*2.3. Mean of Parameter Point Clouds*

The main idea of the traditional K-means algorithm is that for a given data cloud with the scale *m*,

$$\mathbb{C}\_{m} = \{p\_i \in \mathbb{R}^n \mid i = 1, \dots, m\}\_{\nu}$$

which is abbreviated as *C*, by using the clustering algorithm, we divide the point cloud into *K* classes. The effect of the traditional K-means algorithm is mainly affected by the selection of initial cluster centers, the expression of data and the difference function.

In order to avoid the shortage of the traditional K-means algorithm, we will consider the clustering algorithm on the Riemannian space instead of the Euclidean space so that we can use the geodesic distance and KL divergence but the Euclidean distance and obtain better clustering results.

Now, we give the definition of the geometric mean of point cloud *C* in N*<sup>n</sup>* under different difference functions *D*.

**Definition 5.** *The geometric mean c*(*C*) *of point cloud C* = {(*μ*1, Σ1), ··· ,(*μm*, Σ*m*)} *in* N*<sup>n</sup> is*

$$c(\mathsf{C}) := \operatorname\*{argmin}\_{\left(\boldsymbol{\mu}, \boldsymbol{\Sigma}\right) \in \mathcal{N}\_n} \frac{1}{m} \sum\_{i=1}^m D\left(\left(\boldsymbol{\mu}\_i, \boldsymbol{\Sigma}\_i\right)\_\prime \left(\boldsymbol{\mu}, \boldsymbol{\Sigma}\right)\_\prime\right).$$

In practical problems, the calculation of the geometric mean of some difference functions may be very complicated; thus, we will use the arithmetic mean instead of the geometric mean.

**Definition 6.** *The parameter space* <sup>R</sup>*<sup>n</sup>* <sup>×</sup> SPD(*n*) *of* <sup>N</sup>*<sup>n</sup> is a convex set. Hence, the arithmetic mean c*(*C*) *of the parameter point cloud C* = {(*μ*1, Σ1), ··· ,(*μm*, Σ*m*)} *in* N*<sup>n</sup> can be defined as*

$$\mathbb{E}(\tilde{C}) = \frac{1}{m} \sum\_{i=1}^{m} (\mu\_{i\prime} \Sigma\_i) \dots$$

Now, we introduce the geometric mean of the point cloud *C* with respect to the KL divergence.

From (5), we can obtain the following proposition [16].

**Proposition 2.** *The geometric mean of the point cloud C with respect to the KL divergence exists and is unique, and is equal to the arithmetic mean in the above natural coordinates.*

Furthermore, we can see that the geometric mean of the point cloud *C* with respect to the Bregman divergence *Bϕ* exists and is unique, and it is equal to the arithmetic mean in natural coordinates, hence the geometric mean of point cloud *C* about KL divergence exists and is unique, and it is equal to the arithmetic mean in natural coordinates, that is,

$$\mathbf{c}(\mathbb{C}) = \operatorname\*{argmin}\_{P \in \mathcal{N}\_{\mathbf{z}}} \frac{1}{m} \sum\_{i=1}^{m} D\_{KL}(P \| P\_i) = P\left(\mathbf{x}\_1, \mathbf{x}\_2; \frac{1}{m} \sum\_{i=1}^{m} \theta\_i\right). \tag{6}$$

In the following K-means algorithm with KL divergence as the difference function, the Proposition 2 ensures that the geometric mean of the parameter point cloud can be explicitly given by the arithmetic mean after parameter transformation.
