量子聚类算法（matlab实现）

VOLUME 88, N

UMBER 1 PHYSICAL REVIEW LETTERS 7J

ANUARY 2002

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

David Horn and Assaf Gottlieb

School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University,

Tel Aviv 69978, Israel

(Received 16 July 2001; published 20 December 2001)

We propose a novel clustering method that is based on physical intuition derived from quantum me-

recognition, covered in textbooks such as [1–3]. The prob-

lem we are looking at is defining clusters of data solely by

the proximity of data points to one another. This problem is

one of unsupervised learning, and is in general ill defined.

Solutions to such problems can be based on intuition de-

rived from physics. A good example of the latter is the

algorithm by [4] that is based on associating points with

Potts spins and formulating an appropriate model of sta-

tistical mechanics. We propose an alternative that is also

a Gaussian with each of the

space of dimension d and summing over all of them. This

can be represented, up to an overall normalization, by

vector clustering (SVC) [6] that is based on a Hilbert-space

analysis. In SVC, one defines a transformation from data

space to vectors in an abstract Hilbert space. SVC pro-

ceeds to search for the minimal sphere surrounding these

states in Hilbert space. We will also associate data points

with states in Hilbert space. Such states may be repre-

sented by Gaussian wave functions, whose sum is c共x兲.

This is the starting point of our quantum clustering (QC)

method. We will search for the Schrödinger potential for

1 V共x兲

∂

c 苷 Ec . (2)

Here we rescaled H and V of the conventional quantum

mechanical equation to leave only one free parameter, s.

For comparison, the case of a single point at x

thus coinciding with the ground state of the harmonic os-

cillator in quantum mechanics.

Given c for any set of data points we can solve Eq. (2)

for V :

V共x兲 苷 E 1

s

X

i

E 苷 2 min

s

(4)

and determines V 共x兲 uniquely. E has to be positive since

V is a non-negative function. Moreover, since the last term

function of the operator H in Eq. (2), its eigenvalue E is the

lowest eigenvalue of H, i.e., it describes the ground state.

All higher eigenfunctions have nodes whose numbers in-

crease as their energy eigenvalues increase. (In quantum

mechanics, where one interprets jcj

tribution, all eigenfunctions of H have physical meaning.

Although this approach could be adopted, we have chosen

c as the probability distribution because of the simplicity

VOLUME 88, N

UMBER 1 PHYSICAL REVIEW LETTERS 7J

ANUARY 2002

region, and to exhibit one or several local minima within

the region. We identify these minima with cluster centers,

which seems natural in view of the opposite roles of the

two terms in Eq. (2): Given a potential function, it attracts

the data distribution function

c to its minima, while the

Laplacian drives it away. The diffused character of the

distribution is the balance of the two effects.

taken from Ripley’s book [7]. These data, given in a

surface of V 共x兲, roughly the same clustering assignment

with s 苷 1 兾

circles) that are interpreted as cluster centers. The contours of

the topographic map are set at values of V 共x兲兾E 苷 0.2, 0.4, 0.6,

0.8, 1.

018702-2 018702-2