## 3 Proposed Method  
In the presented method, the RPCA algorithm is applied to the matrix organized by the speech feature temporal sequence in order to extract the embedded noise-robust component. The procedure for this method is split into the following two steps:  
**Step 1: Create the baseline features, FBANK and MFCC:**  
Any time-domain utterance \(\{\text{x}[\ell]\}\) in the training and test sets is first passed through a high-pass pre-emphasis filter, and the operations of framing and windowing are performed in turn. Then, each windowed frame signal is converted to the acoustic frequency domain via short-time Fourier transform (STFT) to create the corresponding acoustic spectrum. Next, each frame-wise acoustic spectrum is converted to the FBANK or MFCC features. The magnitude of the acoustic spectrum associated with each frame is weighted by a Mel-frequency filter bank and then processed with the logarithmic operation to produce the FBANK features. Moreover, the MFCC features are derived after the application of the discrete cosine transform (DCT) to FBANK. **Step 2: Use RPCA to extract the sparse part of the FBANK/MFCC matrix**  
Let \(\{\mathbf{v}_{\text{m}};0\leq\text{m}\leq\text{M}-1\}\) denote the frame time series of FBANK or MFCC vectors for the time-domain signal \(x[\ell]\) obtained from Step 1, where m is the frame index and M is the total number of frames. Then a matrix V is created by assigning the \(\text{m}^{\text{th}}\) column vector of V to be \(\mathbf{v}_{\text{m}}\). Therefore, the matrix V is termed as the feature matrix of the time-domain signal \(\text{x}[\ell]\). Next, the RPCA algorithm stated in Section II is used to decompose the feature matrix V,  
\[\text{V}=\text{V}_{\text{L}}+\text{V}_{\text{S}}, \tag{3}\]  
where \(\text{V}_{\text{L}}\) and \(\text{V}_{\text{S}}\) denote the low-rank and sparse component matrices of V. Finally, we discard the low-rank part \(\text{V}_{\text{L}}\) while the sparse part \(\text{V}_{\text{S}}\) is preserved and treated as the new feature matrix for the subsequent processing in training and testing. Because the main idea of the aforementioned method is to extract the sparse component of the feature matrix via RPCA, we will use the notation \"RPCA-SPC\" to denote this new method hereafter for the ease of discussion. The primary idea of RPCA-SPC is as follows: For a noise-corrupted utterance and the corresponding speech feature sequence, the embedded noise part often varies more slowly with time relative to the clean-speech part. In other words, clean speech is likely to be more non-stationary than noise. This phenomenon can be easily observed when analyzing the spectrogram of an utterance. The spectral structure of pure noise is usually fixed or slow-varying, while the speech component changes quickly with time. Such an assumption implies that the noise part appears to be of low-rank, while the clean-speech part is sparse. Therefore, extracting the sparse component of the speech feature matrix tends to enhance the speech and alleviate noise. Here we provide two examples to reveal that RPCA tends to highlight the clean-speech part in the noise-corrupted data:  
First, Figure 1a depicts the spectrogram of a noisy utterance, and Figure 1b,c depicts the RPCA-derived sparse and low-rank partitions of the spectrogram, respectively, shown in Figure 1a. From these figures, it is obvious that the sparse part contains rich speech clues, while the low-rank counterpart corresponds to relatively less speech information and more about noise. Next, Figures 2a-c and 3a-c, respectively, show the time series of the sixth and eighth MFCC features of a noisy utterance as well as the corresponding sparse and low-rank components. Likewise, Figures 4a-c and 5a-c correspond to the original, sparse, and low-rank versions of the sixth and eighth FBANK coefficients of a noisy utterance.