A Quantum Approximation Scheme for k-Means

• URL: http://arxiv.org/abs/2308.08167v2
• Date: Fri, 24 May 2024 07:01:12 GMT
• Title: A Quantum Approximation Scheme for k-Means
• Authors: Ragesh Jaiswal,
• Abstract summary: We give a quantum approximation scheme for the classical $k$-means clustering problem in the QRAM model. Our quantum algorithm runs in time $tildeO left( 2tildeO(frackvarepsilon) eta2 dright)$. Unlike previous works on unsupervised learning, our quantum algorithm does not require quantum linear algebra subroutines.
• Score: 0.16317061277457
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We give a quantum approximation scheme (i.e., $(1 + \varepsilon)$-approximation for every $\varepsilon > 0$) for the classical $k$-means clustering problem in the QRAM model with a running time that has only polylogarithmic dependence on the number of data points. More specifically, given a dataset $V$ with $N$ points in $\mathbb{R}^d$ stored in QRAM data structure, our quantum algorithm runs in time $\tilde{O} \left( 2^{\tilde{O}(\frac{k}{\varepsilon})} \eta^2 d\right)$ and with high probability outputs a set $C$ of $k$ centers such that $cost(V, C) \leq (1+\varepsilon) \cdot cost(V, C_{OPT})$. Here $C_{OPT}$ denotes the optimal $k$-centers, $cost(.)$ denotes the standard $k$-means cost function (i.e., the sum of the squared distance of points to the closest center), and $\eta$ is the aspect ratio (i.e., the ratio of maximum distance to minimum distance). This is the first quantum algorithm with a polylogarithmic running time that gives a provable approximation guarantee of $(1+\varepsilon)$ for the $k$-means problem. Also, unlike previous works on unsupervised learning, our quantum algorithm does not require quantum linear algebra subroutines and has a running time independent of parameters (e.g., condition number) that appear in such procedures.

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