## 2 Methods
### Adjusting Spread Factor for Balance between Distance-Stress and Local Topology
On the other hand, Sammon mapping and non-metric MDS caused data-points to be localised as highly overlapping strata according to eight groups. Using K-means clustering and calculating purity of classification provided higher NMI and ARI score for TPDS compared to other tested methods (Figure 2). We confirmed the improvement provided by TPDS in clustering using DBSCAN (see Table 1). However, the mixing of classes hinted towards unknown co-variates. Therefore we again used TPDS to represent the high dimensional protein expression data set using three dimensions. When data was reduced to three dimension, the 3D scatter plot of TPDS could show clear separability between trisomic and normal mice. Whereas other methods could not provide comparable separability, such as TPDS (Figure 3). Overall, with mouse cortex protein expression data set, we realised that TPDS has better potential to reduce dimension to align similar class data-points together and provide separability between non-similar groups. Figure 1: Dimension reduction of [PERSON]’s data set by [PERSON] et al. [14] using four different methods. The distance stress-based cost is represented here as MDS-cost. of children [15]. Applying our approach and other three methods revealed clear separability for a few classes in reduced dimension (Figure 4). However, evaluating the separability after dimension reduction of SCADI data set, revealed that the classification of TPDS output resulted in substantially higher clustering purity than other tested methods (see Figure 4 and Table 1). Finally, we used single-cell expression data to evaluate the potential of TPDS. The single-cell expression data set used here was published by [PERSON] et al. [17]. This data set consists of read-count of more than 57,242 genes as features and 562 cells as data-points. For dimension reduction of such data-sets with such large features, principal component decomposition (or SVD) are often used, especially in case of t-SNE. Therefore we provided loading on top 30 principal components to all the methods. The single-cell data-sets used here had labels for each cell and they could be categorised to 7 cell-types which we took as classes. The visualization of t-SNE and TPDS outputs showed distinguishable loci for cells belonging to different types (classes). In terms of clustering purity using k-means and DBSCAN, TPDS had better performance than other distance preserving methods (Sammon mapping, non-metric MDS) (see Figure 5). Notice that at different values of perplexity the performance of t-SNE changes. Nevertheless, TPDS, which uses MDS function, also has comparable performance to t-SNE for single-cell expression data-set. Surprisingly, for single-cell data-set, the distance stress cost (MDS-cost) for TPDS was lower than other tested methods despite decent visualization through the preservation of local topology among cells. It hints about possible convergence issue of Sammon mapping and non-metric MDS method due to outlier cells. In Figure 5, it is clearly visible that a few cells (represented by red color) are outliers which caused the collapse of locations of other cell types with Sammon mapping and non-metric MDS. Notice that, even if TPDS uses non-metric MDS method as the last step, the collapse of data-points do not happen due to distance scaling to preserve local topology. Thus, TPDS seem to have avoided local minima by providing less weightage to large distances of outlier data-point. Figure 4: Dimension reduction of SCADI data set.The adjusted Rand Index and Normalized mutual information (NMI) were calculated after performing k-mean clustering using k = 7. ## 4 Discussion

Various procedures of analysis of large data-sets such as classification, regression, and anomaly detection can be improved using dimension reduction techniques. Given the diversity of Biomedical data-set, there could be multiple factors which influence classification.