Related papers: Polynomial Threshold Functions of Bounded Tree-Wid…
Here, we find the characteristics polynomial of normalized Laplacian of a tree. The coefficients of this polynomial are expressed by the higher order general Randi\'c indices for matching, whose values depend on the structure of the tree.…
We consider the problem of sensitivity of threshold risk, defined as the probability of a function of a random variable falling below a specified threshold level $\delta >0.$ We demonstrate that for polynomial and rational functions of that…
In the Fully Leafed Induced Subtrees, one is given a graph $G$ and two integers $a$ and $b$ and the question is to find an induced subtree of $G$ with $a$ vertices and at least $b$ leaves. This problem is known to be NP-complete even when…
Phylogenetic networks are a generalization of phylogenetic trees to leaf-labeled directed acyclic graphs that represent ancestral relationships between species whose past includes non-tree-like events such as hybridization and horizontal…
Given an $\mathbb{N}$-weighted tree automaton, we give a decision procedure for exponential vs polynomial growth (with respect to the input size) in quadratic time, and an algorithm that computes the exact polynomial degree of growth in…
Bayesian Optimization (BO) has shown significant success in tackling expensive low-dimensional black-box optimization problems. Many optimization problems of interest are high-dimensional, and scaling BO to such settings remains an…
Advances in neural architecture search, as well as explainability and interpretability of connectionist architectures, have been reported in the recent literature. However, our understanding of how to design Bayesian Deep Learning (BDL)…
Deep networks often exhibit a preference for "simple" solutions, and such a simplicity bias is widely believed to play a key role in generalization. Yet a broadly applicable, quantitative measure of simplicity remains elusive. We introduce…
Treewidth is a measure of how tree-like a graph is. It has many important algorithmic applications because many NP-hard problems on general graphs become tractable when restricted to graphs of bounded treewidth. Algorithms for problems on…
The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in [B. Despr\'es, Numer.…
We consider random binary trees that appear as the output of certain standard algorithms for sorting and searching if the input is random. We introduce the subtree size metric on search trees and show that the resulting metric spaces…
Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that…
A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…
We develop several notions of multiplicity for linear factors of multivariable polynomials over different arithmetics (hyperfields). The key example is multiplicities over the hyperfield of signs, which encapsulates the arithmetic of…
We prove that the Fourier dimension of any Boolean function with Fourier sparsity $s$ is at most $O\left(s^{2/3}\right)$. Our proof method yields an improved bound of $\widetilde{O}(\sqrt{s})$ assuming a conjecture of…
We study the expressivity and learning process for polynomial neural networks (PNNs) with monomial activation functions. The weights of the network parametrize the neuromanifold. In this paper, we study certain neuromanifolds using tools…
Treewidth is a parameter that measures how tree-like a relational instance is, and whether it can reasonably be decomposed into a tree. Many computation tasks are known to be tractable on databases of small treewidth, but computing the…
The study of Markov processes and broadcasting on trees has deep connections to a variety of areas including statistical physics, graphical models, phylogenetic reconstruction, Markov Chain Monte Carlo, and community detection in random…
We obtain upper bounds, independent of the ambient dimension, for the number of realizable zero-nonzero patterns and (over ordered fields) sign conditions of a finite family of polynomials $\mathcal P$ restricted to an algebraic subset $V$…
The definition of $n$-width of a bounded subset $A$ in a normed linear space $X$ is based on the existence of $n$-dimensional subspaces. Although the concept of an $n$-dimensional subspace is not available for metric trees, in this paper,…