Related papers: The Computational Complexity of Understanding Netw…
We consider the Stochastic Boolean Function Evaluation (SBFE) problem where the task is to efficiently evaluate a known Boolean function $f$ on an unknown bit string $x$ of length $n$. We determine $f(x)$ by sequentially testing the…
We provide two sufficient and necessary conditions to characterize any $n$-bit partial Boolean function with exact quantum 1-query complexity. Using the first characterization, we present all $n$-bit partial Boolean functions that depend on…
We present completeness results for inference in Bayesian networks with respect to two different parameterizations, namely the number of variables and the topological vertex separation number. For this we introduce the parameterized…
Fixing an arbitrary set $\mathcal{F}$ of complex-valued functions over Boolean variables yields a counting problem $\#\mathcal{F}$. Taking only functions from $\mathcal{F}$ to form a tensor network as the problem's input, the counting…
The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such…
A fundamental issue in real-world systems, such as sensor networks, is the selection of observations which most effectively reduce uncertainty. More specifically, we address the long standing problem of nonmyopically selecting the most…
Let $p$ be a prime. Given a polynomial in $\F_{p^m}[x]$ of degree $d$ over the finite field $\F_{p^m}$, one can view it as a map from $\F_{p^m}$ to $\F_{p^m}$, and examine the image of this map, also known as the value set. In this paper,…
A nearest neighbor representation of a Boolean function is a set of positive and negative prototypes in $R^n$ such that the function has value 1 on an input iff the closest prototype is positive. For $k$-nearest neighbor representation the…
We say that a function is rare-case hard against a given class of algorithms (the adversary) if all algorithms in the class can compute the function only on an $o(1)$-fraction of instances of size $n$ for large enough $n$. Starting from any…
A Bayesian network is a directed acyclic graph that represents statistical dependencies between variables of a joint probability distribution. A fundamental task in data science is to learn a Bayesian network from observed data.…
We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let $f$ be an $m$-bit Boolean function and consider an $n$-bit function $F$ obtained by applying $f$ to conjunctions of…
In nearest-neighbor classification problems, a set of $d$-dimensional training points are given, each with a known classification, and are used to infer unknown classifications of other points by using the same classification as the nearest…
In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of f is $\|\hat{f}\|_1=\sum_{\alpha}|\hat{f}(\alpha)|$). Specifically, we prove the following results for functions $f:\{0,1\}^n \to…
This paper investigates the learnability of the nonlinearity property of Boolean functions using neural networks. We train encoder style deep neural networks to learn to predict the nonlinearity of Boolean functions from examples of…
We show that two related classes of algorithms, stable algorithms and Boolean circuits with bounded depth, cannot produce an approximate sample from the uniform measure over the set of solutions to the symmetric binary perceptron model at…
We prove that any non-adaptive algorithm that tests whether an unknown Boolean function $f: \{0, 1\}^n\to \{0, 1\}$ is a $k$-junta or $\epsilon$-far from every $k$-junta must make $\widetilde{\Omega}(k^{3/2} / \epsilon)$ many queries for a…
We consider an unconstrained continuous optimization problem where, in each iteration, gradient estimates may be arbitrarily corrupted with a probability greater than 1/2. Additionally, function value estimates may exhibit heavy-tailed…
A probability distribution over the Boolean cube is monotone if flipping the value of a coordinate from zero to one can only increase the probability of an element. Given samples of an unknown monotone distribution over the Boolean cube, we…
We examine the necessary and sufficient complexity of neural networks to approximate functions from different smoothness spaces under the restriction of encodable network weights. Based on an entropy argument, we start by proving lower…
Determinantal Point Processes (DPPs) are a widely used probabilistic model for negatively correlated sets. DPPs have been successfully employed in Machine Learning applications to select a diverse, yet representative subset of data. In…