Related papers: Polynomial Representations of Threshold Functions …
We present an amelioration of current known algorithms for optimal spectral partitioning problems. The idea is to use the advantage of a representation using density functions while decreasing the computational time. This is done by…
Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and…
The $2 \rightarrow q$ norm of a matrix $X \in \mathbb{R}^{n \times d}$ is defined as $\lVert X \rVert_{2 \rightarrow q} = \sup_{\lVert v \rVert_2 = 1} \lVert Xv \rVert_q$. We give polynomial-time multiplicative approximation algorithms for…
We study a graph partitioning problem motivated by the simulation of the physical movement of multi-body systems on an atomistic level, where the forces are calculated from a quantum mechanical description of the electrons. Several advanced…
We enhance the approximation capabilities of algebraic polynomials by composing them with homeomorphisms. This composition yields families of functions that remain dense in the space of continuous functions, while enabling more accurate…
Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in…
Symmetric submodular functions are an important family of submodular functions capturing many interesting cases including cut functions of graphs and hypergraphs. Maximization of such functions subject to various constraints receives little…
In this paper, we study first-order methods on a large variety of low-rank matrix optimization problems, whose solutions only live in a low dimensional eigenspace. Traditional first-order methods depend on the eigenvalue decomposition at…
The problem of testing low-degree polynomials has received significant attention over the years due to its importance in theoretical computer science, and in particular in complexity theory. The problem is specified by three parameters:…
We formalize the problem of selecting the optimal set of options for planning as that of computing the smallest set of options so that planning converges in less than a given maximum of value-iteration passes. We first show that the problem…
This paper presents a novel stochastic barrier function (SBF) framework for safety analysis of stochastic systems based on piecewise (PW) functions. We first outline a general formulation of PW-SBFs. Then, we focus on PW-Constant (PWC) SBFs…
Polynomial functions have plenty of useful analytical properties, but they are rarely used as learning models because their function class is considered to be restricted. This work shows that when trained properly polynomial functions can…
The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the…
One fundamental goal of high-dimensional statistics is to detect or recover planted structure (such as a low-rank matrix) hidden in noisy data. A growing body of work studies low-degree polynomials as a restricted model of computation for…
We develop a pseudo-random generator to fool degree-$d$ polynomial threshold functions with respect to the Gaussian distribution. For $c>0$ any constant, we construct a pseudo-random generator that fools such functions to within $\epsilon$…
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…
We set new speed records for multiplying long polynomials over finite fields of characteristic two. Our multiplication algorithm is based on an additive FFT (Fast Fourier Transform) by Lin, Chung, and Huang in 2014 comparing to previously…
It is well known that, using fast algorithms for polynomial multiplication and division, evaluation of a polynomial $F \in \mathbb{C}[x]$ of degree $n$ at $n$ complex-valued points can be done with $\tilde{O}(n)$ exact field operations in…
The fastest known algorithm for factoring a degree $n$ univariate polynomial over a finite field $\mathbb{F}_q$ runs in time $O(n^{3/2 + o(1)}\text{polylog } q)$, and there is a reason to believe that the $3/2$ exponent represents a…
When computing bounds, spatial branch-and-bound algorithms often linearly outer approximate convex relaxations for non-convex expressions in order to capitalize on the efficiency and robustness of linear programming solvers. Considering…