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We study network loss tomography based on observing average loss rates over aset of paths forming a tree -- a severely underdetermined linear problem forthe unknown link loss probabilities. We examine in detail the role of sparsityas a regularising principle, pointing out that the problem is technicallydistinct from others in the compressed sensing literature. While sparsity hasbeen applied in the context of tomography, key questions regarding uniquenessand recovery remain unanswered. Our work exploits the tree structure of pathmeasurements to derive sufficient conditions for sparse solutions to be uniqueand the condition that $\ell_1$ minimization recovers the true underlyingsolution. We present a fast single-pass linear algorithm for $\ell_1$minimization and prove that a minimum $\ell_1$ solution is both unique andsparsest for tree topologies. By considering the placement of lossy linkswithin trees, we show that sparse solutions remain unique more often than iscommonly supposed. We prove similar results for a noisy version of the problem.

Sparsity without the Complexity: Loss Localisation using Tree Measurements