The Method of Quantum Clustering

2001

We propose a novel clustering method that is based on physical intuition derived from quantum mechanics. Starting with given data points, we construct a scale-space probability function. Viewing the latter as the lowest eigenstate of a Schrodinger equation, we use simple analytic operations to derive a potential function whose minima determine cluster centers. The method has one parameter, determining the

Physical Review Letters, 2001

We propose a novel clustering method that is based on physical intuition derived from quantum mechanics. Starting with given data points, we construct a scale-space probability function. Viewing the latter as the lowest eigenstate of a Schrödinger equation, we use simple analytic operations to derive a potential function whose minima determine cluster centers. The method has one parameter, determining the scale over which cluster structures are searched. We demonstrate it on data analyzed in two dimensions (chosen from the eigenvectors of the correlation matrix). The method is applicable in higher dimensions by limiting the evaluation of the Schrödinger potential to the locations of data points.

Mathematics, 2017

Data clustering is a vital tool for data analysis. This work shows that some existing useful methods in data clustering are actually based on quantum mechanics and can be assembled into a~powerful and accurate data clustering method where the efficiency of computational quantum chemistry eigenvalue methods is therefore applicable. These methods can be applied to scientific data, engineering data and even text.

ArXiv, 2019

Quantum Clustering is a powerful method to detect clusters in data with mixed density. However, it is very sensitive to a length parameter that is inherent to the Schrodinger equation. In addition, linking data points into clusters requires local estimates of covariance that are also controlled by length parameters. This raises the question of how to adjust the control parameters of the Schrodinger equation for optimal clustering. We propose a probabilistic framework that provides an objective function for the goodness-of-fit to the data, enabling the control parameters to be optimised within a Bayesian framework. This naturally yields probabilities of cluster membership and data partitions with specific numbers of clusters. The proposed framework is tested on real and synthetic data sets, assessing its validity by measuring concordance with known data structure by means of the Jaccard score (JS). This work also proposes an objective way to measure performance in unsupervised learni...

International Journal of Data Mining & Knowledge Management Process, 2021

Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a σ value, a hyper-parameter which can be manually defined and manipulated to suit the application. Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an outstanding task because normally such expressions are impossible to solve analytically. However, we prove that if the points are all included in a square region of size σ, there is only one minimum. This bound is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new numerical approach "per block". This technique decreases the number of particles by approximating some groups of particles to weighted particles. These findings are not only useful to the quantum clustering problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics and other applications.

Physica A: Statistical Mechanics and its Applications, 2001

We discuss novel clustering methods that are based on mapping data points to a Hilbert space by means of a Gaussian kernel. The ÿrst method, support vector clustering (SVC), searches for the smallest sphere enclosing data images in Hilbert space. The second, quantum clustering (QC), searches for the minima of a potential function deÿned in such a Hilbert space. In SVC, the minimal sphere, when mapped back to data space, separates into several components, each enclosing a separate cluster of points. A soft margin constant helps in coping with outliers and overlapping clusters. In QC, minima of the potential deÿne cluster centers, and equipotential surfaces are used to construct the clusters. In both methods, the width of the Gaussian kernel controls the scale at which the data are probed for cluster formations. We demonstrate the performance of the algorithms on several data sets.

IEEE International Conference on Image Processing 2005, 2005

This paper introduces a new nonparametric estimation approach that can be used for data that is not necessarily Gaussian distributed. The proposed approach employs the Shrödinger partial differential equation. We assume that each data sample is associated with a quantum physics particle that has a radial field around its value. We consider a statistical estimation approach for finding the size of the influence field around each data sample. By implementing the Shrödinger equation we obtain a potential field that is assimilated with the data density. The regions of minima in the potential are determined by calculating the local Hessian on the potential hypersurface. The quantum clustering approach is applied for blind separation of signals and for segmenting SAR images of terrain based on surface normal orientation.

Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a value, a hyper-parameter which can be manually defined and manipulated to suit the application. Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an outstanding task because normally such expressions are impossible to solve analytically. However, we prove that if the points are all included in a square region of size , there is only one minimum. This bound is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new numerical approach "per block". This technique decreases the number of particles (or samples) by approximating some groups of particles to weighted particles. These findings are not only useful to the quantum clustering problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics and other applications.

arXiv (Cornell University), 2023

In this paper, two novel measurement-based clustering algorithms are proposed based on quantum parallelism and entanglement. The Euclidean distance metric is used as a measure of 'similarity' between the data points. The first algorithm follows a divisive approach and the bound for each cluster is determined based on the number of ancillae used to label the clusters. The second algorithm is based on unsharp measurements where we construct the set of effect operators with a gaussian probability distribution to cluster similar data points. We specifically implemented the algorithm on a concentric circle data set for which the classical clustering approach fails. It is found that the presented clustering algorithms perform better than the classical divisive one; both in terms of clustering and time complexity which is found to be O(kN logN) for the first and O(N 2) for the second one. Along with that we also implemented the algorithm on the Churrtiz data set of cities and the Wisconsin breast cancer dataset where we found an accuracy of approximately 97.43% which For the later case is achieved by the appropriate choice of the variance of the gaussian window.

arXiv: Quantum Physics, 2018

We present an algorithm for quantum-assisted cluster analysis (QACA) that makes use of the topological properties of a D-Wave 2000Q quantum processing unit (QPU). Clustering is a form of unsupervised machine learning, where instances are organized into groups whose members share similarities. The assignments are, in contrast to classification, not known a priori, but generated by the algorithm. We explain how the problem can be expressed as a quadratic unconstrained binary optimization (QUBO) problem, and show that the introduced quantum-assisted clustering algorithm is, regarding accuracy, equivalent to commonly used classical clustering algorithms. Quantum annealing algorithms belong to the class of metaheuristic tools, applicable for solving binary optimization problems. Hardware implementations of quantum annealing, such as the quantum annealing machines produced by D-Wave Systems, have been subject to multiple analyses in research, with the aim of characterizing the technology&...