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COCO
2009
Springer

Improved Approximation of Linear Threshold Functions

10 years 7 months ago
Improved Approximation of Linear Threshold Functions
We prove two main results on how arbitrary linear threshold functions f(x) = sign(w · x − θ) over the n-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every n-variable threshold function f is -close to a threshold function depending only on Inf(f)2 · poly(1/ ) many variables, where Inf(f) denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut’s well-known theorem [Fri98], which states that every Boolean function f is -close to a function depending only on 2O(Inf(f)/ ) many variables, for the case of threshold functions. We complement this upper bound by showing that Ω(Inf(f)2 + 1/ 2) many variables are required for -approximating threshold functions. Our second result is a proof that every n-variable threshold function is -close to a threshold function with integer weights at most poly(n)·2 ˜O(1/ 2/3). This is a significant improvement, in the dependence on the e...
Ilias Diakonikolas, Rocco A. Servedio
Added 26 May 2010
Updated 26 May 2010
Type Conference
Year 2009
Where COCO
Authors Ilias Diakonikolas, Rocco A. Servedio
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