Nearly optimal solutions for the chow parameters problem and low-weight approximation of halfspaces

The Chow parameters of a Boolean function f : {−1, 1}n → {−1, 1} are its n + 1 degree-0 and degree-1 Fourier coefficients. It has been known since 1961 [Cho61, Tan61] that the (exact values of the) Chow parameters of any linear threshold function f uniquely specify f within the space of all Boolean functions, but until recently [OS11] nothing was known about efficient algorithms for reconstructing f (exactly or approximately) from exact or approximate values of its Chow parameters. We refer to this reconstruction problem as the Chow Parameters Problem. Our main result is a new algorithm for the Chow Parameters Problem which, given (sufficiently accurate approximations to) the Chow parameters of any linear threshold function f, runs in time ˜O(n2 )· (1/ǫ)O(log2 (1/ǫ)) and with high probability outputs a representation of an LTF f′ that is ǫ-close to f. The only previous algorithm [OS11] had running time poly(n) · 22 ˜O(1/ǫ2) . As a byproductof our approach, we show t...

• Type: Journal
• Year: 2012
• Where: STOC
• Authors: Anindya De, Ilias Diakonikolas, Vitaly Feldman, Rocco A. Servedio
• Keywords: Accurate Approximations, Algorithms, Linear Threshold, STOC 2012, Threshold Function