# Kernel Density Estimation on the Siegel Space with an Application to Radar Processing +
## 3 Non Parametric Density Estimation on the Siegel Space
### Kernels
Since the space is homogeneous, the computation can be made at the origin \(il\in\mathcal{H}_{n}\) or \(0\in\mathcal{D}_{n}\) and transported to the whole space. In the present work, the random variable lays in \(\mathcal{D}_{n}\). However, in the literature, the Cartan and Iwasawa decompositions are usually given for the isometry group of \(\mathcal{H}_{n}\). Thus, our computation starts in \(\mathcal{H}_{n}\) before moving to \(\mathcal{D}_{n}\). The Killing form on the Lie algebra \(\mathfrak{sp}(n,\mathbb{R})\) of the isometry group of \(\mathcal{H}_{n}\) induces a scalar product on \(\mathfrak{p}\). This scalar product can be transported on \(exp(\mathfrak{p})\) by left multiplication. This operation gives \(exp(\mathfrak{p})\) a Riemannian structure. It can be shown that on this Riemannian manifold, the Riemannian exponential at the identity coincides with the group exponential. Furthermore,  
\[ \begin{array}{ccccc}\phi&:&exp(\mathfrak{p})&\rightarrow&\mathcal{H}_{n}\\ &&g&\mapsto&g.iI\end{array}  \tag{10}\]  
is a bijective isometry, up to a scaling factor. Since the volume change \(\theta_{p}\) is invariant under rescaling of the metric, this scaling factor has no impact. Thus, \(\mathcal{H}_{n}\) can be identified with \(exp(\mathfrak{p})\) and \(exp_{il\in\mathcal{H}_{n}}\) with \(exp_{|\mathfrak{p}}\). The expression of the Riemannian exponential is difficult to obtain in general; however, it boils down to the group exponential in the case of symmetric spaces. This is the main element of the computation of \(\theta_{p}\). The Riemannian volume measure on \(exp(\mathfrak{p})\) is noted \(vol^{\prime}\). Let  
\[ \begin{array}{ccccc}\psi&:&K\times\mathfrak{a}&\rightarrow&\mathfrak{p}\\ &&(k,H)&\mapsto&Ad_{k}(H).\end{array} \]  
Let \(\mathfrak{a}^{+}\) be the diagonal matrices with strictly decreasing positive eigenvalues. Let \(\Lambda^{+}\) be the set of positive roots of \(\mathfrak{sp}(n,\mathbb{R})\) as described in p. 282 in [20],  
\[\Lambda^{+}=\{e_{i}+e_{j},i\leq j\}\cup\{e_{i}-e_{j},i<j\},\]  
where \(e_{i}(H)\) is the \(i\)-th diagonal term of the diagonal matrix \(H\). Let \(C_{c}(E)\) be the set of continuous compactly supported functions on the space \(E\). In [27], at page 73, it is given that for all \(t\in C_{c}(\mathfrak{p})\), there exists \(c_{1}>0\) such that  
\[\int_{\mathfrak{p}}t(Y)dY=c_{1}\int_{K}\int_{\mathfrak{a}^{+}}t(\psi(k,H))\prod _{\lambda\in\Lambda^{+}}\lambda(H)dkdH, \tag{11}\]where \(dY\) is a Lebesgue measure on the coefficients of \(Y\). Let \(\tilde{\mathfrak{p}}=\psi(K\times\mathfrak{a}^{+})\). \(\lambda\in\Lambda^{+}\) never vanishes on \(\mathfrak{a}^{+}\) and \(\mathfrak{p}\setminus\tilde{\mathfrak{p}}\) has a null measure. Thus,  
\[\int_{\tilde{\mathfrak{p}}}t(Y)\prod_{\lambda\in\Lambda^{+}}\frac{1}{\lambda( H_{Y})}dY=c_{1}\int_{K^{\prime}}\int_{\mathfrak{a}^{+}}t(Ad_{k}(H))dkdH, \tag{12}\]  
where \(H_{Y}\) is the point in \(\mathfrak{a}^{+}\) such that there exists \(k\) in \(K\) such that \(\psi(k,H_{Y})=Y\). Calculation in p. 73 in [27] also gives that for all \(t\in C_{c}(\mathfrak{p})\), there exists \(c_{2}>0\), such that  
\[\int_{Sp(n,\mathbb{R})}t(g)dg=c_{2}\int_{K}\int_{\mathfrak{a}^{+}}\int_{K}t(k _{2}.exp(Ad_{k_{1}}(H)))J(H)dk_{1}dHdk_{2}, \tag{13}\]  
where \(dg\) is the Haar measure on \(Sp(n,\mathbb{R})\) and  
\[J(H) = \prod_{\lambda\in\Lambda^{+}}e^{\lambda(H)}-e^{-\lambda(H)}\] \[= 2^{|\Lambda^{+}|}\prod_{\lambda\in\Lambda^{+}}\sinh(\lambda(H)).\]  
Thus, for all \(t\in C_{c}(Sp(n,\mathbb{R})/K)\),  
\[\int_{Sp(n,\mathbb{R})/K}t(x)dx=c_{2}\int_{K}\int_{\mathfrak{a}^{+}}t(exp(Ad_{ k}(H)))J(H)dkdH, \tag{14}\]  
where \(dx\) is the invariant measure on \(Sp(n,\mathbb{R})/K\). After identifying \(Sp(n,\mathbb{R})/K\) and \(exp(\mathfrak{p})\), the Riemannian measure on \(exp(\mathfrak{p})\) coincides with the invariant measure on \(Sp(n,\mathbb{R})/K\).