Near-Optimal Quantum Coreset Construction Algorithms for Clustering
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k-Clustering in ℝ^d (e.g., k-median and k-means) is a fundamental machine learning problem. While near-linear time approximation algorithms were known in the classical setting for a dataset with cardinality n, it remains open to find sublinear-time quantum algorithms. We give quantum algorithms that find coresets for k-clustering in ℝ^d with Õ(√(nk)d^3/2) query complexity. Our coreset reduces the input size from n to poly(kϵ^-1d), so that existing α-approximation algorithms for clustering can run on top of it and yield (1 + ϵ)α-approximation. This eventually yields a quadratic speedup for various k-clustering approximation algorithms. We complement our algorithm with a nearly matching lower bound, that any quantum algorithm must make Ω(√(nk)) queries in order to achieve even O(1)-approximation for k-clustering.
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