Related papers: Polynomial Representations of Threshold Functions …
We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of degree-$d$ polynomial threshold functions (PTFs). These bounds hold both for PTFs over the Boolean hypercube and for PTFs over $\R^n$ under the…
When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given…
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…
Representations of Boolean functions by real polynomials play an important role in complexity theory. Typically, one is interested in the least degree of a polynomial p(x_1,...,x_n) that approximates or sign-represents a given Boolean…
An efficient algorithm is presented to compute the characteristic polynomial of a threshold graph. Threshold graphs were introduced by Chv\'atal and Hammer, as well as by Henderson and Zalcstein in 1977. A threshold graph is obtained from a…
The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in…
In this article we present new results on neural networks with linear threshold activation functions. We precisely characterize the class of functions that are representable by such neural networks and show that 2 hidden layers are…
The degrees of polynomials representing or approximating Boolean functions are a prominent tool in various branches of complexity theory. Sherstov recently characterized the minimal degree deg_{\eps}(f) among all polynomials (over the…
In pursuit of a deeper understanding of Boolean Promise Constraint Satisfaction Problems (PCSPs), we identify a class of problems with restricted structural complexity, which could serve as a promising candidate for complete…
Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
We study the efficient learnability of low-degree polynomial threshold functions (PTFs) in the presence of a constant fraction of adversarial corruptions. Our main algorithmic result is a polynomial-time PAC learning algorithm for this…
We consider positive solutions to parametrized systems of generalized polynomial equations (with real exponents and positive parameters). By a fundamental result obtained in parallel work, polynomial systems are determined by geometric…
We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of polynomial threshold functions. More specifically, for a Boolean function f on n variables equal to the sign of a real, multivariate polynomial…
We present fully polynomial approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures…
This work aims to introduce the framework of polynomial optimization theory to solve fractional polynomial problems (FPPs). Unlike other widely used optimization frameworks, the proposed one applies to a larger class of FPPs, not…
Recent years have witnessed the introduction and development of extremely fast rational function algorithms. Many ideas in this realm arose from polynomial-based linear-algebraic algorithms. However, polynomial approximation is occasionally…
The tree-width of a multivariate polynomial is the tree-width of the hypergraph with hyperedges corresponding to its terms. Multivariate polynomials of bounded tree-width have been studied by Makowsky and Meer as a new sparsity condition…
We develop catalytic algorithms for fundamental problems in algorithm design that run in polynomial time, use only $\mathcal{O}(\log(n))$ workspace, and use sublinear catalytic space matching the best-known space bounds of non-catalytic…
In this work we design graph neural network architectures that capture optimal approximation algorithms for a large class of combinatorial optimization problems, using powerful algorithmic tools from semidefinite programming (SDP).…
We give polynomial-time approximation schemes for monotone maximization problems expressible in terms of distances (up to a fixed upper bound) and efficiently solvable in graphs of bounded treewidth. These schemes apply in all fractionally…