TDApplied (version 3.0.3)

diagram_kernel: Calculate persistence Fisher kernel value between a pair of persistence diagrams.

Description

Returns the persistence Fisher kernel value between a pair of persistence diagrams in a particular homological dimension, each of which is either the output from a diagram_to_df function call or from a persistent homology calculation like ripsDiag/calculate_homology/PyH.

Usage

diagram_kernel(D1, D2, dim = 0, sigma = 1, t = 1, rho = NULL)

Value

the numeric kernel value.

Arguments

D1

the first persistence diagram.

D2

the second persistence diagram.

dim

the non-negative integer homological dimension in which the distance is to be computed, default 0.

sigma

a positive number representing the bandwidth for the Fisher information metric, default 1.

t

a positive number representing the scale for the persistence Fisher kernel, default 1.

rho

an optional positive number representing the heuristic for Fisher information metric approximation, see diagram_distance. Default NULL.

Author

Shael Brown -

Details

The persistence Fisher kernel is calculated from the Fisher information metric according to the formula \(k_{PF}(D_1,D_2) = exp(-t*d_{FIM}(D_1,D_2))\), resembling a radial basis kernel for standard Euclidean spaces.

References

Le T, Yamada M (2018). "Persistence fisher kernel: a riemannian manifold kernel for persistence diagrams." https://proceedings.neurips.cc/paper/2018/file/959ab9a0695c467e7caf75431a872e5c-Paper.pdf.

Murphy, K. "Machine learning: a probabilistic perspective", MIT press (2012).

See Also

gram_matrix for Gram (i.e. kernel) matrix calculations.

Examples

if(require("TDAstats"))
{
  # create two diagrams
  D1 <- TDAstats::calculate_homology(TDAstats::circle2d[sample(1:100,20),],
                      dim = 1,threshold = 2)
  D2 <- TDAstats::calculate_homology(TDAstats::circle2d[sample(1:100,20),],
                      dim = 1,threshold = 2)

  # calculate the kernel value between D1 and D2 with sigma = 2, t = 2 in dimension 1
  diagram_kernel(D1,D2,dim = 1,sigma = 2,t = 2)
  # calculate the kernel value between D1 and D2 with sigma = 2, t = 2 in dimension 0
  diagram_kernel(D1,D2,dim = 0,sigma = 2,t = 2)
}