## 4 Experiments and Results
### Reconstruction of 2D NMR Spectra  
Reconstruction experiments with realistic NMR are conducted on 15% NUS 2D \({}^{1}\)H-\({}^{15}\)N HSQC spectra of GB1, 20% NUS 2D \({}^{1}\)H-\({}^{15}\)N best-TROSY spectra of ubiquitin, and 15% NUS 2D \({}^{1}\)H-\({}^{1}\)H solid-state spectra. Figure 3a, Figure 4a, and Figure 5a are fully sampled 2D \({}^{1}\)H-\({}^{15}\)N HSQC spectra of GB1, 2D \({}^{1}\)H-\({}^{15}\)N best-TROSY spectra of ubiquitin, and 2D \({}^{1}\)H-\({}^{1}\)H solid-state spectrum, respectively. Results show that all methods can reconstruct a majority of spectral peaks well. However, some peaks in best-TROSY experiments shown by arrows in Figure 4b,c are lost in LRHM and LRHMF. LRHM and LRHMF also produce some pseudo peaks in the reconstructed GB1 and solid spectra, as shown by arrows in Figure 3b,c and Figure 5b,c. By contrast, SLS and SLSMF methods provide faithful reconstructions. What is more, Figure 6 illustrates that the proposed method performs better in reconstructing low-intensity peaks. For example, LRHM and LRHMF weaken the reconstructed spectral peak near 125 ppm (Figure 6b-1,c-1) ), while SLSMF obtains the spectrum with peak intensity and line shape close to the fully sampled spectrum (Figure 6e-1). Figure 6: 1D traces of reconstructed 2D spectra (**a-1**)–**(**a-3**) is the fully sampled spectrum of the 1D traces of HSQC spectra of GB1, the best-TROSY spectrum of ubiquitin, and solid NMR spectrum, respectively. **(b-1)–**(b-3)**, **(c-1)–**(c-3)**, **(d-1)–**(d-3)**, and **(e-1)–**(e-3)** are reconstructed spectra using LRHM, LRHMF, SLS, and SLSMF, respectively. Note: **(a-1)–**(a-3)** are fully sampled spectra of 1D traces, and their locations in the corresponding fully sampled spectrum are marked with a dotted line in Figures 3–5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **Peak ID** & **1** & **2** & **3** & **4** & **5** \\ \hline LRHM & 0.597 \(\pm\) 0.384 & 0.859 \(\pm\) 0.203 & 0.944 \(\pm\) 0.090 & 0.991 \(\pm\) 0.012 & 0.997 \(\pm\) 0.004 \\ \hline LRHMF & 0.600 \(\pm\) 0.384 & 0.857 \(\pm\) 0.203 & 0.943 \(\pm\) 0.090 & 0.991 \(\pm\) 0.012 & 0.997 \(\pm\) 0.004 \\ \hline SLS & 0.935 \(\pm\) 0.183 & 0.974 \(\pm\) 0.152 & 0.992 \(\pm\) 0.039 & 0.999 \(\pm\) 0.003 & 0.999 \(\pm\) 0.001 \\ \hline SLSMF & 0.936 \(\pm\) 0.181 & 0.974 \(\pm\) 0.152 & 0.993 \(\pm\) 0.035 & 0.999 \(\pm\) 0.002 & 0.999 \(\pm\) 0.001 \\ \hline \end{tabular} Note: Peak intensity correlations and error bars for the reconstructed spectra when 6% of data are acquired. The error bars are the standard deviations of correlations over 100 sampling trials. The numerical form of the table is \(C\pm D\), where C is the mean of correlation, and D is the standard deviation.  
\end{table} 
Table 4: Statistics of peaks correlations and standard deviations. Regression analyses (Figures 7-9) also exhibit that the SLSMF outperforms LRHM and LRHMF. It is shown that the SLSMF method gets a higher correlation than LRHM and LRHMF, especially in low-intensity peaks (Figure 7e-h, Figure 8e-h, and Figure 9e-h). Besides, lower RLNEsby SLSMF can be achieved than in LRHM and LRHMF, and RLNEs of SLSMF decrease much faster (Figure 10). For instance, we can see that after the update of the subspace, the RLNEs of SLSMF drop almost linearly, implying that the introduction of the signal subspace can help reconstruct high-fidelity spectra more quickly and accurately with the same sampling rate. The above observations indicate that in terms of reconstruction time and reconstruction quality, the proposed approach permits considerable improvements from the SLS and low-rank Hankel matrix methods. From Figure 11, we can see that compared to the SLS approach, the proposed method significantly speeds up the reconstruction time about 2-4 times (Figure 11a). If equipped with parallel computing architecture, the acceleration in the reconstruction time of SLSMF would be striking.