We consider the problem of representing Boolean functions exactly by "sparse" linear combinations (over R) of functions from some "simple" class C . In particular, given C we are interested in finding low-complexity functions lacking sparse representations. When C is the set of PARITY functions or the set of conjunctions, this sort of problem has a well-understood answer; the problem becomes interesting when C is "overcomplete" and the set of functions is not linearly independent. We focus on the cases where C is the set of linear threshold functions, the set of rectified linear units (ReLUs), and the set of low-degree polynomials over a finite field, all of which are well-studied in different contexts.Building on the new easy witness lemma of Cody Murray and the author, we provide generic tools for proving lower bounds on representations of this kind. Applying these, we give several new lower bounds for "semi-explicit" Boolean functions. Let α(n) be an unbounded function such that n α(n) is time constructible (e.g. α(n) = log ⋆ (n)). We show: * EECS and CSAIL, MIT. Supported by NSF CCF-1553288. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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