# Kernel Density Estimation on the Siegel Space with an Application to Radar Processing +
## 3 Non Parametric Density Estimation on the Siegel Space
### Kernels  
Let \(\mathcal{K}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\) be a map which verifies the following properties:  
1. \(\int_{\mathbb{R}^{d}}\mathcal{K}(||x||)dx=1\);
2. \(\int_{\mathbb{R}^{d}}x\mathcal{K}(||x||)dx=0\);
3. \(\mathcal{K}(x>1)=0\);
4. \(sup(\mathcal{K}(x))=\mathcal{K}(0)\). Let \(p\in\mathcal{D}_{n}\). Generally, given a point \(p\) on a Riemannian manifold, \(exp_{p}\) defines an injective application only on a neighborhood of 0. On the Siegel space, \(exp_{p}\) is injective on the whole space. When the tangent space \(T_{p}\mathcal{D}_{n}\) is endowed with the local scalar product,  
\[||u||=d(p,exp_{p}(u)),\]  
where \(||.||\) is the Euclidean distance associated with the local scalar product and \(d(.,. )\) is the Riemannian distance. The corresponding Lebesgue measure on \(T_{p}\mathcal{D}_{n}\) is noted \(Leb_{p}\). Let \(exp_{p}^{*}(Leb_{p})\) denote the push-forward measure of \(Lep_{p}\) by \(exp_{p}\). The function \(\theta_{p}\) defined by:  
\[\theta_{p}:q\mapsto\theta_{p}(q)=\frac{dvol}{dexp_{p}^{*}(Leb_{p})}(q) \tag{6}\]  
is the density of the Riemannian measure on \(\mathcal{D}_{n}\) with respect to the Lebesgue measure \(Leb_{p}\) after the identification of \(\mathcal{D}_{n}\) and \(T_{p}\mathcal{D}_{n}\) induced by \(exp_{p}\) (see Figure 1). Given \(\mathcal{K}\) and a positive radius \(r\), the estimator of \(f\) proposed by [1] is defined by:  
\[\hat{f}_{k}=\frac{1}{k}\sum_{i}\frac{1}{r^{n}}\frac{1}{\theta_{x_{i}}(x)} \mathcal{K}\left(\frac{d(x,x_{i})}{r}\right). \tag{7}\]  
The corrective factor \(\theta_{x_{i}}(x)^{-1}\) is necessary since the kernel \(\mathcal{K}\) originally integrates to one with respect to the Lebesgue measure and not with respect to the Riemannian measure. It can be noticed that this estimator is the usual kernel estimator in the case of Euclidean space. When the curvature of the space is negative, which is the case of the Siegel space, the distribution placed over each sample \(x_{i}\) has \(x_{i}\) as intrinsic mean. The following theorem provides convergence rate of the estimator. It is a direct adaptation of Theorem 3.1 of [1]. **Theorem 1**. : _Let \((\mathcal{M},g)\) be a Riemannian manifold of dimension \(n\) and \(\mu\) its Riemannian volume measure. Let \(X\) be a random variable taking its values in a compact subset \(C\) of \((\mathcal{M},g)\). Let \(0<r\leq r_{inj}\), where \(r_{inj}\) is the infimum of the injectivity radius on \(C\). Assume the law of \(X\) has a twice differentiable density \(f\) with respect to  
Figure 1: \(\mathcal{M}\) is a Riemannian manifold, and \(T_{x}\mathcal{M}\) is its tangent space at \(x\). The exponential application induces a volume change \(\theta_{x}\) between \(T_{x}\mathcal{M}\) and \(\mathcal{M}\). the Riemannian volume measure. Let \(\hat{f}_{k}\) be the estimator defined in Equation (7). There exists a constant \(C_{f}\) such that_  
\[\int_{x\in\mathcal{M}}E_{x_{1}\ldots,x_{k}}[(f(x)-\hat{f}_{k}(x))^{2}]d\mu\leq C _{f}(\frac{1}{kr^{n}}+r^{4}). \tag{8}\]  
_If \(r\sim k^{\frac{-1}{n+4}}\),_  
\[\int_{x\in\mathcal{M}}E_{x_{1}\ldots,x_{k}}[(f(x)-\hat{f}_{k}(x))^{2}]d\mu=O(k^ {-\frac{4}{n+4}}). \tag{9}\]  
Proof. : See Appendix A. It can be checked that on the Siegel space \(r_{inj}=+\infty\) and that, for an isometry \(\alpha\), we have:  
\[\hat{f}_{k}(x,x_{1},...,x_{k})=\hat{f}_{k}(\alpha(x),\alpha(x_{1}),...,\alpha( x_{k})).\]  
Each location and direction are processed as similarly as possible. This density estimator can be used for data classification on Riemannian manifolds, see [26]. In order to obtain the explicit expression of the estimator, one must have the explicit expression of the Riemannian exponential, of its inverse, and of the function \(\theta_{p}\) (see Equations (6) and (7)). These expressions are difficult and sometimes impossible to obtain for general Riemannian manifolds. In the case of the Siegel space, the symmetric structure makes the computation possible.