## 2 Methods
### Adjusting Spread Factor for Balance between Distance-Stress and Local Topology
Hence techniques meant only for visualization just as t-SNE may not be optimal for improving the performance of analysis and classification for most of the data-sets. Preserving distance during dimension reduction could have its own advantage in terms of providing clear separability among dissimilar data-points. The success of TPDS in achieving low MDS-cost (distance stress), decent visualization and better clustering purity for tested data-sets hint that having a balance between local-topology and distance preservation could be useful for other analysis procedures also. The approach of TPDS to scale the distances to preserve some information about local topology can also help in better convergence of MDS-like methods. It could be explained as such: the larger distances among data-points could act as outliers and cause hurdle in convergence. The smaller distances among data-points with in the same group could be large in number, so giving them more weightage lead to a better reduction of overall MDS-cost. In this process, some information about local topology could also be preserved for better visualization. Influence of outliers on other distance-preserving methods is so strong that it causes a collapse of data-points and more computation time in convergence. Thus, TPDS shows a clear advantage over the typical distance-preserving method by giving less weightage to outliers. Another important feature in results of TPDS is that besides separability among data-points of different classes they have tighter co-localization among data-points of same classes, that too without the collapse of their location. Such result caused substantially higher classification purity for TPDS results by DBSCAN which exploit such tight co-localization for density-based clustering. Hence TPDS could be used as an alternative dimension reduction method to support such density-based clustering approach also. Besides being robust to outliers and having the potential to provide better classification

Figure 5: Dimension reduction of single-cell expression data. The 7 types of cells are shown with different colors. The adjusted Rand Index (ARI) and Normalized mutual information (NMI) were calculated after k-mean clustering (k = 7). The outlier cells (shown in red color) seem to have different effect with each dimension reduction method. For demonstration purpose perplexity was set to 4 for t-SNE. The performance of TPDS is substantially better than Sammon mapping and non-metric MDS even in terms of MDS-cost minimization. results, TPDS is not so slow (see supplementary Table S1). However, we are trying to optimise it further for better speed. Here we used SOM to learn preliminary estimation of the manifold. Other methods generally use KNN or a modified version of it [3; 19; 20] for the same purpose. We also evaluated the effect of the geometry of SOM on TPDS performance. By default, TPDS uses rectangular geometry, where every node has four neighbors. However, we also tested hexagonal geometry of SOM. The results with hexagonal geometry of SOM was almost similar to default mode (supplementary Figure S1). SOM provides neighborhood information about the cluster of data-points, hence in comparison to KNN, SOM results could be less influenced by noise because of the averaging effect. Such as with protein expression data set it can be seen that data-points belonging to the same class were closer to each other in TPDS due to averaging effect while calculating proximity likelihood. On other other hand t-SNE was not efficient in estimating the likelihood of proximity for data-points in protein expression data, probably due to noise or sample-specific artefacts. For visualization with TPDS, one needs to provide a guess about the number of SOM clusters (grid size), which could have the least effect on downstream analysis steps such as supervised classification and regression. Other methods, such as t-SNE and Sammon mapping, are also dependent on parameters.