The estimates for Boolean surface area of polynomial threshold functions
Nankai University（南开大学）；Institute for Basic Science (IBS)（基础科学研究院）；University of Washington（华盛顿大学）；MSU（密歇根州立大学）；Max Planck Institute for Mathematics（马普数学研究所）；University of South Carolina（南卡罗来纳大学）
Fan Chang, Joseph Slote, Chun-Kai Tseng, Alexander Volberg, Haonan Zhang
Polynomial threshold functions (PTFs) are an important low-complexity class of Boolean functions, with strong connections to learning theory and approximation theory. Recent work on learning and testing PTFs has exploited structural and isoperimetric properties of the class, especially bounds on average sensitivity, one of the central themes in the study of PTFs since the Gotsman--Linial conjecture recently solved by OpenAI, \cite{GLsol}.
In this work we study PTFs through the lens of the Boolean surface area (or Talagrand boundary) \[ \mathbf{BSA}[f]=\mathbb{E}|\nabla f|=\mathbb{E}\sqrt{s_{f}(x)}, \] a natural measure of vertex-boundary complexity on the discrete cube. Our main result is that every degree-$d$ PTF has polylogarithmic Boolean surface area: \[ \mathbf{BSA}[f]\le C_d(\log(en))^{C_d}. \] The proof is based on the PTF Restriction Lemma of Kabanets, Kane, and Lu \cite{KKL2017} and proceeds through a tail bound for the pointwise sensitivity. In particular, it controls all subcritical fractional moments of the sensitivity. We also record a random block partition principle for Boolean surface area and an alternative recursive argument following Kane's work \cite{DK} on average sensitivity, which independently yields the second bound \[ \mathbf{BSA}[f]\le \exp(C(d)\sqrt{\log n}). \] This bound was obtained in \cite{FSVZ} without clarifying the magnitude of $C(d)$, which we do below. For lower moments $\bE |\nabla f|^η \le C(d, η)$ there is no dependence on $n$ for $0<η<1$.
Limits on representing Boolean functions by linear combinations of simple functions: thresholds, ReLUs, and low-degree polynomials
We consider the problem of representing Boolean functions exactly by "sparse" linear combinations (over $\mathbb{R}$ ) of functions from some "simple" class ${\cal C}$ . In particular, given ${\cal C}$ we are interested in finding low-complexity functions lacking sparse representations. When ${\cal C}$ 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 ${\cal C}$ is "overcomplete" and the set of functions is not linearly independent. We focus on the cases where ${\cal C}$ is the set of linear thresh…
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In this report, we show that all n-variable Boolean function can be represented as polynomial threshold functions (PTF) with at most $0.75 \times 2^n$ non-zero integer coefficients and give an upper bound on the absolute value of these coefficients. To our knowledge this provides the best known bound on both the PTF density (number of monomials) and weight (sum of the coefficient magnitudes) of general Boolean functions. The special case of Bent functions is also analyzed and shown that any n-variable Bent function can be represented with integer coefficients less than $2^n$ while also obeying…
Chang's lemma (Duke Mathematical Journal, 2002) is a classical result with applications across several areas in mathematics and computer science. For a Boolean function $f$ that takes values in {-1,1} let $r(f)$ denote its Fourier rank. For each positive threshold $t$, Chang's lemma provides a lower bound on $wt(f):=\Pr[f(x)=-1]$ in terms of the dimension of the span of its characters with Fourier coefficients of magnitude at least $1/t$. We examine the tightness of Chang's lemma w.r.t. the following three natural settings of the threshold: - the Fourier sparsity of $f$, denoted $k(f)$, - the …
Improved Approximation of Linear Threshold Functions
We prove two main results on how arbitrary linear threshold functions $f(x) = \sign(w\cdot x - θ)$ 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 $\eps$ -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 …
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In a recent work, O'Donnell, Servedio and Tan (STOC 2019) gave explicit pseudorandom generators (PRGs) for arbitrary $m$-facet polytopes in $n$ variables with seed length poly-logarithmic in $m,n$, concluding a sequence of works in the last decade, that was started by Diakonikolas, Gopalan, Jaiswal, Servedio, Viola (SICOMP 2010) and Meka, Zuckerman (SICOMP 2013) for fooling linear and polynomial threshold functions, respectively. In this work, we consider a natural extension of PRGs for intersections of positive spectrahedrons. A positive spectrahedron is a Boolean function $f(x)=[x_1A^1+\cdot…
