19 research outputs found
Computational Complexity of the Interleaving Distance
The interleaving distance is arguably the most prominent distance measure in
topological data analysis. In this paper, we provide bounds on the
computational complexity of determining the interleaving distance in several
settings. We show that the interleaving distance is NP-hard to compute for
persistence modules valued in the category of vector spaces. In the specific
setting of multidimensional persistent homology we show that the problem is at
least as hard as a matrix invertibility problem. Furthermore, this allows us to
conclude that the interleaving distance of interval decomposable modules
depends on the characteristic of the field. Persistence modules valued in the
category of sets are also studied. As a corollary, we obtain that the
isomorphism problem for Reeb graphs is graph isomorphism complete.Comment: Discussion related to the characteristic of the field added. Paper
accepted to the 34th International Symposium on Computational Geometr
Approximating Gromov-Hausdorff Distance in Euclidean Space
The Gromov-Hausdorff distance proves to be a useful distance
measure between shapes. In order to approximate for compact subsets
, we look into its relationship with , the
infimum Hausdorff distance under Euclidean isometries. As already known for
dimension , the cannot be bounded above by a constant
factor times . For , however, we prove that
. We also show that the bound is tight. In
effect, this gives rise to an -time algorithm to approximate
with an approximation factor of
Local Equivalence and Intrinsic Metrics between Reeb Graphs
As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent contributions emphasize this aspect, proposing novel distances such as functional distortion or interleaving that are provably more discriminative than the so-called bottleneck distance, being true metrics whereas the latter is only a pseudo-metric. Their main drawback compared to the bottleneck distance is to be comparatively hard (if at all possible) to evaluate. Here we take the opposite view on the problem and show that the bottleneck distance is in fact good enough locally, in the sense that it is able to discriminate a Reeb graph from any other Reeb graph in a small enough neighborhood, as efficiently as the other metrics do. This suggests considering the intrinsic metrics induced by these distances, which turn out to be all globally equivalent. This novel viewpoint on the study of Reeb graphs has a potential impact on applications, where one may not only be interested in discriminating between data but also in interpolating between them
Coarse and bi-Lipschitz embeddability of subspaces of the Gromov-Hausdorff space into Hilbert spaces
In this paper, we discuss the embeddability of subspaces of the
Gromov-Hausdorff space, which consists of isometry classes of compact metric
spaces endowed with the Gromov-Hausdorff distance, into Hilbert spaces. These
embeddings are particularly valuable for applications to topological data
analysis. We prove that its subspace consisting of metric spaces with at most n
points has asymptotic dimension . Thus, there exists a coarse
embedding of that space into a Hilbert space. On the contrary, if the number of
points is not bounded, then the subspace cannot be coarsely embedded into any
uniformly convex Banach space and so, in particular, into any Hilbert space.
Furthermore, we prove that, even if we restrict to finite metric spaces whose
diameter is bounded by some constant, the subspace still cannot be bi-Lipschitz
embedded into any finite-dimensional Hilbert space. We obtain both
non-embeddability results by finding obstructions to coarse and bi-Lipschitz
embeddings in families of isometry classes of finite subsets of the real line
endowed with the Euclidean-Hausdorff distance