Related papers: Log-rank and lifting for AND-functions
An open problem in complexity theory is to find the minimal degree of a polynomial representing the $n$-bit OR function modulo composite $m$. This problem is related to understanding the power of circuits with $\text{MOD}_m$ gates where $m$…
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…
Let $\mathcal{L}$ be a language that can be decided in linear space and let $\epsilon >0$ be any constant. Let $\mathcal{A}$ be the exponential hardness assumption that for every $n$, membership in $\mathcal{L}$ for inputs of length~$n$…
We study a conjecture called "linear rank conjecture" recently raised in (Tsang et al., FOCS'13), which asserts that if many linear constraints are required to lower the degree of a GF(2) polynomial, then the Fourier sparsity (i.e. number…
We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded-width…
Consider a matrix $\mathbf{F} \in \mathbb{K}[x]^{m \times n}$ of univariate polynomials over a field $\mathbb{K}$. We study the problem of computing the column rank profile of $\mathbf{F}$. To this end we first give an algorithm which…
A fuzzy Boolean function is a map $f:\cube^n\to [0,1]$, where $n\in\mathbb N$. We introduce and compare three ways of saying that such a function has bounded complexity. The first is a sampling property: the value $f(x)$ can be recovered,…
Weighted First Order Model Counting (WFOMC) is fundamental to probabilistic inference in statistical relational learning models. As WFOMC is known to be intractable in general ($\#$P-complete), logical fragments that admit polynomial time…
We show that there exists a Boolean function $F$ which observes the following separations among deterministic query complexity $(D(F))$, randomized zero error query complexity $(R_0(F))$ and randomized one-sided error query complexity…
A tensor defined over a finite field $\mathbb{F}$ has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order $d$ tensor has partition rank 1 if it can be written as a product of two…
Type-and-effect systems incorporate information about the computational effects, e.g., state mutation, probabilistic choice, or I/O, a program phrase may invoke alongside its return value. A semantics for type-and-effect systems involves a…
Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated `lifted' function obtained by composing the function with many copies of another function…
The article concerns low-rank approximation of matrices generated by sampling a smooth function of two $m$-dimensional variables. We identify several misconceptions surrounding a claim that, for a specific class of analytic functions, such…
Polynomial approximations to boolean functions have led to many positive results in computer science. In particular, polynomial approximations to the sign function underly algorithms for agnostically learning halfspaces, as well as…
For an analytic and univalent function $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$, the logarithmic coefficients $\gamma_n$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty}…
Spoken word recognition involves at least two basic computations. First is matching acoustic input to phonological categories (e.g. /b/, /p/, /d/). Second is activating words consistent with those phonological categories. Here we test the…
The rank of a finite algebraic structure with a single binary operation is the minimum number of elements needed to express every other element under the closure of the operation. In the case of groups, the previous best algorithm for…
The scope of this work is the constraint-based synthesis of termination arguments for the restricted class of programs called linear lasso programs. A termination argument consists of a ranking function as well as a set of supporting…
A tensor defined over a finite field $\mathbb{F}$ has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order $d$ tensor has partition rank 1 if it can be written as a product of two…
Given an $m\times n$ binary matrix $M$ with $|M|=p\cdot mn$ (where $|M|$ denotes the number of 1 entries), define the discrepancy of $M$ as $\mbox{disc}(M)=\displaystyle\max_{X\subset [m], Y\subset [n]}\big||M[X\times Y]|-p|X|\cdot…