise. For a given f and ε, a homomorphism tester should distinguish whether f is a homomorphism, or if f is ε-far from a homomorphism. When G is Abelian, it was known that the test which picks 0(1/ε) random pairs x,y and tests that f(x) + f(y) = f(x + y) gives a homomorphism tester. Our first result shows that such a test works for all groups G. Next, we consider functions that are close to their self-convolutions. Let A = {ag|g ε G) be a distribution on G. The self-convolution of A, A′ = {a′g|g ε G], is defined by (Equation Presented) It is known that A = A′ exactly when A is the uniform distribution over a subgroup of G. We show that there is a sense in which this characterization is robust - that is, if A is close in statistical distance to A′, then A must be close to uniform over some subgroup of G. Finally, we show a relationship between the question of testing whether a function is close to a homomorphism via the above test and the question of characterizing functions that are close to their self-convolutions.

KW - Convolutions of distributions

KW - Linearity testing

KW - Sublinear time algorithms

UR - http://www.scopus.com/inward/record.url?scp=38049108424&partnerID=8YFLogxK

U2 - 10.1002/rsa.20182

DO - 10.1002/rsa.20182

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AN - SCOPUS:38049108424

SN - 1042-9832

VL - 32

SP - 49

EP - 70

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 1

ER -