Bounding the average sensitivity and noise sensitivity of polynomial threshold functions
Proceedings of the forty-second ACM symposium on Theory of computing
Approximating linear threshold predicates
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Bounded Independence Fools Halfspaces
SIAM Journal on Computing
Approximating Linear Threshold Predicates
ACM Transactions on Computation Theory (TOCT)
Nearly optimal solutions for the chow parameters problem and low-weight approximation of halfspaces
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Low-weight halfspaces for sparse boolean vectors
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We prove two main results on how arbitrary linear threshold functions $f(x) = \sign(w\cdot x - \theta)$ 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 $\eps$-close to a threshold function depending only on $\Inf(f)^2 \cdot \poly(1/\eps)$ 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 \cite{Friedgut:98}, which states that every Boolean function $f$ is $\eps$-close to a function depending only on $2^{O(\Inf(f)/\eps)}$ many variables, for the case of threshold functions. We complement this upper bound by showing that $\Omega(\Inf(f)^2 + 1/\epsilon^2)$ many variables are required for $\epsilon$-approximating threshold functions. Our second result is a proof that every $n$-variable threshold function is $\eps$-close to a threshold function with integer weights at most $\poly(n) \cdot 2^{\tilde{O}(1/\eps^{2/3})}.$ This is a significant improvement, in the dependence on the error parameter $\eps$, on an earlier result of \cite{Servedio:07cc} which gave a $\poly(n) \cdot 2^{\tilde{O}(1/\eps^{2})}$ bound. Our improvement is obtained via a new proof technique that uses strong anti-concentration bounds from probability theory. The new technique also gives a simple and modular proof of the original \cite{Servedio:07cc} result, and extends to give low-weight approximators for threshold functions under a range of probability distributions beyond just the uniform distribution.