undo quadratic transform after using alpha filtration?

[corybrunson]

Hello! I use this package frequently, including as a dependency for others, and am very grateful for its maintenance.

Something pointed out to me years ago, and still noted on this worksheet by a colleague is that the alpha filtration–based engine alphaComplexDiag() returns birth–death pairs that are squared from their associated radii. I understand that the squaring might done internally for efficiency, as in GUDHI. But it seems like the output should be in terms of the original distance. (Please see the example below, adjusted for the squaring but also to convert radii to ball diameters so that the outputs match.)

Thanks for any explanation or action! I'd be happy to submit a pull request if helpful.

library(TDA)
# regular pentagon
t <- seq(0, 4) * pi*2/5
x <- cbind(cos(t), sin(t))
# PH via Vietoris-Rips filtration
ph_r <- ripsDiag(x, maxdimension = 2, maxscale = 2)
# PH via alpha filtration
ph_a <- alphaComplexDiag(x, maxdimension = 1)
# Rips output in radii
ph_r$diagram[, seq(2,3)] <- ph_r$diagram[, seq(2,3)] / 2
print(ph_r)
#> $diagram
#>      dimension     Birth     Death
#> [1,]         0 0.0000000 1.0000000
#> [2,]         0 0.0000000 0.5877853
#> [3,]         0 0.0000000 0.5877853
#> [4,]         0 0.0000000 0.5877853
#> [5,]         0 0.0000000 0.5877853
#> [6,]         1 0.5877853 0.9510565
# raw alpha output
print(ph_a)
#> $diagram
#>      dimension     Birth     Death
#> [1,]         0 0.0000000       Inf
#> [2,]         0 0.0000000 0.3454915
#> [3,]         0 0.0000000 0.3454915
#> [4,]         0 0.0000000 0.3454915
#> [5,]         0 0.0000000 0.3454915
#> [6,]         1 0.3454915 1.0000000
# alpha output transformed to radii
ph_a$diagram[, seq(2,3)] <- sqrt( ph_a$diagram[, seq(2,3)] )
print(ph_a)
#> $diagram
#>      dimension     Birth     Death
#> [1,]         0 0.0000000       Inf
#> [2,]         0 0.0000000 0.5877853
#> [3,]         0 0.0000000 0.5877853
#> [4,]         0 0.0000000 0.5877853
#> [5,]         0 0.0000000 0.5877853
#> [6,]         1 0.5877853 1.0000000

^{Created on 2025-01-27 with reprex v2.1.1}