## 2 Background Knowledge
### Conv-TasNet
#### 2.1.2 Mask Estimation Network  
As shown in the middle part of Figure 2, the mask-estimation layer between the encoder and decoder layers consists of stacked 1-D dilated convolution blocks, which follows the temporal convolutional network (TCN) architecture. In particular, the employed TCN contains \(R\) groups of blocks connected in tandem, each group consisting of \(M\) 1-D convolutional blocks that have dilation factors increasing exponentially from 1 to \(2^{M-1}\). The design of increasing dilation factors provides different context window sizes to catch short- and long-range dependencies of speech utterances. Finally, the output of the TCN goes through a 1-by-1 convolutional layer (viz., pointwise convolution) and a nonlinear activation (sigmoid) to produce the estimated masks for both clean speech and noise.  
Figure 2: A schematic diagram of the Conv-TasNet system.  
Referring to Figure 1, the 1-D convolutional block in the mask-estimation layer has two output paths: a residual path and a skip-connection path. The residual path acts as the input to the next block, while the skip-connection paths of all convolutional blocks are summed up as the output of the TCN. Moreover, a depth-wise separable convolution operation is employed in each convolution block instead of the standard convolution to reduce the computation complexity. A depth-wise separable convolution consists of two consecutive operations and a depth-wise convolution followed by a point-wise convolution. Additionally, the 1-D convolution block uses parametric ReLu (PReLU) [38] as the activation. PReLU modifies ReLU by adding a coefficient that controls the slope of the negative component; this coefficient, along with the other model parameters, is learned from the training set. PReLU prevents zero gradients from being brought on by negative channel outputs, allowing the model to be updated with these outputs. For the ImageNet 2012 benchmark utilizing 10-view testing, substituting PReLU with PReLU reduces the top-1 error from 33.82% to 32.64%--a 1.2% improvement. PReLU has been found to be more beneficial than ReLU, boosting the performance of the learnt network.  
Here, we give some detail about comparing standard convolution and depthwise separable convolution. Let the input contain \(G\) channels with \(M\)-point data and form a matrix \(\mathbf{Y}\in\mathbb{R}^{G\times M}\), and the output is expected to contain \(H\) channels with \(M\)-point data and form a matrix with size \(\mathbb{R}^{H\times M}\). The standard convolution uses a kernel tensor \(\mathbf{K}\in\mathbb{R}^{G\times H\times P}\), where \(P\) is the length of the impulse response for each one-dimension input-output convolution pair. In contrast, the depthwise separable convolution first uses \(G\) impulse responses with length \(P\), individually convolved with each of the \(G\) channels of input data. Next, the overall convolved \(G\) output feature maps pass through a pointwise (1-by-1) convolution layer that has a kernel tensor \(\hat{\mathbf{K}}\in\mathbb{R}^{G\times H\times 1}\) to produce the output matrix with size \(\mathbb{R}^{H\times M}\). Therefore, the standard convolution uses \(G\times H\times P\) parameters, while the depthwise separable convolution uses \(G\times P+G\times H\) parameters. Obviously \(G\times P+G\times H<G\times P\times H\) if \((P-1)(H-1)>1\), and the model size ratio \(\frac{G\times P\times H}{G\times P+G\times H}\) increases as \(H\gg P\) or \(P\gg H\).