**6. KLD between Two Central MCDs**

Plugging (39) and (93) into (5) yields the closed-form expression of the KLD between two central MCDs with pdfs *f* **<sup>X</sup>**<sup>1</sup> (*x*|**Σ**1, *p*) and *f* **<sup>X</sup>**<sup>2</sup> (*x*|**Σ**2, *p*). This result is presented in the following theorem.

**Theorem 1.** *Let X*<sup>1</sup> *and X*<sup>2</sup> *be two random vectors that follow central MCDs with pdfs given, respectively, by f <sup>X</sup>*<sup>1</sup> (*x*|**Σ**1, *p*) *and f <sup>X</sup>*<sup>2</sup> (*x*|**Σ**2, *p*)*. The Kullback–Leibler divergence between central MCDs is*

$$\begin{split} \text{KL}(\mathbf{X}^{1}||\mathbf{X}^{2}) &= -\frac{1}{2}\log\prod\_{i=1}^{p}\lambda\_{i} + \frac{1+p}{2}\Big[\log\lambda\_{p} \\ &- \frac{\partial}{\partial a}\Big\{F\_{D}^{(p)}\left(a,\underbrace{\frac{1}{2},\ldots,\frac{1}{2}}\_{p},a+\frac{1}{2};a+\frac{1+p}{2};1-\frac{\lambda\_{1}}{\lambda\_{p}},\ldots,1-\frac{\lambda\_{p-1}}{\lambda\_{p}},1-\frac{1}{\lambda\_{p}}\right)\Big\}\Big|\_{a=0}\end{split} \tag{94}$$

*where λ*1*,. . . , λ<sup>p</sup> are the eigenvalues of the real matrix* **Σ**1**Σ**−<sup>1</sup> <sup>2</sup> *, and <sup>F</sup>*(*p*) *<sup>D</sup>* (.) *represents the Lauricella D-hypergeometric function defined for p variables.*

Lauricella [39] gave several transformation formulas (see Appendix A), whose relations (A5)–(A7), and (A9) are applied to *<sup>F</sup>*(*p*) *<sup>D</sup>* (.) in (94). The results of transformation are as follows

$$F\_D^{(p)}\left(a,\frac{1}{2},\dots,\frac{1}{2},a+\frac{1}{2};a+\frac{1+p}{2};1-\frac{\lambda\_1}{\lambda\_p},\dots,1-\frac{\lambda\_{p-1}}{\lambda\_p},1-\frac{1}{\lambda\_p}\right)$$

$$\lambda\_p = \lambda\_p^{a+\frac{p}{2}}\prod\_{i=1}^{p-1} \lambda\_i^{-\frac{1}{2}} F\_D^{(p)}\left(\frac{1+p}{2},\frac{1}{2},\dots,\frac{1}{2},a+\frac{1}{2};a+\frac{1+p}{2};1-\frac{\lambda\_p}{\lambda\_1},\dots,1-\frac{\lambda\_p}{\lambda\_{p-1}},1-\lambda\_p\right) \tag{95}$$

$$\hat{\lambda} = \left(\frac{\lambda\_1}{\lambda\_F}\right)^{-a} F\_D^{(p)}\left(a, \frac{1}{2}, \dots, \frac{1}{2}, a + \frac{1}{2}; a + \frac{1+p}{2}; 1 - \frac{\lambda\_p}{\lambda\_1}, \dots, 1 - \frac{\lambda\_2}{\lambda\_1}, 1 - \frac{1}{\lambda\_1}\right) \tag{96}$$

$$\lambda = \lambda\_p^a F\_D^{(p)}\left(a, \frac{1}{2}, \dots, \frac{1}{2}; a + \frac{1+p}{2}; 1 - \lambda\_1, 1 - \lambda\_2, \dots, 1 - \lambda\_p\right) \tag{97}$$

$$=\lambda\_p^a \prod\_{i=1}^p \lambda\_i^{-\frac{1}{2}} F\_D^{(p)}\left(\frac{1+p}{2}, \frac{1}{2}, \dots, \frac{1}{2}; a+\frac{1+p}{2}; 1-\frac{1}{\lambda\_1}, 1-\frac{1}{\lambda\_2}, \dots, 1-\frac{1}{\lambda\_p}\right). \tag{98}$$

