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The problem of fitting distances by tree-metrics has received significantattention in the theoretical computer science and machine learning communitiesalike, due to many applications in natural language processing, phylogeny,cancer genomics and a myriad of problem areas that involve hierarchicalclustering. Despite the existence of several provably exact algorithms fortree-metric fitting of data that inherently obeys tree-metric constraints, muchless is known about how to best fit tree-metrics for data whose structuremoderately (or substantially) differs from a tree. For such noisy data, mostavailable algorithms perform poorly and often produce negative edge weights inrepresentative trees. Furthermore, it is currently not known how to choose themost suitable approximation objective for noisy fitting. Our contributions areas follows. First, we propose a new approach to tree-metric denoising(HyperAid) in hyperbolic spaces which transforms the original data into datathat is ``more'' tree-like, when evaluated in terms of Gromov's $\delta$hyperbolicity. Second, we perform an ablation study involving two choices forthe approximation objective, $\ell_p$ norms and the Dasgupta loss. Third, weintegrate HyperAid with schemes for enforcing nonnegative edge-weights. As aresult, the HyperAid platform outperforms all other existing methods in theliterature, including Neighbor Joining (NJ), TreeRep and T-REX, both onsynthetic and real-world data. Synthetic data is represented by edge-augmentedtrees and shortest-distance metrics while the real-world datasets include Zoo,Iris, Glass, Segmentation and SpamBase; on these datasets, the averageimprovement with respect to NJ is $125.94\%$.

HyperAid: Denoising in hyperbolic spaces for tree-fitting and hierarchical clustering