相关论文: Average sensitivity and noise sensitivity of polyn…

The {\em Total Influence} ({\em Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function \ifnum\plusminus=1 $f: \{\pm1\}^n…

The threshold degree of a Boolean function f:{0,1}^n->{-1,+1} is the least degree of a real polynomial p such that f(x)=sgn p(x). We construct two halfspaces on {0,1}^n whose intersection has threshold degree Theta(sqrt n), an exponential…

In statistical learning theory, generalization error is used to quantify the degree to which a supervised machine learning algorithm may overfit to training data. Recent work [Xu and Raginsky (2017)] has established a bound on the…

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 =…

In this work, we revisit the problem of estimating the mean and covariance of an unknown $d$-dimensional Gaussian distribution in the presence of an $\varepsilon$-fraction of adversarial outliers. The pioneering work of [DKK+16] gave a…

We study the complexity of learning and approximation of self-bounding functions over the uniform distribution on the Boolean hypercube ${0,1}^n$. Informally, a function $f:{0,1}^n \rightarrow \mathbb{R}$ is self-bounding if for every $x…

In this paper we study the separation between two complexity measures: the degree of a Boolean function as a polynomial over the reals and its block sensitivity. We show that separation between these two measures can be improved from $…

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…

We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure, analogous to the Johnson-Lindenstrauss lemma. As…

We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main…

The pure tone hearing threshold is usually estimated from responses to stimuli at a set of standard frequencies. This paper describes a probabilistic approach to the estimation problem in which the hearing threshold is modelled as a smooth…

Understanding the generalization behavior of deep neural networks remains a fundamental challenge in modern statistical learning theory. Among existing approaches, PAC-Bayesian norm-based bounds have demonstrated particular promise due to…

Stochastic Boolean Function Evaluation is the problem of determining the value of a given Boolean function f on an unknown input x, when each bit of x_i of x can only be determined by paying an associated cost c_i. The assumption is that x…

Motivated by engineering applications such as resource allocation in networks and inventory systems, we consider average-reward Reinforcement Learning with unbounded state space and reward function. Recent works studied this problem in the…

The $\epsilon$-approximate degree $deg_\epsilon(f)$ of a Boolean function $f$ is the least degree of a real-valued polynomial that approximates $f$ pointwise to error $\epsilon$. The approximate degree of $f$ is at least $k$ iff there…

To understand the sample-to-sample fluctuations in disorder-generated multifractal patterns we investigate analytically as well as numerically the statistics of high values of the simplest model - the ideal periodic $1/f$ Gaussian noise. By…

Understanding simplicity biases in deep learning offers a promising path toward developing reliable AI. A common metric for this, inspired by Boolean function analysis, is average sensitivity, which captures a model's robustness to…

In this paper we present several heuristic algorithms, including a Genetic Algorithm (GA), for obtaining polynomial threshold function (PTF) representations of Boolean functions (BFs) with small number of monomials. We compare these among…

We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows…

In this paper, we extend the Generalized Finite Difference Method (GFDM) on unknown compact submanifolds of the Euclidean domain, identified by randomly sampled data that (almost surely) lie on the interior of the manifolds. Theoretically,…