## 5 Experimental Results and Discussions
### The Cascade of RPCA with any of the Other Methods  
Next, RPCA-SPC is cascaded with any of MN, MVN, and RASTA in order to see if such a connection is additive to provide even better results than each individual component method. Tables 3 and 4 and Figure 6 show the corresponding results associated with the cases of GMM-HMM and DNN-HMM, respectively. Here the notation \"A + B\" refers to method A followed by method B, while \"B + A\" refers to method B followed by method A, where A and B refer the names of methods to be cascaded. Notably, these two types of combinations are different since at least one of the component methods is a non-linear operation. Comparing the results shown in Tables 3 and 4 and Figure 6 with those in Tables 1 and 2, several observations can be made:  
1. The series connection of RPCA-SPC with either of MN and RASTA behave better than each single component method for Sets 2-7 (containing noise) and Sets 8-14 (containing noise and channel interference). For example, in the case of DNN-HMM, MN + RPCA-SPC, and RPCA-SPC + MN give 25.67% and 30.53% in WER averaged over the 14 Sets, which are lower than those obtained by MN (34.83%) and RPCA-SPC (31.39%). Similarly, RASTA + RPCA-SPC and RPCA-SPC + RASTA give 27.92% and 25.97% in WER averaged over the 14 Sets, which are lower than those obtained by RASTA (35.75%) and RPCA-SPC (31.39%). These results again claim that the newly proposed RPCA-SPC serves as a promising technique that can extract noise-robust components in speech features either preprocessed/post-processed by MN, RASTA or not. In addition, MN + RPCA-SPC provides a relatively low WER value on average for the DNN-HMM scenario, indicating that it is quite suitable for use and development in the deep learning architecture of speech recognition;
2. Unlike MN and RASTA, cascading MVN with RPCA-SPC does not necessarily gives better recognition results than MVN and RPCA-SPC alone. One possible reason is that MVN makes the features of all different channels behave like a common random variable with zero mean and unity variance, and thus the subsequent RPCA-SPC tends to view these features to be low-rank so as to discard a significant amount of them, which somewhat harms the recognition accuracy. In addition, For RPCA-SPC preprocessed features, MVN is likely to bring an over-normalization effect and diminishes the respective discriminating capability. \begin{table}
\begin{tabular}{c c c c} \hline  & **Set 1** & **Sets 2-7** & **Sets 8-14** \\ \hline Baseline & 4.75 & 51.58 & 67.92 \\ MN & 4.17 & 32.58 & 47.98 \\ MVN & 5.38 & 31.59 & 47.09 \\ RASTA & 5.53 & 30.04 & 45.72 \\ RPCA-SPC & 7.42 & 31.70 & 46.30 \\ \hline \end{tabular}
\end{table} 
Table 1: The WER (%) for different sets achieved by the GMM-HMM acoustic models with MFCC features as the baseline and the features processed by any of MN, MVN, RASTA, and RPCA-SPC. \begin{table}
\begin{tabular}{c c c c} \hline  & **Set 1** & **Sets 2-7** & **Sets 8-14** \\ \hline Baseline & 2.97 & 52.81 & 68.81 \\ MN & 2.62 & 28.81 & 44.60 \\ MVN & 2.97 & 26.54 & 43.12 \\ RASTA & 3.27 & 29.46 & 45.77 \\ RPCA-SPC & 3.79 & 27.33 & 38.82 \\ \hline \end{tabular}
\end{table} 
Table 2: The WER (%) for different sets achieved by the DNN-HMM acoustic models with FBANK features as the baseline and the features processed by any of MN, MVN, RASTA, and RPCA-SPC. Finally, the WER values averaged over the 14 Test Sets with respect to all the aforementioned methods are depicted in Figure 6. From this figure, it is reconfirmed that the proposed RPCA-SPC benefits the recognition accuracy under adverse environments, and it can be integrated with MN and RASTA, both of which are simple but effective noise-robust methods, to give rise to further better performance. In addition, RPCA-SPC behaves well for both GMM-HMM and DNN-HMM architectures.