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Sela Fried

Publications and source records attributed to Sela Fried.

29 records · Page 2Linked to original sources
arXiv2023

Further Results on Random Walk Labelings

Recently, we initiated the study of random walk labelings of graphs. These are graph labelings that are obtainable by performing a random walk on the graph, such that each vertex is labeled upon its first visit. In this work, we calculate the number of random walk labelings of several natural graph families: The wheel, fan, barbell, lollipop, tadpole, friendship, and snake graphs. Additionally, we prove several combinatorial identities that emerged during the calculations.

math.CO↗
arXiv2023

Graph Labelings Obtainable by Random Walks

We initiate the study of what we refer to as random walk labelings of graphs. These are graph labelings that are obtainable by performing a random walk on the graph, such that the labeling occurs increasingly whenever an unlabeled vertex is encountered. Some of the results we obtain involve sums of inverses of binomial coefficients, for which we obtain new identities. In particular, we prove that $\sum_{k=0}^{n-1}2^{k}(2k+1)^{-1}\binom{2k}{k}^{-1}\binom{n+k}{k}=\binom{2n}{n}2^{-n}\sum_{k=0}^{n-1}2^{k}(2k+1)^{-1}\binom{2k}{k}^{-1}$, thus confirming a conjecture of Bala.

math.CO↗
arXiv2022

The minimal sum of squares over partitions with a nonnegative rank

Motivated by a question of Defant and Propp (2020) regarding the connection between the degrees of noninvertibility of functions and those of their iterates, we address the combinatorial optimization problem of minimizing the sum of squares over partitions of $n$ with a nonnegative rank. Denoting the sequence of the minima by $(m_n)_{n\in\mathbb{N}}$, we prove that $m_n=Θ\left(n^{4/3}\right)$. Consequently, we improve by a factor of $2$ the lower bound provided by Defant and Propp for iterates of order two.

math.CO↗
arXiv2022

The expected degree of noninvertibility of compositions of functions and a related combinatorial identity

Recently, Defant and Propp [2020] defined the degree of noninvertibility of a function $f\colon X\to Y$ between two finite nonempty sets by $\text{deg}(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. We obtain an exact formula for the expected degree of noninvertibility of the composition of $t$ functions for every $t\in \mathbb{N}$. An equivalent formulation for the definition of the degree of noninvertibility is then the starting point for a generalization yielding a seemingly new combinatorial identity involving the Stirling transform of the signed Stirling numbers of the first kind.

math.CO↗
arXiv2022

Identity testing of reversible Markov chains

We consider the problem of identity testing of Markov chain transition matrices based on a single trajectory of observations under the distance notion introduced by Daskalakis et al. [2018a] and further analyzed by Cherapanamjeri and Bartlett [2019]. Both works made the restrictive assumption that the Markov chains under consideration are symmetric. In this work we relax the symmetry assumption and show that it is possible to perform identity testing under the much weaker assumption of reversibility, provided that the stationary distributions of the reference and of the unknown Markov chains are close under a distance notion related to the separation distance. Additionally, we provide intuition on the distance notion of Daskalakis et al. [2018a] by showing how it behaves under several natural operations. In particular, we address some of their open questions.

math.ST↗
arXiv2021

$Θ$-Hilbertianity and strong $Θ$-Hilbertianity

$Θ$-Hilbertianity and its strengthening, strong $Θ$-Hilbertianity, are two generalizations of Hilbertianity inspired by Jarden's definition of $p$-Hilbertianity and strong $p$-Hilbertianity. Jarden has asked whether the two notions defined by him are actually the same. We address this question in its more general version of $Θ$-Hilbertianity and show that for PRC, and, in particular, for PAC fields, $p$-Hilbertianity and strong $p$-Hilbertianity coincide.

math.NT↗
arXiv2021

On the $α$-lazy version of Markov chains in estimation and testing problems

Given access to a single long trajectory generated by an unknown irreducible Markov chain $M$, we simulate an $α$-lazy version of $M$ which is ergodic. This enables us to generalize recent results on estimation and identity testing that were stated for ergodic Markov chains in a way that allows fully empirical inference. In particular, our approach shows that the pseudo spectral gap introduced by Paulin [2015] and defined for ergodic Markov chains may be given a meaning already in the case of irreducible but possibly periodic Markov chains.

stat.ML↗
arXiv2021

The restrictiveness of the hazard rate order and the moments of the maximal coordinate of a random vector uniformly distributed on the probability n-simplex

Continuing the work of [9] who defined the restrictiveness of stochastic orders and calculated the restrictiveness of the usual stochastic order and the likelihood ratio order, we calculate the restrictiveness of the hazard rate order. Inspired by the works of [17] and [24], we propose a possible application of the restrictiveness results in randomness testing. We then apply a dimension reduction technique, that proved useful in obtaining the restrictiveness results, and provide an alternative proof for Whitworth's formula. By integrating the formula, we derive the moments of the maximal coordinate of a random vector that is uniformly distributed on the probability $n$-simplex.

math.PR↗
arXiv2020

The probability that two random points on the $n$-probability simplex are comparable with respect to the first order stochastic dominance and the monotone likelihood ratio partial orders

First order stochastic dominance and monotone likelihood ratio are two partial orders on the $n$-probability simplex that play an important role in the establishment of structural results for MDPs and POMDPs. We study the strength of those partial orders in terms of how likely it is for two random points on the $n$-probability simplex to be comparable with respect to each of the two partial orders.

math.PR↗