Considering the above equations, it is easy to provide different expressions of KL(**X**1||**X**2) shown in Table 1. The derivative of the Lauricella D-hypergeometric series with respect to *a* goes through the derivation of the following expression

$$\frac{\partial}{\partial a} \left\{ F\_D^{(p)} \left( a, \frac{1}{2}, \frac{1}{2}, \dots, \frac{1}{2}, a + \frac{1}{2}; a + \frac{1+p}{2}; 1 - \frac{\lambda\_1}{\lambda\_p}, \dots, 1 - \frac{\lambda\_{p-1}}{\lambda\_p}, 1 - \frac{1}{\lambda\_p} \right) \right\} \Big|\_{a=0} \tag{99}$$

$$\mathcal{I} = \sum\_{\substack{m\_1,\ldots,m\_r\\m\_p=0}}^{+\infty} \frac{\partial}{\partial a} \left\{ \frac{(a)\_{\sum\_{i=1}^p m\_i} (a + \frac{1}{2})\_{m\_p}}{(a + \frac{1+p}{2})\_{\sum\_{i=1}^p m\_i}} \right\} \Big|\_{a=0} \prod\_{i=1}^{p-1} \left(\frac{1}{2}\right)\_{m\_i} \left(1 - \frac{\lambda\_i}{\lambda\_p}\right)^{m\_i} \frac{1}{m\_i!} \frac{(1 - \lambda\_p^{-1})^{m\_p}}{m\_p!} \tag{100}$$

The derivative with respect to *a* of the Lauricella D-hypergeometric series and its transformations goes through the following expressions (see Appendix B for demonstration)

$$\frac{\partial}{\partial a} \left\{ \frac{(a)\_{\sum\_{i=1}^{p} m\_i} (a + \frac{1}{2})\_{m\_p}}{(a + \frac{1+p}{2})\_{\sum\_{i=1}^{p} m\_i}} \right\} \Big|\_{a=0} = \frac{(\frac{1}{2})\_{m\_p} (1)\_{\sum\_{i=1}^{p} m\_i}}{(\frac{1+p}{2})\_{\sum\_{i=1}^{p} m\_i} (\sum\_{i=1}^{p} m\_i)} \,\tag{101}$$

$$\frac{\partial}{\partial a} \left\{ \frac{(a)\_{\sum\_{i=1}^{p} m\_i}}{(a + \frac{1+p}{2})\_{\sum\_{i=1}^{p} m\_i}} \right\} \Big|\_{a=0} = \frac{(1)\_{\sum\_{i=1}^{p} m\_i}}{(\frac{1+p}{2})\_{\sum\_{i=1}^{p} m\_i} (\sum\_{i=1}^{p} m\_i)} \, \tag{102}$$

$$\frac{\partial}{\partial a} \left\{ \frac{(a + \frac{1}{2})\_{m\_p}}{(a + \frac{1 + p}{2})\_{\sum\_{i=1}^p m\_i}} \right\} \Big|\_{a=0} = \frac{(\frac{1}{2})\_{m\_p}}{(\frac{1 + p}{2})\_{\sum\_{i=1}^p m\_i}} \left[ \sum\_{k=0}^{m\_p - 1} \frac{1}{k + \frac{1}{2}} - \sum\_{k=0}^{\sum\_{i=1}^p m\_i - 1} \frac{1}{k + \frac{1 + p}{2}} \right],\tag{103}$$

$$\frac{\partial}{\partial a} \left\{ \frac{1}{(a + \frac{1+p}{2})\_{\sum\_{i=1}^{p} m\_i}} \right\} \Big|\_{a=0} = \frac{-1}{(\frac{1+p}{2})\_{\sum\_{i=1}^{p} m\_i}} \sum\_{k=0}^{\sum\_{i=1}^{p} m\_i - 1} \frac{1}{k + \frac{1+p}{2}}.\tag{104}$$

