Statistical Analysis of Metric Graph Reconstruction

TL;DR

This paper investigates the statistical problem of reconstructing the topology of metric graphs from noisy data, providing bounds on the minimax risk and analyzing reconstruction algorithms for applications like street maps and neural networks.

Contribution

It offers theoretical bounds on the minimax risk for metric graph reconstruction under noiseless and noisy conditions, extending previous algorithms.

Findings

Derived lower and upper bounds on minimax risk

Analyzed reconstruction algorithm performance

Applicable to real-world filamentary data

Abstract

A metric graph is a 1-dimensional stratified metric space consisting of vertices and edges or loops glued together. Metric graphs can be naturally used to represent and model data that take the form of noisy filamentary structures, such as street maps, neurons, networks of rivers and galaxies. We consider the statistical problem of reconstructing the topology of a metric graph embedded in R^D from a random sample. We derive lower and upper bounds on the minimax risk for the noiseless case and tubular noise case. The upper bound is based on the reconstruction algorithm given in Aanjaneya et al. (2012).