Local Correction with Constant Error Rate

TL;DR

This paper extends local correction techniques to functions with a constant error rate, focusing on juntas and partially symmetric functions, and introduces algorithms with non-adaptive queries that work for most functions in these families.

Contribution

It relaxes the closeness requirement for local correction to a constant error rate and provides algorithms for almost all juntas and partially symmetric functions.

Findings

Almost every k-junta is O(k log^2 k)-locally correctable for epsilon < 0.001.

Similar correction results are shown for partially symmetric functions.

The algorithms use non-adaptive queries and are applicable to most functions in these families.

Abstract

A Boolean function f of n variables is said to be q-locally correctable if, given a black-box access to a function g which is "close" to an isomorphism f_sigma(x)=f_sigma(x_1, ..., x_n) = f(x_sigma(1), ..., x_sigma(n)) of f, we can compute f_sigma(x) for any x in {0,1}^n with good probability using q queries to g. It is known that degree d polynomials are O(2^d)-locally correctable, and that most k-juntas are O(k log k)-locally correctable, where the closeness parameter, or more precisely the distance between g and f_sigma, is required to be exponentially small (in d and k respectively). In this work we relax the requirement for the closeness parameter by allowing the distance between the functions to be a constant. We first investigate the family of juntas, and show that almost every k-junta is O(k log^2 k)-locally correctable for any distance epsilon < 0.001. A similar result is shown for the family of partially symmetric functions, that is functions which are indifferent to any reordering of all but a constant number of their variables. For both families, the algorithms provided here use non-adaptive queries and are applicable to most but not all functions of each family (as it is shown to be impossible to locally correct all of them). Our approach utilizes the measure of symmetric influence introduced in the recent analysis of testing partial symmetry of functions.