## 4 Training Sequence Optimization Based on CGD Algorithm over the Riemannian Manifold  
Optimization on manifolds means finding an optimum solution of a desired function using a smooth finite-dimensional Riemannian manifold. The Riemannian gradient is simply the orthogonal projection of the classical gradient. Conceptually, the key point is to perform as unconstrained nonlinear optimization. Exploiting the Riemannian makes it easy to deal with various types of constraints, which arise in low rank matrices. In particular, optimization on manifolds is a powerful scheme to address the nonlinear optimization problems such as the problem of finding the training sequence matrix that maximizes the achievable sum rate of the FDD massive MIMO systems, which is considered in this present paper. In wireless communication systems and array signal processing, several studies have considered various optimization problems under a unitary matrix constraint [44, 45, 46, 47]. Consequently, advanced gradient-based iterative algorithms have been developed to find efficient solutions with fast convergence rate to such optimization problems [48, 49, 50, 51]. Some of those efficient solutions have been obtained using a geometrical approach based on the smooth parameter space known as the Riemannian manifold. The potential of the Riemannian manifold has motivated the author of this paper to explore the application of such a geometrical approach for optimising the training sequence in an FDD massive MIMO system when the users exhibit distinct spatial correlations. In this section, the CGD iterative algorithm based on the Riemannian manifold is explored to optimize the DL pilot matrix \(\mathbf{S}_{\text{tr}}\) iteratively across multiple users with independent channel covariance matrices. In optimizing (6) for \(\mathbf{S}_{\text{tr}}\), the CGD method uses the almost periodic property of the matrix exponential search [52, 53] over the Riemannian manifold to increase the algorithm convergence speed by lessening the required number of iterations. In particular, the introduction of matrix exponential on the Riemannian manifold aids the proposed CGD algorithm to operate over a smaller dimensional search space and thus provide a fast convergence behaviour. In addition, the existence of matrix exponential parametrization in the proposed CGD method is a sufficient condition to satisfy the semi-unitary matrices \(\mathbf{S}_{\mathrm{tr}}^{\mathrm{H}}\mathbf{S}_{\mathrm{tr}}=\mathbf{I}_{ \mathrm{Tr}}\). Further details on the matrix exponential property on the Riemannian manifold can be found in [52, 53, 54]. This algorithm can also be used to increase the capacity with interference channel [55]. Algorithm 1 summarizes the proposed CGD iterative algorithm. The following steps explain Algorithm 1 in further detail. ```
1:Initialization: \(t=0\) and \(\mu=1,\mathbf{S}_{t}=\mathbf{I}\in\mathbb{C}^{\mathrm{N}\times\mathrm{Tr}}\)
2:Determine the gradient using (8), and set \(\mathbf{G}_{t}=\mathbf{D}_{t}\). 3:If \(\langle\mathbf{D}_{t},\mathbf{D}_{t}\rangle<\epsilon\), then break. 4:Determine the matrix exponential: \(\mathbf{P}_{t}=\exp(-\mu\mathbf{D}_{t})\). 5:If \(f(\mathbf{S}_{t})-f(\mathbf{P}_{t}\,\mathbf{S}_{t})\geq\mu\,\langle\mathbf{D} _{t},\mathbf{D}_{t}\rangle\), double \(\mathbf{P}_{t}\), \(t:=2\mu\). 6:If \(f(\mathbf{S}_{t})-f(\mathbf{P}_{t}\,\mathbf{S}_{t})<0.5\mu\,\langle\mathbf{D} _{t},\mathbf{D}_{t}\rangle\), halve \(\mathbf{P}_{t}\), \(\mu:=0.5\mu\). 7:Update \(\mathbf{S}_{t+1}:=\mathbf{P}_{t}\,\mathbf{S}_{t}\). 8:Update \(\mathbf{G}_{t+1}=\mathbf{D}_{t+1}+v_{t}\mathbf{G}_{t}\), \(t:=t+1\), and go to step 2. ```  
**Algorithm 1** Iterative CGD optimization algorithm. Step 1--Initial step: The proposed iterative gradient algorithm starts by selecting an initial training sequence matrix \(\mathbf{S}_{\mathrm{tr}}\in\mathbb{C}^{\mathrm{N}\times\mathrm{Tr}}\).