Relative persistent homology¶

For a pair of filtrations \((K, L)\) where \(L\) is a subcomplex of \(K\), the relative persistence diagram captures the cells of \(K\) that are not in \(L\) – equivalently, the homology of the quotient \(K / L\).

API¶

import oineus as oin

# fil_K = the big filtration; fil_L = the subcomplex you want to quotient by
dgms_rel = oin.compute_relative_diagrams(fil_K, fil_L, include_inf_points=True)
print(dgms_rel.in_dimension(0))
print(dgms_rel.in_dimension(1))

compute_relative_diagrams does the bookkeeping internally: it walks the two filtrations, identifies which cells of \(K\) are missing from \(L\), and returns the relative diagrams as a oineus.Diagrams object indexed by homology dimension.

When you have two unrelated filtrations on the same vertex set¶

If your inputs are two independent filtrations \(f_1, f_2 \colon K \to \mathbb{R}\) on the same complex, build their pointwise minimum first:

fil_min = oin.min_filtration(fil_1, fil_2)
dgms = oin.compute_relative_diagrams(fil_1, fil_min)

min_filtration assigns each cell the smaller of its two values. The relative diagram of \((f_1, \min(f_1, f_2))\) then captures the cells where \(f_2\) is smaller – a common workhorse in interleaving constructions.

Relative diagrams vs. KICR¶

Both relative persistence and Kernel, image, cokernel persistence deal with pairs of filtrations, but they answer different questions:

Relative diagrams compute the persistence module of \(H_*(K, L)\) – the homology of the quotient.
KICR computes the kernel, image, and cokernel persistence modules of the map \(H_*(L) \to H_*(K)\) – three different modules tied to the long exact sequence of the pair.

A short exact sequence of complexes induces a long exact sequence of persistence modules; relative diagrams give you one piece, KICR gives you three. Pick the one that names the algebraic object you actually want.

Relative persistent homology¶

API¶

When you have two unrelated filtrations on the same vertex set¶

Relative diagrams vs. KICR¶

See also¶