Algorithm for Data Clustering in Pattern Recognition Problems Based on Quantum Mechanics

Abstract

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.

Strange Bedfellows: Quantum Mechanics and Data Mining

“…The construction of this function was first introduced in the paper of Horn and Gottlieb [2] where it was defined by the condition that it is the function V ( x) for which the Parzen estimator satisfies the n-dimensional time-independent Schrödinger equation…”

“…In the orginal quantum clustering approach [2] points lying in the basin of attraction of particular minimum were identified as a single cluster. One way of determining which minimum was closest to a given point was to classically roll the points downhill using the gradient descent method.…”

Clustering Algorithms Optimizer: A Framework for Large Datasets

“…The QC algorithm [7] uses the Schrödinger equation to provide an effective clustering description of the data. It requires one parameter, σ, a Parzen window width.…”

“…Some algorithms such as CLICK [2], CTWC [3,4] and CAST [5] were primarily developed for large sets of biological data while others were adopted from other fields (e.g., K-Means, Fuzzy C-means [6], Agglomerative Hierarchical Clustering, Self Organized Maps). One of the algorithms that we will expand on is Quantum Clustering (QC), the effectiveness of which has been demonstrated on gene-expression data [7,8].…”

“…Amongst them are (i) the CAST algorithm [5], in which the affinity threshold determines the number of clusters, (ii) the CLICK algorithm [2], in which the homogeneity value determines K by controlling the kernels and the definition of singletons. (iii) The CTWC algorithm [4] where some parameters (such as stability threshold and minimal group size) determine K, and (iv) QC [7] where the Parzen window size (σ) determines the number of clusters.…”