## 3 Traditional KRLS Algorithm
### Traditional KRLS Algorithm
In our proposed SSW-KRLS algorithm, we took similar operations to avoid an increase in computation, which is discussed in detail in Section 4. ### Extensions to KRLS Algorithm

The kernel recursive least-squares method is one of the most efficient online kernel methods, and it achieves good performance in nonlinear fitting and prediction. However, the bottleneck problem is that the network structure grows with the training samples, which leads to insufficient memory and computational complexities when processing continuously incoming signals. In order to solve this problem and render the online kernel method feasible, researchers have proposed various sparseness methods or criteria, such as approximate linear dependency (ALD), the novelty criterion (NC), the surprise criterion (SC) and the coherence criterion (CC). On the basis of these sparseness methods or criteria, only new input samples that meet the prerequisites are used as training samples. Thus, the growth of network structure is effectively slowed down. Table 1 shows several common sparseness criteria. The performance of the above methods in filtering samples largely depends on the selected threshold. As time goes by, the number of samples increases slowly and lastly stabilizes, while the update of model parameters tends to be slow. This is not conducive to updating time-varying channels. Furthermore, when the user moves into a new environment, the outdated samples are reserved, which is not conducive to channel prediction. An effective way to keep the dictionary updated while precisely controlling its size is using the sliding window. A simple implementation is discarding the oldest sample directly each time a new sample is collected, but it lacks the judgement of correlation between samples. Sparseness simply based on time information is unreliable and may result in unstable prediction performance when the window is small. In order to solve this problem, we propose a new algorithm, SSW-KRLS, where we take both the time value and correlation between samples into consideration. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Criterion** & **Indicators** & **Handling Method** \\ \hline \multirow{4}{*}{ALD} & Determine whether the kernel function of the new sample can be linearly represented & If \(\delta(n)<\tau\), \\  & by the kernel function of the existing & discard new samples. \\  & samples in the dictionary: & If \(\delta(n)\geq\tau\), \\  & \(\delta(n)=\min\limits_{a}\left\|\sum\limits_{i=1}^{m}a(i)\varphi(x_{i})-\varphi (x_{n})\right\|^{2}\) & add the new sample \\  & \(\text{Calculate the minimal distance between}\) & to the dictionary. \\ \hline \multirow{4}{*}{NC} & Calculate the new and existing & If \(\text{{dis}}<\tau\), \\  & the new and existing & discard new samples. \\  & samples in the kernel dictionary: & If \(\text{{dis}}\geq\tau\), \\  & \(\text{{dis}}=\min\limits_{a}\left\|\sum\limits_{i=1}^{m}a(i)\varphi(x_{i})- \varphi(x_{n})\right\|^{2}\) & add the new sample \\  & \(\text{{dis}}=\min\limits_{a}\left\|\sum\limits_{i=1}^{m}a(i)\varphi(x_{i})- \varphi(x_{n})\right\|^{2}\) & to the dictionary. \\ \hline \multirow{4}{*}{SC} & According to information theory, \\  & based on prior joint Gaussian \\  & distribution, the amount of information \\  & brought by the new sample is: \\  & \(S(n)=-\ln\ p(x_{n},d_{n}|D_{n-1})\), \\  & where \(p(x_{n},d_{n}|D_{n-1})\) is posterior probability \\  & distribution of \((x_{n},d_{n})\) \\ \hline \multirow{4}{*}{CC} & Calculate the maximal kernel function of \\  & the new and existing samples : \\  & \(\mu=\max\limits_{i}|k(x_{i},x_{n})|\) & \\ \cline{1-1}  & \(\text{{dis}}=\min\limits_{a}|k(x_{i},x_{n})|\) & \\ \cline{1-1}  & & \\ \end{tabular}
\end{table} 
Table 1: Some common sparseness criteria. ## 4 Proposed SSW-KRLS Algorithm

Assume that there is a set of input-output pairs \(D=\{u_{i},y_{i}\}|_{i=1}^{L}\) where \(L\) is the total number of samples. \(u_{i}\in R^{m}\) are m-dimensional input vectors, and \(y_{i}\in R\) is the output.