On Efficient Low Distortion Ultrametric Embedding

Jul 12, 2020

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A classic problem in unsupervised learning and data analysis is to find simpler and easy-to-visualize representations of the data that preserve its essential properties. A widely-used method to preserve the underlying hierarchical structure of the data while reducing its complexity is to find an embedding of the data into a tree or an ultrametric. The most popular algorithms for this task are the classic "linkage" algorithms (single, average, or complete). However, these methods exhibit a quite prohibitive running time of Θ(n^2). In this paper, we provide a new algorithm which takes as input a set of points P in R^d, and for every c> 1, runs in time n^1+O(1/c^2) to output an ultrametric Δ such that for any two points u,v in P, we have Δ(u,v) is within a multiplicative factor of 5c to the distance between u and v in the "best" ultrametric representation of P. Here, the best ultrametric is the ultrametric Δ^* that minimizes the maximum distance distortion with respect to the ℓ_2 distance, namely that minimizes max_u,v ∈ PΔ^*(u,v)/||u-v||_2." We complement the above result by showing that under popular complexity theoretic assumptions, for every constant ϵ>0, no algorithm with running time n^2-ϵ can distinguish between inputs that admit isometric embedding and inputs that can incur a distortion of 3/2 in L∞ -metric. Finally, we present empirical evaluation on classic machine learning datasets and show that the output of our algorithm is comparable to the output of the linkage algorithms while achieving a much faster running time.

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About ICML 2020

The International Conference on Machine Learning (ICML) is the premier gathering of professionals dedicated to the advancement of the branch of artificial intelligence known as machine learning. ICML is globally renowned for presenting and publishing cutting-edge research on all aspects of machine learning used in closely related areas like artificial intelligence, statistics and data science, as well as important application areas such as machine vision, computational biology, speech recognition, and robotics. ICML is one of the fastest growing artificial intelligence conferences in the world. Participants at ICML span a wide range of backgrounds, from academic and industrial researchers, to entrepreneurs and engineers, to graduate students and postdocs.

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