Persistence Landscape — persistence

Computes the persistence landscape (Bubenik 2015) from a persistence diagram. Each (birth, death) pair contributes a tent function $$\Lambda_{(b,d)}(t) = \max(0, \min(t - b, d - t)).$$ The $k$-th landscape function $\lambda^{(k)}(t)$ is the $k$-th largest of $\{\Lambda_{(b_i,d_i)}(t)\}_i$ at each $t$. Landscapes are stable under bottleneck distance and form a Banach-space embedding of persistence diagrams.

Usage

persistence_landscape(ph, k_max = 5L, dimension = 1L, t_grid = NULL)

Arguments

ph: A persistent_homology object or a data.frame with columns dimension, birth, death.
k_max: Maximum landscape index to compute (default 5). Must be a single positive integer.
dimension: Integer scalar – which homology dimension to compute the landscape for. Default 1.
t_grid: Numeric vector of evaluation points. NULL (default) uses an even grid of 200 points covering the union of pair intervals.

Value

A persistence_landscape object with:

landscape: Data frame: k, t, value.
dimension: Integer scalar.
k_max: Integer scalar.
t_grid: Numeric vector.

References

Bubenik, P. (2015). Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research 16, 77-102.

Examples

mat <- matrix(c(0, .6, .5, .6, 0, .4, .5, .4, 0), 3, 3)
rownames(mat) <- colnames(mat) <- c("A","B","C")
ph <- persistent_homology(mat, n_steps = 5)
pl <- persistence_landscape(ph, k_max = 3, dimension = 0)