[Paper Review] q-means: A quantum algorithm for unsupervised machine learning

Iordanis Kerenidis, Jonas Landman+2 more|arXiv (Cornell University)|Dec 10, 2018

Quantum Computing Algorithms and ArchitectureComputer Science21 references116 citations

TL;DR

q-means is a quantum clustering algorithm that mirrors delta-k-means, outputting centroids with high probability and achieving sublinear dependence on N with runtimes polylogarithmic in N, linear in d, and polynomial in k under QRAM assumptions.

ABSTRACT

Quantum machine learning is one of the most promising applications of a full-scale quantum computer. Over the past few years, many quantum machine learning algorithms have been proposed that can potentially offer considerable speedups over the corresponding classical algorithms. In this paper, we introduce q-means, a new quantum algorithm for clustering which is a canonical problem in unsupervised machine learning. The $q$-means algorithm has convergence and precision guarantees similar to $k$-means, and it outputs with high probability a good approximation of the $k$ cluster centroids like the classical algorithm. Given a dataset of $N$ $d$-dimensional vectors $v_i$ (seen as a matrix $V \in \mathbb{R}^{N imes d})$ stored in QRAM, the running time of q-means is $\widetilde{O}\left( k d \fracη{δ^2}κ(V)(μ(V) + k \fracηδ) + k^2 \frac{η^{1.5}}{δ^2} κ(V)μ(V) ight)$ per iteration, where $κ(V)$ is the condition number, $μ(V)$ is a parameter that appears in quantum linear algebra procedures and $η= \max_{i} ||v_{i}||^{2}$. For a natural notion of well-clusterable datasets, the running time becomes $\widetilde{O}\left( k^2 d \frac{η^{2.5}}{δ^3} + k^{2.5} \frac{η^2}{δ^3} ight)$ per iteration, which is linear in the number of features $d$, and polynomial in the rank $k$, the maximum square norm $η$ and the error parameter $δ$. Both running times are only polylogarithmic in the number of datapoints $N$. Our algorithm provides substantial savings compared to the classical $k$-means algorithm that runs in time $O(kdN)$ per iteration, particularly for the case of large datasets.

Motivation & Objective

• Motivate clustering as a canonical unsupervised learning problem and address its scalability for large datasets.
• Develop a quantum analogue of k-means (delta-k-means) that preserves convergence and approximation guarantees.
• Provide a runtime analysis showing polylogarithmic dependence on the number of data points N and linear dependence on feature dimension d.
• Ensure the algorithm outputs classical centroids that can be used in subsequent classical or quantum tasks.

Proposed method

• Define q-means as a quantum counterpart to delta-k-means for clustering N vectors in R^d stored in QRAM.
• Use quantum subroutines for distance estimation, minimum selection, matrix multiplication, and tomography to update centroids.
• Provide runtime bounds per iteration that depend on k, d, η (max row norm), δ (robustness parameter), κ(V) (condition number), and μ(V) from quantum linear algebra procedures.
• Leverage amplitude estimation and median-based amplification to obtain reliable distance estimates.
• Employ vector state tomography to recover classical centroid vectors from quantum states created during the algorithm.
• Show that q-means outputs centroids consistent with delta-k-means with high probability.

Experimental results

Research questions

• RQ1 Can q-means reproduce the behavior and guarantees of the classical delta-k-means in a quantum setting?
• RQ2 What are the per-iteration and overall runtime requirements of q-means in terms of dataset parameters (N, d, k, η, δ, κ(V), μ(V))?
• RQ3 How does the well-clusterable data model affect the theoretical guarantees and running time of q-means?
• RQ4 Are the produced centroids usable as classical objects for downstream tasks, and how does accuracy compare to classical k-means?

Key findings

• Per-iteration runtime for general data: ~O~(kd η/(δ^2) κ(V)(μ(V)+k η/δ) + k^2 η^1.5/δ^2 κ(V) μ(V)).
• Per-iteration runtime under well-clusterable data: ~O~(k^2 d η^2.5/δ^3 + k^2.5 η^2/δ^3).
• Running times are polylogarithmic in N and linear in d, with polynomial dependence on k, η, and 1/δ.
• Algorithm outputs classical centroids corresponding to the delta-k-means solution with high probability.
• Uses QRAM data structures and quantum linear algebra subroutines to achieve speedups over the classical kdN bound per iteration.
• Simulations suggest q-means can achieve similar accuracy to k-means or better runtimes for large datasets.