Algorithm 1 Base graph optimization for a fixed structure of the elastic graph

(1)Initialize the graph G, its elastic matrix EM(G) and the map φ (2)Compute the matrix $\tilde{L}\left(EM\left(G\right)\right)$ (3)Partition the data by proximity to the embedded nodes of the graph (i.e., compute the mapping K:{X} $\to${V} of a data point i to a graph node j) (4) Solve the following system of linear equations to determine the new map φ $${{\displaystyle \sum }}_{j=1}^{\left|V\right|}\left(\frac{{{\displaystyle \sum }}_{\left\{K\left(i\right)=j\right\}}{w}_i}{{{\displaystyle \sum }}_{i=1}^{\left|V\right|}{w}_i}{\delta }_{ij}+\tilde{L}{\left(EM\left(G\right)\right)}_{ij}\right)\varphi \left(V_j\right)=\frac{1}{{{\displaystyle \sum }}_{i=1}^{\left|V\right|}{w}_i}{{\displaystyle \sum }}_{K\left(i\right)=j}^{}{w}_i{X}_i,$$ where ${\delta }_{ij}$ is the Kronecker delta. Iterate 3–4 till the map φ does not change more than ε in some appropriate measure of difference between two consecutive iterations.