Sparsity without the Complexity: Loss Localisation using Tree Measurements

TL;DR

This paper investigates network loss tomography on tree-structured paths, demonstrating that sparsity can be effectively used for loss localization with guarantees of uniqueness and recovery through $ ext{l}_1$ minimization, even in noisy conditions.

Contribution

The work provides theoretical conditions for unique sparse solutions in tree-based loss tomography and introduces a fast linear algorithm for $ ext{l}_1$ minimization tailored to tree structures.

Findings

Sparse solutions are often unique in tree topologies.

A fast single-pass linear algorithm for $ ext{l}_1$ minimization is proposed.

Sparse solutions remain unique more often than previously assumed, even with lossy links and noise.

Abstract

We study network loss tomography based on observing average loss rates over a set of paths forming a tree -- a severely underdetermined linear problem for the unknown link loss probabilities. We examine in detail the role of sparsity as a regularising principle, pointing out that the problem is technically distinct from others in the compressed sensing literature. While sparsity has been applied in the context of tomography, key questions regarding uniqueness and recovery remain unanswered. Our work exploits the tree structure of path measurements to derive sufficient conditions for sparse solutions to be unique and the condition that $\ell_1$ minimization recovers the true underlying solution. We present a fast single-pass linear algorithm for $\ell_1$ minimization and prove that a minimum $\ell_1$ solution is both unique and sparsest for tree topologies. By considering the placement of lossy links within trees, we show that sparse solutions remain unique more often than is commonly supposed. We prove similar results for a noisy version of the problem.