## 2 Methods
### Adjusting Spread Factor for Balance between Distance-Stress and Local Topology  
The spread factor \(\rho\) in Equation (1) indirectly controls the level of scaling of distances. If the value of \(\rho\) is low, the likelihood of proximity for larger distances become negligible and the local topology neighborhood is also small. Whereas if the value of \(\rho\) is high, larger distances are not ignored while scaling and lead to preservation of global structure of distances but may suppress local topology information. Hence optimization for \(\rho\) is an important step for TPDS. For optimization of \(\rho\) we use a combination of two well-known costs, namely distance-stress and a local-topology preserving strain. TPDS chooses the value of spread factor \(\rho\) which minimizes the cost function, which is defined as  
\[CF=C+\lambda E \tag{5}\]  
where \(\lambda\) is the Lagrange multiplier, \(C\) is distance-based stress which measures the preservation of global distances pattern, and \(E\) is a local-topology preserving constraint. TPDS uses a default value of \(\lambda=1\); however, a different value of \(\lambda\) could also be used. Please note that we do not differentiate the cost function mentioned in Equation (5) to learn \(\rho\) but perform a grid search with a fixed step size to find the value of \(\rho\) at which \(CF\) is lowest. The grid search is done through use of a multiplying factor multiplied to standard deviation of distance matrix as explained in Algorithm-1. It is just meant to choose the correct scaled distance matrix to be provided to the non-metric MDS method for dimension reduction. Thus, TPDS also provides flexibility in choosing the level of preservation of local topology through \(\lambda\). Distance-based stress is the cost function used by MDS, which can be defined as the sum of differences among distances in high dimension and reduced dimension  
\[C_{ij}=\sum_{i\
eq j}[d_{ij}-d^{\prime}_{ij}]^{2} \tag{6}\]  
Here \(d_{ij}\) is the distance between two data-points \(x_{i}\) and \(x_{j}\) in higher dimension space and \(d^{\prime}_{ij}\) is distance in reduced dimension. The cost represented in Equation (5) is often called at distance stress, which we term here as MDS-cost. The strain of local topology preservation could be formulated using different approaches based on hard KNN or smooth neighborhood preservation [12; 13]. To test our model, we have used the cost function associated with symmetric SNE. As proposed by [PERSON] et al. [2] symmetric-SNE is a modification of SNE-based approach. The symmetric-SNE cost function consists of single Kullback-Leibler divergence between a joint probability distribution, P, in the high-dimensional space and a joint probability distribution, Q, in the low-dimensional space  
\[E=KL(P|Q)=\sum_{i}\sum_{j}p_{ij}log\frac{p_{ij}}{q_{ij}} \tag{7}\]  
where \(p_{ii}\) and \(q_{ii}\) is set to zero. Here it is called symmetric SNE because \(p_{ij}=p_{ji}\) and \(q_{ij}=q_{ji}\) as these pair wise similarities are defined as  
\[p_{j|i}=\frac{exp(-\|x_{i}-x_{j}\|^{2}/2\sigma_{i}^{2})}{\sum_{k\
eq i}exp(- \|x_{i}-x_{k}\|^{2}/2\sigma_{i}^{2})} \tag{8}\]  
and  
\[p_{ij}=(p_{j|i}+p_{i|j})/2n \tag{9}\]  
in a higher dimension. Here \(n\) is the number of data-points. We keep the value of \(\sigma_{i}\) at perplexity = 15 [2]. In the lower dimension, the likelihood of distances is determined as  
\[q_{ij}=\frac{exp(-\|y_{i}-y_{j}\|^{2}}{\sum_{k\
eq i}exp(-\|y_{i}-y_{k}\|^{2})} \tag{10}\]  
Here \(y_{i}\) and \(y_{j}\) represent coordinates of data-points in lower dimension.