Optimal string clustering based on a Laplace-like mixture and EM algorithm on a set of strings

In this study, we address the problem of clustering string data in an unsupervised manner by developing a theory of a mixture model and an EM algorithm for string data based on probability theory on a topological monoid of strings developed in our previous studies. We first construct a parametric distribution on a set of strings in the motif of the Laplace distribution on a set of real numbers and reveal its basic properties. This Laplace-like distribution has two parameters: a string that represents the location of the distribution and a positive real number that represents the dispersion. It is difficult to explicitly write maximum likelihood estimators of the parameters because their log likelihood function is a complex function, the variables of which include a string; however, we construct estimators that almost surely converge to the maximum likelihood estimators as the number of observed strings increases and demonstrate that the estimators strongly consistently estimate the parameters. Next, we develop an iteration algorithm for estimating the parameters of the mixture model of the Laplace-like distributions and demonstrate that the algorithm almost surely converges to the EM algorithm for the Laplace-like mixture and strongly consistently estimates its parameters as the numbers of observed strings and iterations increase. Finally, we derive a procedure for unsupervised string clustering from the Laplace-like mixture that is asymptotically optimal in the sense that the posterior probability of making correct classifications is maximized.