To derive the closed-form expression of *d*KL(**X**1, **X**2) we have to evaluate the expression of KL(**X**2||**X**1). The latter can be easily deduced from KL(**X**1||**X**2) as follows

$$\begin{split} \text{KL}(\mathbf{X}^{2}||\mathbf{X}^{1}) &= \frac{1}{2} \log \prod\_{i=1}^{p} \lambda\_{i} - \frac{1+p}{2} \Big[ \log \lambda\_{p} \\ &+ \frac{\partial}{\partial a} \Big\{ F\_{D}^{(p)} \left( a, \frac{1}{2}, \dots, \frac{1}{2}, a + \frac{1}{2}; a + \frac{1+p}{2}; 1 - \frac{\lambda\_{p}}{\lambda\_{1}}, \dots, 1 - \frac{\lambda\_{p}}{\lambda\_{p-1}}, 1 - \lambda\_{p} \right) \Big\} \Big|\_{a=0} . \end{split} \tag{105}$$

Proceeding in the same way by using Lauricella transformations, different expressions of KL(**X**2||**X**1) are provided in Table 1. Finally, given the above results, it is straightforward to compute the symmetric KL similarity measure *d*KL(**X**1, **X**2) between **X**<sup>1</sup> and **X**2. Technically, any combination of the KL(**X**1||**X**2) and KL(**X**2||**X**1) expressions is possible to compute *d*KL(**X**1, **X**2). However, we choose the same convergence region for the two divergences for the calculation of the distance. Some expressions of *d*KL(**X**1,**X**2) are given in Table 1.

**Table 1.** KLD and KL distance computed when **X**<sup>1</sup> and **X**<sup>2</sup> are two random vectors following central MCDs with pdfs *f* **<sup>X</sup>**<sup>1</sup> (*x*|**Σ**1, *p*) and *f* **<sup>X</sup>**<sup>2</sup> (*x*|**Σ**2, *p*).

$$\begin{aligned} \text{KL}(\mathbf{X}^{\text{i}}||\mathbf{X}^{\text{i}})\\ \text{s.t.} \quad &= -\frac{1}{2}\log\prod\_{i=1}^{p}\lambda\_{i} + \frac{1+p}{2}\left[\log\lambda\_{P} - \frac{\partial}{\partial a}\left\{F\_{D}^{(p)}\left(a,\underbrace{1,\ldots,1}\_{p},a+\frac{1}{2};a+\frac{1+p}{2};1-\frac{\lambda\_{1}}{\lambda\_{P}},\ldots,1-\frac{\lambda\_{P-1}}{\lambda\_{P}},1-\frac{1}{\lambda\_{P}}\right)\right\}\right]\_{a=0}\end{aligned} \tag{106}$$

$$\hat{\lambda}\_{i} = -\frac{1}{2}\log\prod\_{i=1}^{p}\lambda\_{i} - \frac{1+p}{2}\lambda\_{p}^{\frac{p}{2}}\prod\_{i=1}^{p-1}\lambda\_{i}^{-\frac{1}{2}}\frac{\partial}{\partial a}\left\{\hat{r}\_{D}^{(p)}\left(\frac{1+p}{2},\underbrace{\frac{1}{2},\ldots,\frac{1}{2}}\_{p},a+\underbrace{\frac{1}{2};a+\frac{1+p}{2}}\_{p};1-\frac{\lambda\_{p}}{\lambda\_{1}},\ldots,1-\frac{\lambda\_{p}}{\lambda\_{p-1}},1-\lambda\_{p}\right)\right\}\bigg|\_{a=0}\tag{107}$$

$$\hat{\lambda} = -\frac{1}{2}\log\prod\_{i=1}^{p}\lambda\_{i} + \frac{1+p}{2}\left[\log\lambda\_{1} - \frac{\partial}{\partial a}\left\{F\_{B}^{(p)}\left(a,\underbrace{\frac{1}{2},\ldots,\frac{1}{2}}\_{p},a+\underbrace{\frac{1}{2}+p}{2};1-\frac{\lambda\_{1}}{\lambda\_{1}},\ldots,1-\frac{\lambda\_{2}}{\lambda\_{1}},1-\frac{1}{\lambda\_{1}}\right)\right\}\right]\_{\mathbf{x}=0}\tag{108}$$

