Related papers: Approximate degree, secret sharing, and concentrat…
When belief propagation (BP) converges, it does so to a stationary point of the Bethe free energy $F$, and is often strikingly accurate. However, it may converge only to a local optimum or may not converge at all. An algorithm was recently…
A polynomial threshold function (PTF) $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is a function of the form $f(x) = \mathsf{sign}(p(x))$ where $p$ is a polynomial of degree at most $d$. PTFs are a classical and well-studied complexity class…
In analogy with the regularity lemma of Szemer\'edi, regularity lemmas for polynomials shown by Green and Tao (Contrib. Discrete Math. 2009) and by Kaufman and Lovett (FOCS 2008) modify a given collection of polynomials \calF =…
This paper concerns algorithms that give correct answers with (asymptotic) density $1$. A dense description of a function $g : \omega \to \omega$ is a partial function $f$ on $\omega$ such that $\left\{n : f(n) = g(n)\right\}$ has density…
An algebraic approximation, of order $K$, of a polyhedron correlation function (CF) can be obtained from $\gamma\pp(r)$, its chord-length distribution (CLD), considering first, within the subinterval $[D_{i-1},\, D_i]$ of the full range of…
We show nearly quadratic separations between two pairs of complexity measures: 1. We show that there is a Boolean function $f$ with $D(f)=\Omega((D^{sc}(f))^{2-o(1)})$ where $D(f)$ is the deterministic query complexity of $f$ and $D^{sc}$…
A seminal result of Nisan and Szegedy (STOC, 1992) shows that for any total Boolean function, the degree of the real polynomial that computes the function, and the minimal degree of a real polynomial that point-wise approximates the…
We investigate the structure of polynomials of degree four in many variables over a fixed prime field $\mathbb{F}=\mathbb{F}_{p}$. In 2007, Green and Tao proved that if a polynomial $f:\mathbb{F}^{n}\rightarrow\mathbb{F}$ is poorly…
A fractional matching of $G$ is a function $f: E(G)\to [0,1]$ such that $\sum_{e\in E_G(v_i)}f(e)\le 1$ for any $v_i\in V(G)$, where $E_G(v_i)=\{e: e\in E(G) \ \textrm{and}\ e \ \textrm{is incident with} \ v_i\}$. Let $\alpha_f(G)$ denote…
What kinds of functions are learnable from their satisfying assignments? Motivated by this simple question, we extend the framework of De, Diakonikolas, and Servedio [DDS15], which studied the learnability of probability distributions over…
We design a new, fast algorithm for agnostically learning univariate probability distributions whose densities are well approximated by piecewise polynomial functions. Let $f$ be the density function of an arbitrary univariate distribution,…
We consider the problem of learning a discrete distribution in the presence of an $\epsilon$ fraction of malicious data sources. Specifically, we consider the setting where there is some underlying distribution, $p$, and each data source…
We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function $F:\{0,1\}^n \rightarrow \{0,1\}$, define its $k$-lift to be $F_k:\{0,1\}^n \rightarrow…
We give a highly efficient "semi-agnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let $p$ be an arbitrary distribution over an interval $I$ which is…
We study the Densest At-Least-$k$-Subgraph (DAL$k$S) problem, in which we are given an undirected graph $G$ and an integer $k$, and the goal is to find a subgraph of $G$ with at least $k$ vertices with maximum density. The best-known…
We prove a new structural lemma for partial Boolean functions $f$, which we call the seed lemma for DNF. Using the lemma, we give the first subexponential algorithm for proper learning of DNF in Angluin's Equivalence Query (EQ) model. The…
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…
We consider the problem of clustering mixtures of mean-separated Gaussians in high dimensions. We are given samples from a mixture of $k$ identity covariance Gaussians, so that the minimum pairwise distance between any two pairs of means is…
We reprove that the approximate degree of the OR function on n bits is Omega(sqrt(n)). We consider a linear program which is feasible if and only if there is an approximate polynomial for a given function, and apply the duality theory. The…
We prove a simple, nearly tight lower bound on the approximate degree of the two-level $\mathsf{AND}$-$\mathsf{OR}$ tree using symmetrization arguments. Specifically, we show that $\widetilde{\mathrm{deg}}(\mathsf{AND}_m \circ…