Deterministic Error Bounds for Euclidean Clustering
Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
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Keywords: euclidean, subspace, davis, kahan
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TL;DR: This paper presents a closed-form solution to k-means clustering and a deterministic universal error bound
Abstract: This paper gives a closed-form solution to Euclidean clustering, also known as k-means clustering. The key observation behind our solution is that the features of clustered data lie near a subspace whose projection operator encodes the clustering. In contrast to classical alternating approaches like Lloyd's algorithm or K-means++, which suffer local minima, our solution can be trivially computed with a singular value decomposition. Moreover, we show that if the distinct clusters are sufficiently well-defined (meaning different clusters are sufficiently separated, and data in each cluster not too scattered), our solution is deterministically guaranteed to be correct. We corroborate our theoretical findings with a comprehensive array of experiments, showing that simple relaxations of our solution yield algorithms that not only rival but also surpass the current state-of-the-art in a wide variety of settings, both in terms of accuracy and speed.
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Submission Number: 2110
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