$$\hat{\lambda}\_i = -\frac{1}{2}\log\prod\_{i=1}^p \lambda\_i - \frac{1+p}{2}\frac{\partial}{\partial a}\left\{\hat{\mathbf{r}}\_D^{(p)}\left(a, \underbrace{\frac{1}{2}, \dots, \frac{1}{2}}\_{p}; a + \frac{1+p}{2}; 1 - \lambda\_1, \dots, 1 - \lambda\_p\right)\right\}\Big|\_{a=0} \tag{109}$$

$$\lambda\_i = -\frac{1}{2}\log\prod\_{i=1}^p \lambda\_i - \frac{1+p}{2}\prod\_{i=1}^p \lambda\_i^{-\frac{1}{2}} \frac{\partial}{\partial a} \left\{ F\_D^{(p)}\left(\frac{1+p}{2}, \underbrace{\frac{1}{2}, \dots, \frac{1}{2}}\_p; a + \frac{1+p}{2}; 1 - \frac{1}{\lambda\_1}, \dots, 1 - \frac{1}{\lambda\_p}\right) \right\}\Big|\_{a=0} \tag{110}$$

$$\begin{aligned} \text{KL}(\mathbf{X}^{2}||\mathbf{X}^{1}) \\ = \left. \frac{1}{2} \log \prod\_{i=1}^{p} \lambda\_{i} - \frac{1+p}{2} \right[ \log \lambda\_{\mathcal{P}} + \frac{\partial}{\partial a} \left\{ F\_{D}^{(p)} \left( a, \underbrace{\frac{1}{\lambda\_{1}}, \dots, \frac{1}{2}}\_{p}, a + \frac{1+p}{2}; a + \frac{1+p}{\lambda\_{1}}; 1 - \frac{\lambda\_{p}}{\lambda\_{1}}, \dots, 1 - \frac{\lambda\_{p}}{\lambda\_{p-1}}, 1 - \lambda\_{p} \right) \right\} \Bigg|\_{a=0} \end{aligned} \tag{11}$$

$$\lambda = -\frac{1}{2}\log\prod\_{i=1}^{p}\lambda\_{i} - \frac{1+p}{2}\lambda\_{p}^{-\frac{p}{2}}\prod\_{i=1}^{p-1}\lambda\_{i}^{\frac{1}{2}}\frac{\partial}{\partial t}\left\{F\_{D}^{(p)}\left(\frac{1+p}{2},\underbrace{\frac{1}{2},\dots,\frac{1}{2}}\_{p},\mathfrak{a}+\frac{1+p}{2};\mathfrak{a}+\frac{\lambda\_{1}}{2};1-\frac{\lambda\_{1}}{\lambda\_{p}},\dots,1-\frac{\lambda\_{p-1}}{\lambda\_{p}},1-\frac{1}{\lambda\_{p}}\right)\right\}\Big|\_{\mathfrak{a}=0}\tag{112}$$

$$\lambda\_i = -\frac{1}{2}\log\prod\_{i=1}^p \lambda\_i - \frac{1+p}{2}\left[\log\lambda\_1 + \frac{\partial}{\partial a}\left\{F\_D^{(p)}\left(a, \underbrace{\frac{1}{2}, \dots, \frac{1}{2}}\_{p}, a + \frac{1}{2}; a + \frac{1+p}{2}; 1 - \frac{\lambda\_1}{\lambda\_p}, \dots, 1 - \frac{\lambda\_1}{\lambda\_2}, 1 - \lambda\_1\right)\right\}\right]\_{a=0}\tag{113}$$

$$\hat{\lambda}\_i = \left. \frac{1}{2} \log \prod\_{i=1}^p \lambda\_i - \frac{1+p}{2} \frac{\partial}{\partial a} \left\{ F\_D^{(p)} \left( a, \underbrace{1}\_p, \dots, \underbrace{1+p}\_p; a + \frac{1+p}{2}; 1 - \frac{1}{\lambda\_1}, \dots, 1 - \frac{1}{\lambda\_p} \right) \right\} \right|\_{a=0} \tag{114}$$

