## 2 Background Fundamentals
### Randomized Low-Rank Approximations  
Data approximation plays a major role in reducing complex calculations into less complicated ones. Most data modeling works better for certain and less complex datasets. An appropriate approximation model helps in performing the decision-making process better with complex datasets. If the knowledge base has a huge number of attributes with various ranges of data, the computation complexity can be reduced by finding the rank of the space. Thus, we utilized the randomized low-rank approximation (RLRA) technique in the research model [24; 25]. The method of approximating a matrix by a relatively lower rank matrix is known as a low-rank approximation. The goal is to achieve a further compact representation of the original data set with limited loss of information. In this section, let us discuss the basics of the RLRA technique. **(i)**: **Singular Value Decomposition (SVD)**: Any real matrix of arbitrary size can be factored or decomposed into a product of orthonormal matrices and diagonal matrices, such as Equation (2):  
\[[Y]_{p\times s}=U_{p\times p}\Sigma_{p\times q}V_{q\times q}^{T} \tag{2}\]  
where \(U\) and \(V\) both have orthonormal columns, and whose columns contain, respectively, the left and right singular vectors. The diagonal matrix \(\Sigma\) contains singular values corresponding to the singular vectors. We denote the \(i\)th singular value as \(\sigma_{i}(Y)\) and or \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Summary**} & \multicolumn{5}{c}{**Attribute**} \\  & \(a_{1}\) & \(a_{2}\) & \(a_{3}\) & \(a_{4}\) & \(a_{5}\) & \(a_{6}\) & \(a_{7}\) \\ \hline Multiple R & 0.927 & 0.681 & 0.425 & 0.433 & 0.819 & 0.399 & 0.895 \\ R Square & 0.860 & 0.464 & 0.181 & 0.187 & 0.671 & 0.159 & 0.802 \\ Adjusted R Square & 0.438 & \(-\)0.429 & \(-\)0.638 & \(-\)0.300 & 0.561 & 0.039 & 0.207 \\ Standard Error & 0.224 & 10.686 & 4.369 & 6.059 & 6.135 & 1.926 & 1.006 \\ VIF value & 7.120 & 1.866 & 1.221 & 1.231 & 3.037 & 1.190 & 5.047 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: VIF number with the corresponding attribute values. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(a_{2}\) & \(a_{3}\) & \(a_{4}\) & \(a_{5}\) & \(a_{6}\) \\ \hline
31.33 & 5.3 & 19.98 & 38.2 & 2.67 \\
42.44 & 4.7 & 6.01 & 35.8 & 4.97 \\
43.13 & 9.4 & 8.89 & 17 & 3.11 \\
22.45 & 9.5 & 12.25 & 11.3 & 6.34 \\
52.87 & 9.8 & 3.2 & 32.5 & 4.81 \\
43.36 & 0.9 & 17.1 & 26.2 & 7.32 \\
29.19 & 9.2 & 8.5 & 27.1 & 2.64 \\
34.56 & 2.7 & 6.62 & 15.6 & 1.37 \\
37.23 & 4.4 & 16.21 & 15.3 & 2.22 \\
41.78 & 4.1 & 1.98 & 11.02 & 3.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Reduced information system. der them and their corresponding singular vectors into \(U\) and \(V\) such that \(\Sigma=\text{diag}\)\((\sigma_{1}(Y),\ \sigma_{2}(Y),\ \ldots)\) and \(\sigma_{i}(Y)\geq\sigma_{i+1}(Y)\) for all \(i\). This is represented as Equation (3),  
\[Y=\sum_{i=1}^{\min\{p,q\}}\sigma_{i}u_{i}v_{i}^{T} \tag{3}\]  
which is a decomposition of \(Y\) into a sum of rank-l matrices. Here, \(Y^{T}Y=V\Sigma^{2}V^{T}\) and \(YY^{T}=U\Sigma^{2}U^{T}\); note that the eigenvectors of \(Y^{T}Y\) are \(\{v_{i}\}\), and the eigenvectors of \(YY^{T}\) are \(\{u_{i}\}\). Each one has the eigenvalues \(\lambda_{i}=\sigma_{i}^{2}\). **(ii) Orthogonal Projections**  
Assume we have an orthonormal basis for a linear subspace that has been stacked into a matrix: \(Q=[q_{1}q_{2}\ldots q_{k}]\). Then, \(QQ^{T}\) is a projection matrix that, when applied to a matrix \(A_{R}\), projects it orthogonally onto the subspace spanned by \(Q\), which we denote as \(\hat{A}_{Ap}\) in Equation (4):  
\[\hat{A}_{Ap}\equiv QQ^{T}A_{R} \tag{4}\]  
**(iii) Norms**  
Two matrix norms are commonly used to keep things simple and easy to compare--the Frobenius norm, Equation (5), and the \(L_{2}\) norm, Equation (6).