# A Novel Generalized Group-Sparse Mixture Adaptive Filtering Algorithm
## 3 The Proposed GGS-MAF Algorithms
### The GGS-MAF Algorithms  
In the well-known SGM, we attempt to seek out the minimum of the cost functions regardless of the unknown parameter space. Herein, we use a Riemannian metric structure (RMS) presented in [24] instead of the Euclidean space for providing a rapid convergence within the Pt updating method. In the RMS, the cost functions are given by \(J(d,\mathbf{x},\mathbf{\hat{h}})\)[24], where the distance between \(\mathbf{\hat{h}}(n+1)\) and \(\mathbf{\hat{h}}(n)\) becomes  
\[ \begin{array}{l}D\left[\mathbf{\hat{h}}(n+1),\mathbf{\hat{h}}(n)\right]\\ =\left[\mathbf{\hat{h}}(n+1)-\mathbf{\hat{h}}(n)\right]^{T}\mathbf{G}^{-1}(n) \left[\mathbf{\hat{h}}(n+1)-\mathbf{\hat{h}}(n)\right]\\ =\left\|\mathbf{\hat{h}}(n+1)-\mathbf{\hat{h}}(n)\right\|_{\mathbf{G}^{-1}(n)} ^{2}\end{array}  \tag{10}\]  
with Riemannian metric tensor \(\mathbf{G}^{-1}(n)\)[24]. According to the proportionate-type AF updating scheme, we can write the reported proportionate-type algorithms via finding out the solution of the following generic updating [25,26]  
\[\mathbf{\hat{h}}(n+1)=\mathbf{\hat{h}}(n)-\mu_{1}\frac{\mathbf{G}(n)\
abla_{ \mathbf{\hat{h}}}J(d,\mathbf{x},\mathbf{\hat{h}})}{\mathbf{x}^{T}(n)\mathbf{G }(n)\mathbf{x}(n)} \tag{11}\]  
where \(\mu_{1}\) represents the step-size. To make the MEC suitable for exploiting the group-sparse characteristics in the blocked echo systems, we introduce proportionate group-update approach to the MEC algorithm for further utilizing the prior group-sparse information. In this case, the GGS-MAF algorithms aim to seek a solution by solving the giving equation  
\[\mathbf{\hat{h}}(n+1)=\mathbf{\hat{h}}(n)-\mu_{1}\frac{\mathbf{G}_{1}(n) \
abla_{\mathbf{\hat{h}}}(d,\mathbf{x},\mathbf{\hat{h}})}{\mathbf{x}^{T}(n) \mathbf{G}_{1}(n)\mathbf{x}(n)}-\mu_{1}\frac{\mathbf{G}_{2}(n)\
abla_{ \mathbf{\hat{h}}}(d,\mathbf{x},\mathbf{\hat{h}})}{\mathbf{x}^{T}(n)\mathbf{G }_{2}(n)\mathbf{x}(n)} \tag{12}\]  
where \(J_{\mathbf{A}}(d,\mathbf{x},\mathbf{\hat{h}})=\frac{\alpha}{2}\log\left[1+ \frac{e^{2}(n)}{2\tau^{2}}\right]\) and \(J_{\mathbf{B}}(d,\mathbf{x},\mathbf{\hat{h}})=\frac{1-\alpha}{p}|e(n)|^{p}\). Then, the mixed-norm constraint coefficient vectors are developed to obtain the \(\mathbf{G}_{1}(n)\) and \(\mathbf{G}_{2}(n)\) which are obtained from \(\left\|\mathbf{\hat{h}}\right\|_{2,\mathcal{L}}\), (c = 1,0). Then, we have  
\[\left\|\mathbf{\hat{h}}\right\|_{2,0}=\left\|\left[\left\|\mathbf{\hat{h}}_{[ 1]}\right\|_{2^{\prime}}\left\|\mathbf{\hat{h}}_{[2]}\right\|_{2^{\prime}} \cdots\right.\left\|\mathbf{\hat{h}}_{[N]}\right\|_{2}\right]^{T}\right\|_{0} \approx\sum_{i=1}^{N}\left(1-e^{-\beta\left\|\mathbf{\hat{h}}_{[i]}\right\|_{2}}\right) \tag{13}\]  
and  
\[\left\|\mathbf{\hat{h}}\right\|_{2,1}=\left\|\left[\left\|\mathbf{\hat{h}}_{[ 1]}\right\|_{2^{\prime}}\left\|\mathbf{\hat{h}}_{[2]}\right\|_{2^{\prime}} \cdots\right.\left\|\mathbf{\hat{h}}_{[N]}\right\|_{2}\right]^{T}\right\|_{1} =\sum_{i=1}^{N}\left\|\mathbf{\hat{h}}_{[i]}\right\|_{2} \tag{14}\]  
Herein, the approximation \(l_{0}\)-norm scheme in [27] is considered to implement \(\left\|\mathbf{\hat{h}}\right\|_{2,0^{\prime}}\), and \(N\) denotes the number of groups defined as \(N=L/B\), and \(B\) denotes the length of the group, and \(\beta>0\) is a regularization parameter. Considering the combination of the \(\mathbf{G}_{1}(n)\) and \(\mathbf{G}_{2}(n)\), we can have four different GGS-MAF algorithms, named GGS-MAF-1, GGS-MAF-2, GGS-MAF-3 and GGS-MAF-4 algorithms, which are  
\[ \begin{array}{l}\mathbf{\hat{h}}(n+1)=\mathbf{\hat{h}}(n)+\mu a \frac{e(n)\mathbf{G}_{1}(n)\mathbf{x}(n)}{(e^{2}(n)+2\tau^{2})\mathbf{x}^{T}(n )\mathbf{G}_{1}(n)\mathbf{x}(n)}\\ +\mu(1-\alpha)\frac{|e(n)|^{p-1}\text{sgn}(e(n))\mathbf{G}_{2}(n)\mathbf{x}( n)}{\mathbf{x}^{T}(n)\mathbf{G}_{2}(n)\mathbf{x}(n)}.\end{array}  \tag{15}\]  
Then, we can achieve GGS-MAF-1 when \(\mathbf{G}_{1}=\mathbf{G}_{2}\) is implemented by using (13). The GGS-MAF-2 algorithm is obtained when \(\mathbf{G}_{1}\) is achieved from (13) and \(\mathbf{G}_{2}\) is from (14).