# Laplacian Eigenmaps Dimensionality Reduction Based on Clustering-Adjusted Similarity
## 3 Experiment Setup
### Evaluation Metrics  
To evaluate the performance of different dimensionality reduction methods, we adopt four frequently used clustering-effect evaluation index: Fowlkes and Mallows Index (FMI) [32], F-measure [33] and Purity (PU) [34].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Datasets** & **Instance** & **Dimensions** \\ \hline Msplice & 3175 & 240 \\ W1a & 2477 & 300 \\ Soccer-sub1 & 1735 & 279 \\ Madelon & 2600 & 500 \\ FG-NET & 1002 & 262 \\ ORL & 400 & 1024 \\ Musk & 6598 & 223 \\ CNAE-9 & 1080 & 857 \\ SECOM & 1567 & 591 \\ DrivFace & 1606 & 525 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Statistics of datasets used for experiments.  
For dataset \(\mathbf{X}=[\mathbf{x}_{1};\mathbf{x}_{2};\ldots;\mathbf{x}_{\mathbf{n}}]\), it is assumed that the cluster obtained through clustering is divided into \(\mathcal{C}=\{\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{m}\}\), and the cluster given by the reference model is divided into \(\mathcal{C}^{*}=\{\mathcal{C}_{1}^{*},\mathcal{C}_{2}^{*},\ldots,\mathcal{C}_ {m}^{*}\}\). Accordingly, let \(\lambda\) and \(\lambda^{*}\) respectively represent the cluster marker vectors corresponding to \(\mathcal{C}\) and \(\mathcal{C}^{*}\). We consider the sample pairwise, as defined below: \(\mathbf{a}=|\mathbf{SS}|,\mathbf{SS}=\left\{\left(\mathbf{x}_{i},\mathbf{x}_{ j}\right)|\lambda_{i}=\lambda_{j},\lambda_{i}^{*}=\lambda_{j}^{*},i<j\right\}\), \(\mathbf{b}=|\mathbf{SD}|,\mathbf{SD}=\left\{\left(\mathbf{x}_{i},\mathbf{x}_{ j}\right)|\lambda_{i}=\lambda_{j},\lambda_{i}^{*}\
eq\lambda_{j}^{*},i<j\right\}\), \(\mathbf{c}=|\mathbf{DS}|,\mathbf{DS}=\left\{\left(\mathbf{x}_{i},\mathbf{x}_{ j}\right)|\lambda_{i}=\lambda_{j},\lambda_{i}^{*}=\lambda_{j}^{*},i<j\right\}\), \(\mathbf{d}=|\mathbf{DD}|,\mathbf{DD}=\left\{\left(\mathbf{x}_{i},\mathbf{x}_{ j}\right)|\lambda_{i}=\lambda_{j},\lambda_{i}^{*}\
eq\lambda_{j}^{*},i<j\right\}\).  
\(|\mathbf{SS}|\) set contains sample pairs which belong to the same cluster in \(\mathcal{C}\) are still belong to the same cluster in \(\mathcal{C}^{*}\), \(|\mathbf{SD}|\) set contains sample pairs which belong to the same cluster in \(\mathcal{C}\) are not belong to the same cluster in \(\mathcal{C}^{*}\). \(|\mathbf{DS}|\) set contains sample pairs which not belong to the same cluster in \(\mathcal{C}\) are belong to the same cluster in \(\mathcal{C}^{*}\), \(|\mathbf{DD}|\) set contains sample pairs which not belong to the same cluster in \(\mathcal{C}\) are still not belong to the same cluster in \(\mathcal{C}^{*}\). a, b, c, d represents the number of data pairs in set \(|\mathbf{SS}|\), \(|\mathbf{SD}|\), \(|\mathbf{DS}|\), \(|\mathbf{DD}|\). Thus,  
\[FMI=\sqrt{\frac{a}{a+b}\cdot\frac{a}{a+c}} \tag{16}\]  
F-measure is based on precision and recall of clustering results:  
\[F-\text{measure}=\frac{2\times\text{Precision}\times\text{Recall}}{\text{ Precision}\times\text{Recall}} \tag{17}\]  
Purity represents the number of samples with correct clustering in the total number of samples. \(\mathbf{N}\) represents total number of samples, \(\mathcal{C}=\{\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{m}\}\) is the collection of cluster, \(\mathcal{C}_{h}\) represents \(h\)-th cluster, \(\mathbf{X}=\{\mathbf{X}_{1},\mathbf{X}_{2},\ldots,\mathbf{X}_{n}\}\) is the collection of sample, \(\mathbf{X}_{i}\) represents \(i\)-th sample. Computational formula is shown as below:  
\[PU(\mathcal{C},\mathbf{X})=\frac{1}{\mathbf{N}}\sum_{h}\max_{i}|\mathcal{C}_ {h}\cap\mathbf{X}_{i}| \tag{18}\]