*2.1. Multivariate Normal Distribution Manifold*

Information geometry is used to solve some nonlinear and stochastic problems in the information field, because compared with the treatment in the Euclidean space, the one of Riemannian manifold can often achieve precise results. The statistical manifold is a set of all probability density functions with some regular conditions. In addition, by introducing the Fisher information matrix as a Riemannian metric, the statistical manifold becomes a Riemannian manifold. It is well known that the Kullback–Leibler divergence is a suitable difference function measuring the difference of two points on the statistical manifold, even though it is not a real distance function [10,11]. The manifold of a family of multivariate normal distributions is an important statistical manifold and is widely applied to the researches of signal processing, image processing, neural networks and so on. The K-means algorithm on statistical manifolds introduced in this paper is to transform the data point cloud in Euclidean space into the parameter point cloud on the statistical manifold of a family of multivariate normal distributions, and then, it applies cluster analysis to the parameter point cloud.

#### **Definition 1.** *We call a set*

$$\mathcal{S} = \{ p(\mathfrak{x}; \theta) \mid \theta \in \Theta \subset \mathbb{R}^n \}$$

*an n-dimensional statistical manifold, where p*(*x*; *θ*) *is the probability density of functions, with some regular conditions.*

Since each *n* multivariate normal distribution density function can be determined by an *n*-dimensional vector (mean) and an *n*-order symmetric positive definite matrix (covariance matrix), the manifold that consists of the family of normal distributions is closely related to manifold of the symmetric positive definite matrices [12].

**Definition 2.** *The manifold of symmetric positive definite matrices* SPD(*n*) *is defined as*

$$\text{SPD}(n) = \left\{ P \in M(n) \mid P^T = P\_\prime \\ \text{and } \mathbf{x}^T P \mathbf{x} > 0, \forall \mathbf{x} \in \mathbb{R}^n - \{0\} \right\},$$

*where M*(*n*) *is the set of n-order matrices and P<sup>T</sup> denotes the transpose of the matrix P. The smooth structure on* SPD(*n*) *is induced as the submanifold of the general linear group GL*(*n*, R)*, which is a set of all non-singular matrices.*

**Definition 3.** *The multivariate normal distribution manifold consists of the probability density functions of all n multivariate normal distributions, which is defined as*

$$\mathcal{N}\_{\mathbf{n}} = \left\{ f \, \middle| \, f(\boldsymbol{\mu}, \boldsymbol{\Sigma}) = \frac{1}{\sqrt{(2\pi)^{n} \det(\boldsymbol{\Sigma})}} \exp \left\{ -\frac{(\boldsymbol{\pi} - \boldsymbol{\mu})^{T} \boldsymbol{\Sigma}^{-1} (\boldsymbol{\pi} - \boldsymbol{\mu})}{2} \right\} \right\},$$

*where <sup>μ</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup> and* <sup>Σ</sup> <sup>∈</sup> SPD(*n*) *are the mean and the covariance matrix of the distributions, respectively, and* (*μ*, Σ) *is called the parameter coordinate of* N*n.*

*It is worth noting that* <sup>N</sup>*<sup>n</sup> is topologically homeomorphic in the product space* <sup>R</sup>*<sup>n</sup>* <sup>×</sup> SPD(*n*)*.*
