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- W2913304003 abstract "We study the approximation of halfspaces $h:{0,1}^nto{0,1}$ in the infinity norm by polynomials and rational functions of any given degree. Our main result is an explicit construction of the hardest halfspace, for which we prove polynomial and rational approximation lower bounds that match the trivial upper bounds achievable for all halfspaces. This completes a lengthy line of work started by Myhill and Kautz (1961). As an application, we construct a communication problem that achieves essentially the largest possible separation, of $O(n)$ versus $2^{-Omega(n)},$ between the sign-rank and discrepancy. Equivalently, our problem exhibits a gap of $log n$ versus $Omega(n)$ between the communication complexity with unbounded versus weakly unbounded error, improving quadratically on previous constructions and completing a line of work started by Babai, Frankl, and Simon (FOCS 1986). Our results further generalize to the $k$-party number-on-the-forehead model, where we obtain an explicit separation of $log n$ versus $Omega(n/4^{n})$ for communication with unbounded versus weakly unbounded error. This gap is a quadratic improvement on previous work and matches the state of the art for number-on-the-forehead lower bounds." @default.
- W2913304003 created "2019-02-21" @default.
- W2913304003 creator A5079755716 @default.
- W2913304003 date "2019-02-05" @default.
- W2913304003 modified "2023-09-27" @default.
- W2913304003 title "The Hardest Halfspace" @default.
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