Simplicial-Haudorff-Distance

Many practical applications in topological data analysis arise from data in the form of point clouds, which then yield simplicial complexes. The combinatorial structure of simplicial complexes captures the topological relationships between the elements of the complex. In addition to combinatorial structure, simplicial complexes possess a geometric realization, providing a concrete way to visualize the complex and understand its geometric properties. As such, comparing these simplicial complexes entails a notion of nearness between the vertices, and hence these vertices need to be considered ambient in some metric space. A simplicial complex may be d-dimensional, but its geometric realization may not “fit” into Euclidean space of a particular fixed dimension. This project proposes an amended Hausdorff distance measure for simplicial complexes that integrates geometric proximity with topological features. The distance measures proximity between abstract simplicial complexes whose respective vertex sets have an embddeding in some fixed Euclidean space.

This code was built to be compatible with the R TDA ripsFiltration routine. Consequently, the "Hausdorff distnance.R" carries the scd function which takes two R lists as input arguments and returns the simplicial Hausdorff distance between them. Each entry list represents a (geometric) simplicial complex with a filtration/ordering of its simplices. This list is at the least, a list of two data structures: the first "cmplx" is a list of integer-valued vectors of varying lengths, where each vector of size $k+1$ is a $k$-simplex and the $k$ integers are the respective indices of its vertices. The $j^{th}$ vector is the $j^{th}$ simplex in the fitration. The second "coordinates" is an $N\times d$ array of the $d$-dimensional Euclidean coordinates of each vertex, where $N$ is the size of the vertex set of this simplicial complex. The $j^{th}$ row contains the coordinates of the $j^{th}$ vertex.An (optional) third entry of the input list is a Boolean variable "increasing" whose default value is "TRUE", indicating that the filtration values are in ascending order. A fourth argument is a vector "values" of the same length as "cmplx", whose entries are the respective filtration values.

The "Reps to complex. R" script carries the reps_to_complex function, which transforms the $3D$ array for the location of a homology cycle, as returned by the R TDA ripsDiag routine, into the simplicial complex list format described in the previous paragraph. This can then serve as input to the scd function to compute the simplicial Hausdorff distance. The "Test sq_dist.R" script provides a small test example for the scd function.

References

• Brittany T. Fasy, Jisu Kim, Fabrizio Lecci, Clement Maria, David L. Millman, and Vincent Rouvreau. TDA: Statistical Tools for Topological Data Analysis. R package version 1.9. 2023
• Nnadi N. and Isaksen I.Simplicial Hausdorff Distance for Topological Data Analysis. Submitted