A function f : double-struck F2n → {-1, 1} is called linear-isomorphic to g if f = g o 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(logs) queries. The upper bound is almost tight since there is indeed a function g close to having spectral norm s whereas testing g-isomorphism 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. © 2013 Springer-Verlag.
CITATION STYLE
Wimmer, K., & Yoshida, Y. (2013). Testing linear-invariant function isomorphism. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7965 LNCS, pp. 840–850). https://doi.org/10.1007/978-3-642-39206-1_71
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