## 1 Introduction  
Dimension reduction of high dimensional data is an important problem in a wide variety of domains. Whether it is the field of genomics, proteomics or medical informatics, dimension reduction always poses a challenge to extract meaningful information in low dimension for visualization, classification and other down-stream analysis. Traditionally, three different kinds of approaches are used for dimension reduction which try to preserve one of three factors: distances between data points, local topology, or the overall information in the data. While principal component analysis is an information preserving method, multidimensional scaling (MDS) and Sammon mapping preserve distance [1], whereas methods such as t-SNE [2] and Locally-Linear embedding (LLE) [3] are nonlinear dimension-reduction techniques which aim to preserve local structure of data [4]. These methods of dimension reduction have their own criteria and cost-function which they try to minimize. Sammon mapping tries to minimize the squared difference among scaled distances in high and low dimension normalised by the original distance in the high-dimensional space. One of the popular methods of dimension reduction, t-SNE aims to minimize cost function similar to SNE (stochastic neighbor embedding). SNE cost function [5] is based on Kullback-Leibler divergences between the conditional probabilities of neighborhood based on Gaussian distribution. The SNE cost function does not allow the collapse of similar data-points and it emphasizes local distanceswhich often leads to loss of information about the global topology. A class of non-linear dimension reduction methods such as Self organising map (SOM) focuses on assigning (or collapsing) data-points to predefined coordinates (lattice). SOM is often used to make an initial prediction of manifold [6]. Even though SOM tends to provide visualization of local neighborhood among groups of data-points, it is not able to properly represent heterogeneity-based local topology or global distance information. Methods which are mainly designed for local topology-based visualization [2; 7; 8] often lose information needed for other downstream analysis. On the other hand, preserving global topology of distances has its own importance in dimension reduction, especially for downstream analysis steps such as clustering, phylogenetic analysis, or regression. However, while optimization of cost function of MDS, most often large distance dominate and cause a collapse of data-points [9] which have some similarity among each-other but are distinct. Given recent trends in biological data-set, the collapse of data-points in lower dimensions could lead to loss of valuable and useful information. Single-cell transcriptome and proteome data-sets are meant to highlight differences among between cell-groups as well as heterogeneity among every cell so that the gradient of cell-states can be studied [10; 11]. Similarly, while studying a cohort of individuals, we would like to visualise a large difference between diseased and normal cases as well as exploit heterogeneity among patients for stratification. Hence there is need for a method which can preserve distance information as well as local topology. Biomedical data analysts could benefit from such methods which could avoid artefact due to outlier effect of large distances as well as avoid collapse of data-points while preserving distances needed for downstream procedures such as clustering and machine learning. Hence we developed an approach to extend pre-existing distance preserving dimension reduction method, MDS, such that local topology can also be preserved to desirable extent to tackle its above mentioned weaknesses. We call our method as Topology preserving distance scaling (TPDS). TPDS uses the approach of distance shrinkage before using the method of non-metric MDS. The objective of TPDS is not only to improve visualization such as t-SNE, but represent data in low dimension such that other procedures such as classification could be made more efficient.