# The Extraction of Maximal-Sum Principal Submatrix and Its Applications
## 7 Conclusions  
We, in this paper, propose a fusion row update accelerated algorithm (MSPS-RU) and present two different applications. The experimental results show that the proposed algorithm avoids repeatedly calculating the sum of the same submatrix and significantly improves computing efficiency compared to the Enum algorithm; the color combination based on MSPS-RU enhances the contrast of the data visualization results; the stock investment portfolio constructed by MSPS-RU can bring higher investment returns but lower risks, avoiding the uncertainty of random investments.  
Our method has limitations, and solving these problems can become future research topics. The improvement of computational efficiency in this paper mainly depends on reducing the number of matrix elements to be summed. However, it can also be achieved  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline Indicator & Ours & Rnd1 & Rnd2 & Rnd3 & Rnd4 & Rnd5 & Rnd6 & Rnd7 & Rnd8 & Rnd9 & Rnd10 \\ \hline \(\sigma\) & 0.0278 & 0.0307 & 0.0260 & 0.0133 & 0.0197 & 0.0343 & 0.0172 & 0.0291 & 0.0293 & 0.0220 & 0.0316 \\ \(Ret\) & 1.0332 & 1.0197 & 1.0300 & 0.9862 & 0.9584 & 1.0240 & 1.0260 & 1.0024 & 1.0029 & 1.0522 & 0.9509 \\ \hline Indicator & Rnd11 & Rnd12 & Rnd13 & Rnd14 & Rnd15 & Rnd16 & Rnd17 & Rnd18 & Rnd19 & Rnd20 & \\ \hline \(\sigma\) & 0.0243 & 0.0229 & 0.0289 & 0.0189 & 0.0251 & 0.0149 & 0.0312 & 0.0302 & 0.0237 & 0.0135 & \\ \(Ret\) & 0.9866 & 1.0366 & 1.0564 & 1.0164 & 1.0185 & 1.0353 & 1.0451 & 0.9961 & 0.9950 & 1.0168 & \\ \hline Indicator & Rnd21 & Rnd22 & Rnd23 & Rnd24 & Rnd25 & Rnd26 & Rnd27 & Rnd28 & Rnd29 & Rnd30 & \\ \hline \(\sigma\) & 0.0260 & 0.0206 & 0.0410 & 0.0287 & 0.0250 & 0.0242 & 0.0206 & 0.0264 & 0.0204 & 0.0169 & \\ \(Ret\) & 1.0243 & 1.0055 & 1.0170 & 1.0244 & 1.0165 & 1.0361 & 1.0388 & 1.0386 & 1.0122 & 0.9902 & \\ \hline \hline \end{tabular}
\end{table}
Table 6: Return and \(\sigma\) of simulated investments for each portfolio after 102 weeks.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline Diff. & Rnd1 & Rnd2 & Rnd3 & Rnd4 & Rnd5 & Rnd6 & Rnd7 & Rnd8 & Rnd9 & Rnd10 \\ \hline \(\Delta\sigma\) & 0.0028 & \(-\)0.0019 & \(-\)0.0146 & \(-\)0.0081 & 0.0064 & \(-\)0.0106 & 0.0013 & 0.0015 & \(-\)0.0058 & 0.0038 \\ \(\Delta Ret\) & \(-\)0.0134 & \(-\)0.0031 & \(-\)0.0470 & \(-\)0.0747 & \(-\)0.0092 & \(-\)0.0072 & \(-\)0.0308 & \(-\)0.0302 & \(-\)0.0190 & \(-\)0.0823 \\ \hline Diff. & Rnd11 & Rnd12 & Rnd13 & Rnd14 & Rnd15 & Rnd16 & Rnd17 & Rnd18 & Rnd19 & Rnd20 \\ \hline \(\Delta\sigma\) & \(-\)0.0035 & \(-\)0.0049 & 0.0011 & \(-\)0.0089 & \(-\)0.0028 & \(-\)0.0129 & 0.0034 & 0.0023 & \(-\)0.0042 & \(-\)0.0143 \\ \(\Delta Ret\) & \(-\)0.0466 & 0.0034 & 0.0232 & \(-\)0.0167 & \(-\by reducing the number of principal submatrices to be calculated, for example, using pruning. The selection of color combinations should consider not only the maximizing color difference but also aesthetic habits and preferences. In addition, more application problems can be modeled by extracting the maximum sum principal submatrix from a raw data matrix.  
Conceptualization, L.L. and H.L.; methodology, Y.Z. and H.L.; software, Y.Z. and H.L.; validation, Y.Z., L.L. and H.L.; formal analysis, H.L.; investigation, Y.Z.; resources, Y.Z.; data curation, Y.Z.; writing--original draft preparation, Y.Z., L.L. and H.L.; writing--review and editing, L.L.; visualization, Y.Z. and H.L.; supervision, L.L. and H.L.; project administration, H.L. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding. Data can be download from [[http://www.data.ac.cn/table/tbc40](http://www.data.ac.cn/table/tbc40)]([http://www.data.ac.cn/table/tbc40](http://www.data.ac.cn/table/tbc40)), accessed on 21 May 2023.  
The author thanks the anonymous referees for their valuable suggestions.  
The authors declare no conflict of interest.