$$\hat{\lambda} = \frac{1}{2} \log \prod\_{i=1}^{p} \lambda\_i - \frac{1+p}{2} \prod\_{i=1}^{p} \lambda\_i^{\frac{1}{2}} \frac{\partial}{\partial t} \left\{ F\_D^{(p)} \left( \frac{1+p}{2}, \underbrace{1, \dots, \frac{1}{2}}\_{p}; a + \frac{1+p}{2}; 1-\lambda\_1, \dots, 1-\lambda\_p \right) \right\} \Big|\_{a=0} \tag{115}$$

*d*KL(**X**1,**X**2) <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>p</sup>* 2 log *<sup>λ</sup><sup>p</sup>* <sup>−</sup> *<sup>∂</sup> ∂a F*(*p*) *D a*, 1 2 ,..., <sup>1</sup> 2 , *a* + 1 <sup>2</sup> *<sup>p</sup>* ; *a* + 1 + *p* <sup>2</sup> ; 1 <sup>−</sup> *<sup>λ</sup>*<sup>1</sup> *λp* ,...,1 <sup>−</sup> *<sup>λ</sup>p*−<sup>1</sup> *λp* , 1 <sup>−</sup> <sup>1</sup> *λp* + + + + *a*=0 − *λ* − *p* 2 *p p*−1 ∏*i*=1 *λ* 1 2 *i* × *∂ ∂a F*(*p*) *D* 1 + *p* <sup>2</sup> , 1 2 ,..., <sup>1</sup> 2 , *a* + 1 <sup>2</sup> *<sup>p</sup>* ; *a* + 1 + *p* <sup>2</sup> ; 1 <sup>−</sup> *<sup>λ</sup>*<sup>1</sup> *λp* ,...,1 <sup>−</sup> *<sup>λ</sup>p*−<sup>1</sup> *λp* , 1 <sup>−</sup> <sup>1</sup> *λp* + + + + *a*=0 (116) <sup>=</sup> <sup>−</sup> <sup>1</sup> <sup>+</sup> *<sup>p</sup>* 2 *∂ ∂a F*(*p*) *D a*, 1 2 ,..., <sup>1</sup> <sup>2</sup> *<sup>p</sup>* ; *a* + 1 + *p* <sup>2</sup> ; 1 <sup>−</sup> *<sup>λ</sup>*1,...,1 <sup>−</sup> *<sup>λ</sup><sup>p</sup>* + + + + *a*=0 + *p* ∏*i*=1 *λ* 1 2 *i ∂ ∂a F*(*p*) *D* 1 + *p* <sup>2</sup> , 1 2 ,..., <sup>1</sup> <sup>2</sup> *<sup>p</sup>* ; *a* + 1 + *p* <sup>2</sup> ; 1 − *λ*1,...,1 − *λ<sup>p</sup>* + + + + *a*=0 (117) <sup>=</sup> <sup>−</sup> <sup>1</sup> <sup>+</sup> *<sup>p</sup>* 2 *<sup>p</sup>* ∏*i*=1 *λ* − 1 2 *i ∂ ∂a F*(*p*) *D* 1 + *p* <sup>2</sup> , 1 2 ,..., <sup>1</sup> <sup>2</sup> *<sup>p</sup>* ; *a* + 1 + *p* <sup>2</sup> ; 1 <sup>−</sup> <sup>1</sup> *λ*1 ,...,1 <sup>−</sup> <sup>1</sup> *λp* + + + + *a*=0 + *∂ ∂a F*(*p*) *D a*, 1 2 ,..., <sup>1</sup> <sup>2</sup> *<sup>p</sup>* ; *a* + 1 + *p* <sup>2</sup> ; <sup>1</sup> <sup>−</sup> <sup>1</sup> *λ*1 ,...,1 <sup>−</sup> <sup>1</sup> *λp* + + + + *a*=0 (118)
