arXiv subjects

Louise Raphael

Publications and source records attributed to Louise Raphael.

3 recordsLinked to original sources

arXiv2024

Asymptotic Lower Bounds for the Feedback Arc Set Problem in Random Graphs

Given a directed graph, the Minimum Feedback Arc Set (FAS) problem asks for a minimum (size) set of arcs in a directed graph, which, when removed, results in an acyclic graph. In a seminal paper, Berger and Shor [1], in 1990, developed initial upper bounds for the FAS problem in general directed graphs. Here we find asymptotic \textit{lower bounds} for the FAS problem in a class of random, oriented, directed graphs derived from the Erd\H{o}s-R\'{e}nyi model $G(n,M)$, with n vertices and M (undirected) edges, the latter randomly chosen. Each edge is then randomly given a direction to form our directed graph. We show that $$Pr\left(\textbf{Y}^* \le M \left( \frac{1}{2} -\sqrt{\frac{\log n}{\Delta_{av}}}\right)\right)$$ approaches zero exponentially in $n$, with $\textbf{Y}^*$ the (random) size of the minimum feedback arc set and $\Delta_{av}=2M/n$ the average vertex degree. Lower bounds for random tournaments, a special case, were obtained by Spencer [12] and de la Vega [13] and these are discussed. In comparing the bound above to averaged experimental FAS data on related random graphs developed by K. Hanauer [7] we find that the approximation $\textbf{Y}^*_{av} \approx M\left( \frac{1}{2} -\frac{1}{2}\sqrt{\frac{\log n}{\Delta_{av}}}\right)$ lies remarkably close graphically to the algorithmically computed average size $\textbf{Y}^*_{av}$ of minimum feedback arc sets.

math.CO↗

arXiv2024

Asymptotics of the Minimal Feedback Arc Set in Erdős-Rényi Graphs

Given a directed graph, the Minimal Feedback Arc Set (FAS) problem asks for a minimal set of arcs which, when removed, results in an acyclic graph. Equivalently, the FAS problem asks to find an ordering of the vertices that minimizes the number of feedback arcs. The FAS problem is considered an algorithmic problem of central importance in discrete mathematics. Our purpose in this paper is to consider the problem in the context of Erdős-Rényi random directed graphs, denoted $D(n,p)$, in which each possible directed arc is included with a fixed probability $p>0$. Our interest is the typical ratio of the number of feedforward arcs to the number of feedback arcs that are removed in the FAS problem. We show that as the number $n$ of vertices goes to infinity the probability that this ratio is greater than $1+ε$ for any fixed $ε> 0$ approaches zero. Similarly, letting $p$ go to zero as $n\rightarrow \infty$ this result remains true if $p>C\log{n}/n$ where $C$ depends on $ε$.

math.CO↗

arXiv2013

Feature vector regularization in machine learning

Problems in machine learning (ML) can involve noisy input data, and ML classification methods have reached limiting accuracies when based on standard ML data sets consisting of feature vectors and their classes. Greater accuracy will require incorporation of prior structural information on data into learning. We study methods to regularize feature vectors (unsupervised regularization methods), analogous to supervised regularization for estimating functions in ML. We study regularization (denoising) of ML feature vectors using Tikhonov and other regularization methods for functions on ${\bf R}^n$. A feature vector ${\bf x}=(x_1,\ldots,x_n)=\{x_q\}_{q=1}^n$ is viewed as a function of its index $q$, and smoothed using prior information on its structure. This can involve a penalty functional on feature vectors analogous to those in statistical learning, or use of proximity (e.g. graph) structure on the set of indices. Such feature vector regularization inherits a property from function denoising on ${\bf R}^n$, in that accuracy is non-monotonic in the denoising (regularization) parameter $α$. Under some assumptions about the noise level and the data structure, we show that the best reconstruction accuracy also occurs at a finite positive $α$ in index spaces with graph structures. We adapt two standard function denoising methods used on ${\bf R}^n$, local averaging and kernel regression. In general the index space can be any discrete set with a notion of proximity, e.g. a metric space, a subset of ${\bf R}^n$, or a graph/network, with feature vectors as functions with some notion of continuity. We show this improves feature vector recovery, and thus the subsequent classification or regression done on them. We give an example in gene expression analysis for cancer classification with the genome as an index space and network structure based protein-protein interactions.

stat.ML↗