The Rips filtration over a finite metric space and its corresponding persistent
homology are prominent methods in topological data analysis to summarise
the “shape” of data. Crucial to their use is the stability result that says if
and
are finite metric spaces then the (bottleneck) distance between the
persistence diagrams constructed via the Rips filtration is bounded by
(where
is the
Gromov–Hausdorff distance). A generalisation of the Rips filtration to any
symmetric
function
was defined by Chazal, de Silva and Oudot (Geom. Dedicata 173 (2014) 193–214), where
they showed it was stable with respect to the correspondence distortion distance.
Allowing asymmetry, we consider four different persistence modules, definable for pairs
where
is any
real valued function. These generalise the persistent homology of the symmetric Rips
filtration in different ways. The first method is through symmetrisation. For each
we can construct a
symmetric function
.
We can then apply the standard theory for symmetric functions and get
stability as a corollary. The second method is to construct a filtration
of ordered tuple
complexes where
if
for all
. Both our
first two methods have the same persistent homology as the standard Rips filtration when
applied to a metric space, or more generally to a symmetric function. We then consider
two constructions using an associated filtration of directed graphs or preorders. For each
we can define a
directed graph
where directed edges
are included in
whenever
(note
this is when
for
a quasimetric). From this we construct a preorder where
if there is a
path from
to
in
. We
build persistence modules using the strongly connected components of the graphs
, which
are also the equivalence classes of the associated preorders. We also consider
persistence modules using a generalisation of poset topology to preorders.
The Gromov–Hausdorff distance, when expressed via correspondence distortions,
can be naturally extended as a correspondence distortion distance to set–function pairs
. We
prove that all these new constructions enjoy the same stability as persistence modules
built via the original persistent homology for symmetric functions.
