## 2 Background Fundamentals
### Randomized Low-Rank Approximations
The Frobenius norm is as follows:  
\[\|Y\|_{F}^{2}=\sum_{i,j}Y(i,\ j)^{2}=tr(Y^{T}Y)=\sum_{i=1}^{\min\{p,q\}}\sigma_ {i}(Y)^{2} \tag{5}\]  
The \(L_{2}-\)_operator norm_ is as follows:  
\[\|Y_{2}\|=\max_{\|v\|_{2}=1}\|Yv\|_{2}=\sigma_{1}(Y) \tag{6}\]  
where, for vectors \(v\), \(\|v\|_{2}\) is the standard \(L_{2}\) (Euclidean) norm \(\|v\|_{2}^{2}=\sum_{i}v_{i}^{2}\). **(iv) Optimal Randomized Low-Rank Decomposition**  
The motivation for using the Frobenius and \(L_{2}\) norm for the RLRA method was supported by the classical result from [26], which states that the best rank-k approximation to \(Y=U\Sigma V^{T}\) is achieved by truncating the singular values (setting \(\sigma_{j}(Y)=0\) for \(j>k\)). There always exists a set of columns and multiplicative factor of those to produce the optimal solution [27]. Table 3 attribute values are approximated by Algorithm 2 and represented in Table 4. By approximation using the RLRA algorithm, minimal information loss is achieved, which ensures that the authenticity of the data remains unchanged. Figure 2 visualizes the heatmap structure of the reduced information system and the RLRA-information system to represent the color variation of numerical data, while preserving the data's authenticity. Figure 3 represents the graphical similarity behavior of the RLRA algorithm to the reduced information system and the approximated information system. ```
Parameters:
1.\([A_{R}]_{p\times q}\): Input reduced matrix. 2.\(\rho[[A_{R}]_{p\times q}]\): Rank of input matrix. 3.\(k\): Rank of target approximated matrix. 4.\(\phi_{q}\): Orthogonal basis. 5.\(s\): Subsample. 6.\([Prob_{q_{i}}]\): Probability of orthogonal basis using \(Frobenius-L_{2}\) norm. 7.\([Y]_{p\times s}\): Resample vector. 8.\(U_{1},U_{2},...,U_{k}\): Orthonormal basis of decomposed resampled vector. 9.\(Q\): Orthonormal basis using random projection. 10.\(\tilde{A}_{AP}\): Low-rank approximation. 11.\(\rho(k)\): Rank of the low-rank approximated vector. ```
Input: Input matrix \([A_{R}]_{p\times q}\). Output: Reduced matrix \([\tilde{A}_{AP}]_{p\times q}\). Procedure:
1: Load input matrix \([A_{R}]_{p\times q}\) with \(p\) rows and \(q\) columns. 2: Compute the rank of \([A_{R}]_{p\times q}\) given as \(\rho[[A_{R}]_{p\times q}]=r\). 3: Let \(k\) be the rank of the approximated matrix. 4: Suppose \(\phi_{q}\) is an orthogonal basis for the subsamples \(s\), where \(s\in[A_{R}]_{p\times q}\) for the Frobenius-\(L_{2}\) norm with the probability \([Prob_{q_{i}}]\) as \([Prob_{q_{i}}]=\frac{\|A_{R}(\cdot\phi_{q_{i}})\|_{L_{2}}^{2}}{(\|A_{R}\|_{ \mathrm{Frobenius}}^{2})}\). 5: Resample the vector \([Y_{p\times s}]=max(Prob_{q_{i}})\), where \(1\leq i\leq s\). 6: Apply the resampled vector to the orthonormal basis for the best approximation. Decompose \([Y]_{p\times s}=U\Sigma V^{T}\) when \(U_{k}=U_{1}\cdot U_{2}\cdot\ldots U_{k}\) is the orthonormal basis of \([Y]\), where \(U_{1}\cdot U_{2}\cdot\ldots U_{k}\longrightarrow\) is a decomposition of a resampled vector. 7: Compute \(Q=[q_{1}\cdot q_{2}\cdot\ldots q_{k}]\), then top-rank \(k\) of a singular vector of \([Y]\) via SVD. 8: Low-rank approximation \(\tilde{A}_{AP}\) onto basis \(Q:\tilde{A}_{AP}=Q\cdot Q^{T}\cdot A_{R}=\rho(k)\). ```

**Algorithm 2** Approximation using RLRA. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(a_{2}\) & \(a_{3}\) & \(a_{4}\) & \(a_{5}\) & \(a_{6}\) \\ \hline
31.48 & 5.61 & 20.55 & 37.19 & 3.12 \\
43.15 & 5.48 & 6.42 & 36.05 & 4.55 \\
42.70 & 10.39 & 8.49 & 15.62 & 3.57 \\
23.11 & 10.97 & 12.04 & 13.25 & 5.86 \\
54.37 & 10.02 & 3.28 & 31.38 & 4.62 \\
42.65 & 1.32 & 16.96 & 28.22 & 7.70 \\
30.28 & 9.55 & 8.82 & 25.56 & 3.57 \\
33.14 & 2.55 & 6.39 & 18.89 & 1.95 \\
39.07 & 4.60 & 15.78 & 16.38 & 3.17 \\
43.00 & 4.71 & 1.46 & 10.31 & 2.68 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Approximate information system. ### Ensemble Learning

In general, ensemble learning is based on the baseline classifiers for dependent or independent methods.