Testing Linear-Invariant Function Isomorphism

Abstract

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.

Tight Lower Bounds for Testing Linear Isomorphism

“…It has been considered by Chakraborty et al [20] who show a lower bound of Ω(k) for testing L(f ) for a function f that is far from having (Fourier) dimension k − 1. In line with testing juntas, a previous result of [25] implicitly proves an upper bound of O(k2 k ) for linear isomorphism to functions that are very close to having dimension k; the "very" here is exponentially small in k. Wimmer and Yoshida [43] give an constant-query algorithm for linear isomorphism to any function close to having dimension k by giving a tolerant tester for functions of dimension k. The technique is an extension of the work of [25], and it applies to functions close to having low spectral norm. They also show lower bounds for testing linear isomorphism, but these lower bounds are no better than Ω(n) for any fixed function.…”

Fast Fourier Sparsity Testing

“…An active line of previous work focuses on tolerant testing under Hamming distance. Wimmer and Yoshida [WY13] showed that the general approach of [GOS + 11] can be extended to yield tolerant testers for Fourier s-sparsity of Boolean functions. Specifically, they give an algorithm that distinguishes between functions that are ǫ/3-close to Fourier s-sparse from those that are ǫ-far from Fourier s-sparse under Hamming distance, using poly(s) queries.…”

“…When f has at most s non-zero Fourier coefficients, we say that it is Fourier s-sparse, or just s-sparse for short. The Fourier sparsity of functions plays an important role in many different areas of computer science, including error-correcting codes [GL89,AGS03], learning theory [KM93,LMN93], communication complexity [ZS09, BC99, MO09, TWXZ13], property testing [GOS + 11, WY13], and parity decision tree complexity [ZS10,STlV14].…”

Institute of Child Health, Kolkata, 1956–2022

“…Recently, [4] gave a characterization of Linear invariant properties that are constant query testable with a one-sided error setup. Testing linear isomorphism was studied by [34] who gave a polynomial time query algorithm in terms of the spectral norm of a function. Linear isomorphism of Boolean functions has also been studied in the context of combinatorial circuit design [12,3,36], error-correcting codes [14,19,27] and cryptography [13,10,18].…”