Related papers: Boolean Functions with Biased Inputs: Approximatio…
Universal approximation theorem suggests that a shallow neural network can approximate any function. The input to neurons at each layer is a weighted sum of previous layer neurons and then an activation is applied. These activation…
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…
The interest in "Physically Unclonable Function"-devices has increased rapidly over the last few years, as they have several interesting properties for system security related applications like, for example, the management of cryptographic…
An algorithm for computing the nonlinearity of a Boolean function from its algebraic normal form (ANF) is proposed. By generalizing the expression of the weight of a Boolean function in terms of its ANF coefficients, a formulation of the…
Estimating expected polynomials of density functions from samples is a basic problem with numerous applications in statistics and information theory. Although kernel density estimators are widely used in practice for such functional…
Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well-posed and convergent, by…
Robust loss functions are essential for training deep neural networks with better generalization power in the presence of noisy labels. Symmetric loss functions are confirmed to be robust to label noise. However, the symmetric condition is…
Filtered Poisson processes are often used as reference models for intermittent fluc- tuations in physical systems. Such a process is here extended by adding a noise term, either as a purely additive term to the process or as a dynamical…
We consider the problem of deciding whether an $n$-qubit unitary (or $n$-bit Boolean function) is $\varepsilon_1$-close to some $k$-junta or $\varepsilon_2$-far from every $k$-junta, where $k$-junta unitaries act non-trivially on at most…
This note addresses the question of optimally estimating a linear functional of an object acquired through linear observations corrupted by random noise, where optimality pertains to a worst-case setting tied to a symmetric, convex, and…
In this note we consider Boolean functions defined on the discrete cube equipped with a biased product probability measure. We prove that if the spectrum of such a function is concentrated on the first two Fourier levels, then the function…
A common approach for minimizing a smooth nonlinear function is to employ finite-difference approximations to the gradient. While this can be easily performed when no error is present within the function evaluations, when the function is…
We propose a new class of Bayesian neural networks (BNNs) that can be trained using noisy data of variable fidelity, and we apply them to learn function approximations as well as to solve inverse problems based on partial differential…
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…
When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given…
This paper presents a finite difference quasi-Newton method for the minimization of noisy functions. The method takes advantage of the scalability and power of BFGS updating, and employs an adaptive procedure for choosing the differencing…
We study the work fluctuations of a particle, confined to a moving harmonic potential, under the influence of friction and external Poissonian shot noise. The asymmetry of the noise induces an effective nonlinearity in the potential, which…
We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some methods of this approximation for functions in a class induced by the convolution with a given function,…
The sparse polynomial approximation of continuous functions has emerged as a prominent area of interest in function approximation theory in recent years. A key challenge within this domain is the accurate estimation of approximation errors.…
We reinterpret multiplicative noise in neural networks as auxiliary random variables that augment the approximate posterior in a variational setting for Bayesian neural networks. We show that through this interpretation it is both efficient…