## 4 Training Sequence Optimization Based on CGD Algorithm over the Riemannian Manifold
For the sake of notational simplicity, in the algorithm, the training sequence matrix notation \(\mathbf{S}_{\mathrm{t}}\) is used instead, where the subscript \(t\) corresponds to the number of iterations. Step 2--Gradient evaluation and projection on the Riemannian parameter space: In this step, the gradient of the sum MSE cost function defined in (5) is determined, and the projection on the Riemannian manifold is computed to find the descent direction and set \(\mathbf{G}_{t}=\mathbf{D}_{t}\). \[\mathbf{D}_{t}=\mathbf{\Gamma}_{t}\,\mathbf{S}_{t}^{\mathrm{H}}-\mathbf{S}_{t} \,\mathbf{\Gamma}_{t}^{\mathrm{H}} \tag{8}\]  
where \(\mathbf{D}_{t}\) is the descent direction for iteration \(t\) and \(\mathbf{\Gamma}_{t}\) corresponds to the Euclidean gradient with respect to the training sequence matrix as given in (7). Step 3--Stopping criteria: Check the gradient on the Riemannian parameter space (i.e., \(\mathbf{D}_{t}\)) whether it reaches convergence or not according to:  
\[\langle\mathbf{D}_{t},\mathbf{D}_{t}\rangle=\frac{1}{2}\mathrm{Re}\big{\{} \mathrm{tr}\big{(}\mathbf{D}_{t}^{\mathrm{H}}\mathbf{D}_{t}\big{)}\big{\}} \tag{9}\]  
where Re stands for the real value and \(\epsilon\) in step 3 of Algorithm 1 denotes the error tolerance. If the steepest direction is sufficiently small, which implies a close to local minimum value of the cost function the algorithm stops. Step 4--Matrix exponential computation: Determine the local parametrisation based on the matrix exponential on the Riemannian manifold, i.e., \(\mathbf{P}_{t}=\exp(-\mu\mathbf{D}_{t})\), where \(\mu\) corresponds to the step size that controls the steepest direction. The complex matrix exponential is determined by the convergent power series [52, 53], where a Matlab function _expm_ is used for this purpose. Steps 5 and 6--Step size evaluation: The step size \(\mu\) is required to be tuned in order to ensure an appropriate steepest direction movement towards the training solution. In particular, in steps 5 and 6, the sum MSE cost function of different sequences are evaluated based on different possibilities of \(\mu\). If \(\mu\) is too small and the condition of the cost function is not met, then \(\mu\) is doubled, as given in step 5. If \(\mu\) is too large and the condition of the cost function is not met, then \(\mu\) is halved, as provided in step 6. Step 7--Updating the training sequence and the gradient direction: In this step, the pilot matrix and the gradient direction are updated according to the obtained matrix exponential \(\mathbf{P}_{t}\). Due to the unitary invariant rotation of the cost function, the obtained training sequence has unitary columns. Step 8--The new conjugate gradient direction is obtained by updating \(\mathbf{G}_{t+1}=\mathbf{D}_{t+1}+v_{t}\mathbf{G}_{t}\), where \(\mathbf{D}_{t+1}\) is determined by substituting the obtained sequence matrix into (8), and the parameter \(v_{t}\) is given as [53]:  
\[v_{t}=\frac{\langle\mathbf{D}_{t+1}-\mathbf{D}_{t},\,\mathbf{D}_{t+1}\rangle}{ \langle\mathbf{D}_{t},\mathbf{D}_{t}\rangle} \tag{10}\]  
where \(v_{t}\) denotes the Polak-Ribiere formula, as defined in [53]. If the stopping criteria is met, the algorithm is executed and the sequence is optimized. In contrast, if the stopping criteria is not met, a new iteration \((t+1)\) is started. We provide the flops calculation (obtained by counting the number of multiplications and additions per iteration) for the proposed CSD algorithm and the existing SCMI and SMSE algorithms. Table 1 summarises the complexity analysis in flops [56] per iteration for the three training designs considered. Parameters (\(t_{dg}\), \(t_{hg}\)) and (\(t_{d}\), \(t_{h}\)) represent the number of iterations required for doubling and halving the step size in the proposed CGD method and the SMSE iterative algorithm, respectively. In this paper, the gradient descent based approach is applied, which can be defined as a first-order iterative optimization solution.