An Improved Algorithm For Learning Sparse Parities In The Presence Of Noise

We revisit the Learning Sparse Parities with Noise (LSPN) problem on k out of n variables for k << n, and present the following findings.1. For true parity size k = n(u) for any 0 < u < 1, and noise rate eta < 1/2, the first algorithm solves the (n,k,eta)-LSPN problem with constant probability and time/sample complexity n((1-u+o(1))k)/(1/2-eta)(2).2. For any 1/2 < c(1) < 1, k = o(eta n/logn), and eta <= n (-c1)/4, our second algorithm solves the (n,k,eta)-LSPN problem with constant probability and time/sample complexity n(2(1-c1+o(1))k).3. We show a "win-win" result about reducing the number of samples. If there is an algorithm that solves (n, k, eta)-LSPN problem with probability Omega(1), time/sample complexity n(O(k)) for k = o(n(1-c)), any noise rate eta = n(1-2C)/3 and 1/2 <= c < 1. Then, either there exists an algorithm that solves the (n, k, mu)-LSPN problem under lower noise rate mu = n(-c)/3 using only 2n samples, or there exists an algorithm that solves the (n, k ', mu)-LSPN problem for a much larger k ' = n(1-c) with probability n(-O(k))/poly(n), and time complexity poly(n) center dot n(O(k)), using only n samples.Our algorithms are simple in concept by combining a few basic techniques such as majority voting, reduction from the LSPN problem to its decisional variant, Goldreich-Levin list decoding, and computational sample amplification. (C) 2021 Elsevier B.V. All rights reserved.