## 4 Experiments and Results  
We compare the proposed SLSMF method with the other three state-of-the-art NMR reconstruction methods, including the LRHM, the LRHM with matrix factorization (LRHMF) [26], and the SLS. Spectra reconstruction experiments were conducted on both synthetic 1D NMR spectra (Figure 2) and real 2D NMR spectra (Figures 3-5). All reconstruction algorithms were performed using MATLAB 2017b (Mathworks Inc., Natick, MA, USA) on a computational server with two E5-2650v4 CPUs (12 core/CPU) and 160 GB RAM. The important spectra parameters, including two 2D spectra of small, large, and intrinsically disordered proteins and one 2D spectrum of solid NMR, are listed in Table 2. More details can be found in the experimental descriptions below. \begin{table}
\begin{tabular}{l l} \hline
**Initialization**: Input \(\mathbf{y}\), \(\mathcal{R}\), \(\mathcal{U}\), set outer maximal iterations times \(L=5\), convergence condition \(\eta_{\text{tol}}=10^{-6}\), \\  & and maximal inner number of iterations \(K=10^{3}\). Initialize the solution \(\mathbf{x}_{0}=\mathcal{U}^{T}\mathbf{y}\), the dual variable \(\mathbf{D}_{0}=1\), \\  & the number of iterations \(k=0\), and \(\eta_{0}=1\). \\
**Main**: \\
**While** (\(\eta_{l}\geq\eta_{\text{tol}}\)) or (\(l<L\)), **do**: \\ (1) & Estimate \(\mathbf{A}_{\Omega,l}\) and \(\mathbf{B}_{\Omega,l}\) from the first \(r\) columns of \(\mathbf{A}_{l}\) and \(\mathbf{B}_{l}\), such that \(\mathcal{R}\mathbf{x}_{l}=\mathbf{A}_{l}\mathbf{\Lambda}\mathbf{B}_{l}^{H}\) \\ (2) & **While** (\(\eta_{k}\geq\eta_{\text{tol}}\)) or (\(k<K\)), **do**; \\  & (a) & Update \(\mathbf{x}_{k+1}\) according to Equation (10); \\  & (b) & Update \(\mathbf{P}_{k+1}\) according to Equation (12); \\  & (c) & Update \(\mathbf{Q}_{k+1}\) according to Equation (14); \\  & (d) & Update \(\mathbf{D}_{k+1}\) according to Equation (15); \\  & (e) & Compute \(\eta_{k+1}=\|\mathbf{x}_{k+1}-\mathbf{x}_{k}\|/\|\mathbf{x}_{k}\|\) and \(k\gets k+1\); \\  & **End while**; \\ (3) & Set \(k=0\),\(\eta_{l}=1\) and \(l\gets l+1\); \\ (4) & Update \(\mathbf{x}_{l+1}\leftarrow\mathbf{x}_{k+1}\), \(\eta_{l+1}=\|\mathbf{x}_{l+1}-\mathbf{x}\|/\|\mathbf{x}_{l}\|\); \\
**End for**; \\
**Output**: The reconstructed FID \(\hat{\mathbf{x}}\leftarrow\mathbf{x}_{l+1}\). \\ \hline \end{tabular}
\end{table} 
Table 1: The reconstruction algorithm of self-learning subspace matrix factorization (SLSMF). Figure 1: Schematic diagram of parallel computing of NMR spectroscopy reconstruction. Note: The blue dot represents the sampled data point and the white one represents unsampled data point; “Core j” represents the \(j^{th}\) CPU (Central Processing Unit) core of a multi-core CPU for parallel computation. The non-uniform sampling (NUS) is performed along the indirect dimension (\(t_{1}\)), and each 1D vector (in red rectangle) located at each data point in the direct dimension (\(t_{2}\)) can be reconstructed separately. Figure 3: Reconstructed 2D HSQC spectra of GB1 with 15% NUS data. (**a**) The fully sampled spectrum; (**b**–**e**) reconstructed spectra using the LRHM, LRHMF, SLS, and SLSMF, respectively. Note: The contours of spectra are at the same level. Figure 2: Reconstructed synthetic 1D spectra with five peaks from 6% acquired data. (**a**) The fully sampled spectrum; (**b**) the noisy fully sampled spectrum; (**c**–**f**) reconstructed spectra using the low-rank Hankel matrix (LRHM), LRHM with matrix factorization (LRHMF), self-learning subspace (SLS) and SLSMF, respectively. The 2D \({}^{1}\)H-\({}^{15}\)N HSQC spectrum of GB1 was prepared with following conditions: The GB1 object sample is 2 mM U-\({}^{15}\)N, 20%-\({}^{13}\)C GB1 in 25 mM PO\({}_{4}\), pH 7.0 with 150 mM NaCl, and 5% D\({}_{2}\)O. Data were collected using a phase-cycle selected HSQC (hsqcfp3 gpphwg in Bruker library) at 298 K on a Bruker