## 3 Proposed Model
### Loss Function  
We assume \(\mathbf{s}\) is the clean speech, \(\mathbf{n}\) is the noise, so the noisy speech signal is \(\mathbf{x}=\mathbf{s}+\mathbf{n}\). The enhanced speech is set to \(\hat{\mathbf{x}}\).  
Most enhancement models based on U-Net use Mean Squared Error (MSE) or Mean Absolute Error (MAE) between enhanced speech and clean speech [4; 13; 14; 15]. However, MSE has poor robustness in dealing with abnormal points in the signal. Previous research has proved that MAE can significantly improve the denoising performance [14; 15]. Therefore, we adopt the MAE of enhanced speech and clean speech as loss, as shown in Equation (4), where \(N\) is the number of samples of speech and \(\|\cdot\|_{1}\) is the \(L_{1}\) norm.  
\[L_{MAE}(\mathbf{s},\hat{\mathbf{x}})=\frac{1}{N}\|\;(\hat{\mathbf{x}}-\mathbf{s})\;\|_{1}=\frac {1}{N}\sum_{i=1}^{N}\lvert\hat{\mathbf{x}}_{i}-s_{i}\rvert \tag{4}\]  
In order to further remove noise in the contaminated speech signal, we define a new loss as shown in Equation (5) by using the measurements as the optimizable target, called CS loss, where \(N\) denotes the number of samples of measurements and \(\|\cdot\|_{1}\) is \(L_{1}\) norm. The measurements of enhanced speech and clean speech are obtained directly by multiplying with the same measurement matrix \(\mathbf{A}\), without relying on sparsity constraint [22]. Measurements can retain the main information and even some unobvious features of original speech. Therefore, the enhancement performance of model can be improved by optimizing the MAE between measurements of clean speech and measurements of enhanced speech.  
\[L_{CS}(\mathbf{s},\hat{\mathbf{x}})=\frac{1}{N}\|\;(\mathbf{A}\hat{\mathbf{x}}-\mathbf{A}\mathbf{s})\; \|_{1} \tag{5}\]  
Furthermore, we also optimize clean speech and enhanced speech in time-frequency domain, which helps our model to learn the time-frequency characteristics of speech. The time-frequency domain loss function applies a multi-resolution Short-Time Fourier  
Figure 3: Shuffle Attention Mechanism, with Channel Attention (C-Att) and Spatial Attention (S-Att). GAP represents global average pooling, and FC represents Full Connection (i.e., \(\mathbf{\omega}\mathbf{x}+\mathbf{b}\)). Channel shuffle is similar to reshape operation.  
Transform (STFT) loss function, which is the sum of STFT losses under different transform parameters [15; 29]. The STFT loss is defined as follows:  
\[\begin{array}{l}L_{STFT}(\mathbf{s},\hat{\mathbf{x}})=\mathbb{E}\\ \begin{array}{l}\mathbf{s}\sim p(\mathbf{s})\\ \hat{\mathbf{x}}\sim p(\hat{\mathbf{x}})\end{array}\end{array}\left[\begin{array}{l} \frac{\parallel\left|STFT(\mathbf{s})\right|-\left|STFT(\hat{\mathbf{x}})\right| \parallel_{F}}{\parallel\left|STFT(\mathbf{s})\right|\parallel_{F}}\\ +\frac{1}{N}\parallel log|STFT(\mathbf{s})-log|STFT(\hat{\mathbf{x}})|\parallel_{1} \end{array}\right]\end{array} \tag{6}\]  
where \(\parallel\cdot\parallel_{F}\) denotes the Frobenius norm and \(\parallel\cdot\parallel_{1}\) is \(L_{1}\) norm. \(\left|STFT(\cdot)\right|\) stands for STFT magnitudes of speech. \(N\) is the number of elements in the magnitude. In Equation (6), the former is called spectral convergence loss, and the latter is called log STFT magnitude loss [29]. The multi-resolution STFT loss function can be expressed as shown in Equation (7), where \(M\) represents the number of STFT loss.  
\[L_{MSTFT}(\mathbf{s},\hat{\mathbf{x}})=\frac{1}{M}\sum_{m=1}^{M}L_{STFT}^{(m)}(\mathbf{s},\hat{\mathbf{x}}) \tag{7}\]  
The influence of loss function on the enhancement results will be compared through the ablation experiment in the experimental part.