## 2 Related Work  
Mathematically, a 1D FID can be modeled as the finite sum of damping exponential functions as follows [28; 29; 30; 31]:  
\[x_{n}=\sum_{j=1}^{J}\left(A_{j}e^{i\phi_{j}}\right)e^{-\frac{n\Delta t}{\gamma _{j}}}e^{in\Delta t\alpha_{ij}}, \tag{1}\]  
where \(J\) is the number of spectral peaks, \(\Delta t\) is the time interval, and \(A_{j},\phi_{j},\tau_{j},\omega_{j}\) are the amplitude, phase, decay time, and frequency of the \(j^{\text{th}}\) spectral peak, respectively. If there are \(N\) sampled data points, the sampled signal can be represented as a vector \(\mathbf{x}=[x_{1},x_{2},x_{3},\cdots,x_{n},\cdots,x_{N}]^{T}\), where \(\cdot^{T}\) represents vector transpose. The reconstruction model of the LRHM method is [21]:  
\[\min_{\mathbf{x}}\|\mathcal{R}\mathbf{x}\|_{+}+\frac{\lambda}{2}\|\mathbf{y}- \mathcal{U}\mathbf{x}\|_{2}^{2} \tag{2}\]  
where \(\mathcal{R}\) denotes an operator converting the undersampled FID \(\mathbf{x}\) into a Hankel matrix \(\mathcal{R}\mathbf{x}\), \(\mathcal{U}\) is an undersampling operator, \(\|\cdot\|_{\text{L}}\), represents the matrix nuclear norm as the sum of singular values, \(\|\cdot\|_{2}\) represents the \(l_{2}\) norm that measures data consistency, and \(\lambda\) is a regularization parameter that balances the low rankness and the data consistency. However, the LRHM model may sacrifice low-intensity peak recovery, which is related to small singular values. As we know, the information of a spectral peak includes its intensity, central frequency, and line shape. The nuclear norm only focuses on intensity. In order to utilize the extra information, including central frequency and line shape, hidden in the Hankel matrix, the SLS method was proposed [24]. The SLS method built a subspace framework to separate the original signal into a relatively strong signal subspace and a weak one. Then, peaks that lie in the weak subspace are particularly protected by enforcing them to have prior spectral peak shapes. The self-learning subspace reconstruction model is expressed as [24]:  
\[\min_{\mathbf{x}}P_{\Omega}(\mathbf{x})+\frac{\lambda}{2}\|\mathbf{y}- \mathcal{U}\mathbf{x}\|_{2}^{2}, \tag{3}\]where:  
\[P_{\Omega}(\mathbf{x})=\|\mathcal{P}\mathbf{x}\|_{*}-Tr\Big{(}\mathbf{A}_{\Omega }^{H}\mathcal{R}\mathbf{x}\mathbf{B}_{\Omega}\Big{)}, \tag{4}\]  
where \(\Omega\) represents strong signal space, \(Tr(\cdot)\) is the trace function and defined as the sum of the main diagonal elements, \(\mathbf{A}_{\Omega}\in\mathbb{C}^{r\times p}\) and \(\mathbf{B}_{\Omega}\in\mathbb{C}^{r\times Q}\) are two matrices formed by the first \(r\) columns of the left and right unitary matrixes, satisfying SVD \(\mathcal{R}\mathbf{x}=\mathbf{A}\mathbf{\Lambda}\mathbf{B}^{H}\). The function \(P_{\Omega}(\mathbf{x})\) sums up the least \(r\) singular values. We can estimate \(\mathbf{A}_{\Omega}\) and \(\mathbf{B}_{\Omega}\) from the initial solution \(\mathbf{x}_{0}\), and we can improve the accuracy of the subspace as it iterates continuously. For example, the subspace \(\mathbf{A}_{\Omega,l}\), \(\mathbf{B}_{\Omega,l}\) of the \(l^{\text{th}}\) outer iteration is more accurate than the initial estimate. Machine learning typically includes supervised learning, semi-supervised learning, and unsupervised learning, which use all, partial, and non-labeled samples, respectively. Transduction learning [32] is another type that predicts specific test samples by learning specific training samples. The proposed method belongs to unsupervised learning since no labeled samples or specific test samples are used. The singular value decomposition is adopted to learn the subspace of the Hankel matrix. The self-learning process means no prior or label information is provided in advance, and we only learn the subspace space information from the initial time-domain signal that is filled with zeros, and we update this information from the intermediate reconstruction. In practice, prior or label spectra are not accessible in the biomedical magnetic resonance experiment.