We present a new approach to clustering, based on the physical properties of an inhomogeneous ferromagnet. No assumption is made regarding the underlying distribution of the data. We assign a Potts spin to each data point and introduce an interaction between neighboring points, whose strength is a decreasing function of the distance between the neighbors. This magnetic system exhibits three phases. At very low temperatures, it is completely ordered; all spins are aligned. At very high temperatures, the system does not exhibit any ordering, and in an intermediate regime, clusters of relatively strongly coupled spins become ordered, whereas different clusters remain uncorrelated. This intermediate phase is identified by a jump in the order parameters. The spin-spin correlation function is used to partition the spins and the corresponding data points into clusters. We demonstrate on three synthetic and three real data sets how the method works. Detailed comparison to the performance of other techniques clearly indicates the relative success of our method.
Genome-wide association studies (GWASs) seek to identify genetic variants associated with a trait, and have been a powerful approach for understanding complex diseases. A critical challenge for GWASs has been the dependence on individual-level data that typically have strict privacy requirements, creating an urgent need for methods that preserve the individual-level privacy of participants. Here, we present a privacy-preserving framework based on several advances in homomorphic encryption and demonstrate that it can perform an accurate GWAS analysis for a real dataset of more than 25,000 individuals, keeping all individual data encrypted and requiring no user interactions. Our extrapolations show that it can evaluate GWASs of 100,000 individuals and 500,000 single-nucleotide polymorphisms (SNPs) in 5.6 h on a single server node (or in 11 min on 31 server nodes running in parallel). Our performance results are more than one order of magnitude faster than prior state-of-the-art results using secure multiparty computation, which requires continuous user interactions, with the accuracy of both solutions being similar. Our homomorphic encryption advances can also be applied to other domains where large-scale statistical analyses over encrypted data are needed.
We present a new approach for clustering, based on the physical properties of an inhomogeneous ferromagnetic model. We do not assume any structure of the underlying distribution of the data. A Potts spin is assigned to each data point and short range interactions between neighboring points are introduced. Spin-spin correlations, measured (by Monte Carlo procedure) in a superparamagnetic regime in which aligned domains appear, serve to partition the data points into clusters. Our method outperforms other algorithms for toy problems as well as for real data. [S0031-9007(96)00104-4]
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