A new approach to clustering based on statistical physics is presented. The problem is formulated as fuzzy clustering and the association probability distribution is obtained by maximizing the entropy at a given average variance. The corresponding Lagrange multiplier is related to the "temperature" and motivates a deterministic annealing process where the free energy is minimized at each temperature. Critical temperatures are derived for phase transitions when existing clusters split. It is a hierarchical clustering estimating the most probable cluster parameters at various average variances.

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at Santa Barbara, as respectively an assistant professor. His research interests are in pattern recognition, Since 1967, he has been at the Nuclear Research information theory, neural networks, image coding and processing, and Center-Negev, Beer Sheva, Israel, doing research nonconvex optimization in general. on neutron scattering, magnetic structures group Dr. Rose received the 1990 William R. Bennett Prize Paper Award of theory, and phase transitions. During the past five the IEEE Communications Society for his paper "Enhancement of One-years, his research has been concentrated in the Dimensional Variable-Length DPCM Images Corrupted by Transmission areas of computational physics, optimization, parallel processing, and image Errors" (with Arie Heiman) in the April 1989 IEEE TRANSACTIONS ON processing COMMUNICATIONS. Dr Gurewitz is a member of the Israel Physical Society.

J. IntroductionClustering is usually formulated as an optimization problem by defining a cost function to be minimized. Most of these cost functions are nonconvex and have several local minima Traditional techniques [1,2,3,4] are essentially decent algorithms, as the cost is reduced at each iteration. Therefore, they tend to get trapped in local minima.Simulated annealing, or stochastic relaxation [5], is a known technique for avoiding local minima of nonconvex optimization problems. However, the process to escape local minima requires very slow schedules [6] which are not realistic for many practical applications.In a previous work [7,8] we proposed the concept of deterministic annealing for the problem of clustering and vector quantization. Our approach is strongly motivated by the physical analogy and baeed on principals of statistical mechanics or information theory [9,10]. In this paper we extend our clustering method to the constraint clustering method. Sections 2 and 3 are a brief presentation of our clustering approach. Section 4 presents the "constraint clustering approach", and sections 5 and 6 are two examples for which this approach can be applied. Clustering by Deterministic AnnealingIn previous works [7,8] we have suggested a deterministic annealing approach to clustering and vector quantization. This approach is briefly summarized in this section and the following one.The method uses a fuzzy formulation where each data point z is associated in probability with each cluster Cj. The cluster C, is represented by its "cluster centroid" yj. The energy (cost or distortion) of associating the data point z to the cluster C, is d ( z , y , ) . If the set Y = {yj} of the clusters representatives is given, the expected energy of the system is where P(z E Cj) is the probability that the data point z belongs to the cluster Cj. Since we do nof have any prior knowledge about the data probability distribution function, we apply the principle of maximum entropy.As is well known, the probability distribution which maximizes the entropy under the expectation constraint (1) is e-Dd(z,Vj)where Zx is the partition function Z, = cj e-pd(zJ'J).The Lagrange multiplier p is determined by a given value of E in (l), and is inversely proportional to the "temperature". For a given set of cluster representatives {y,}, it is assumed that the probabilities relating different data point to their clusters are independent. Hence the total partition function is Z = n, Z,, and the free energy F is given by F = --logz(Y) = -C l o g Ce-@d(Z,Yj)

New neutron-diffraction measurements on single crystals of CeAl2 confirm the previously reported incommensurate antiferromagnetic ground state with peaks at qr = (~+@, 2gn,~). In addition, weaker superlattice peaks are observed at q, =(&, -2, 2) and their position and temperature dependence can be explanined in terms of a multiple-q structure in CeA12. This appears to be the first definitive evidence for a multiple-q' magnetic structure.The rare-earth metals and compounds have served as invaluable proving grounds for testing many of the fundamental ideas of magnetism. 'One of the least understood elements of this series is Ce. This is surprising since it contains only one f electron and therefore might be considered as the simplest of the rare-earth metals. The complicated behavior arises because of the la.rge spatial extent of the 4f wave function of Ce, in contrast to the other rare-earth elements. This, in turn, leads to hybridization effects between the localized 4f electrons and the broadband s and d electrons which are manifested in (1) the Kondo effect in numerous dilute Ce systems'; (2) mixed valence in numerous concentrated systems (e.g. , Ce-based fcc systems'); and (3) Kondo-like behavior in other concentrated systems (e.g. , CeAl, and CeAl, ). ' In fact, CeAl, has been considered as a concentrated Kondo system with a high Kondo temperature (TK. -10 K). ' At lower temperatures (T &4.0 K) the system was recently shown to order in an incommensurate antiferromagnetic structure. ' The modulated moment in this system at low temperatures was postulated to be due to the strong singlet coupling between the f electron and the conduction electrons. 4 We present below neutron-scattering studies performed on single crystals of CeA12 and show that the magnetic ground state of this system is a clear example of a multiple-q structure and differs conceptually from that originally proposed. 4 A triple-q structure was first proposed to explain the observations on the magnetic structure of Nd. However, ambiguities remain in the experimental confirmation of this type of structure in Nd. ' In the present experiment on CeAl" the position and the temperature dependence of newly observed peaks is consistent with a tripleq-type structure.The single crystals were prepared by remelting boules of CeAl, in a sealed tantalum tube followed by a travelingheat-zone solidification. The neutron-diffraction studies were performed on a triple-axis spectrometer at the Brookhaven National Laboratory high-flux-beam reactor. Most measurements were performed with an incident energy of E, . =13.5 meV and collimation of 20'-40'-40'-40' between reactor and monochromator, monochromator and sample, and sample analyzer and analyzer-detector, respectively. Pyrolytic graphite (PG) was used as a monochromator and analyzer in addition to the use of two 5-cm-long PG filters to eliminate any higher-order contamination. The intensity of several of the peaks was measured as a function of incident energy to determine whether multiple scattering contamin...

It is shown that the insulating oxide Fe 2 TiO s exhibits anisotropic (uniaxial) spin-glass behavior below 55 K 0 Extensive experimental results supporting this conclusion, including principal magnetic susceptibility, neutron scattering, ultrasonic, specific heat, and oriented single-crystal Mossbauer measurements, are described.

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