1,506 research outputs found
On Rectangle-Decomposable 2-Parameter Persistence Modules
This paper addresses two questions: (1) can we identify a sensible class of 2-parameter persistence modules on which the rank invariant is complete? (2) can we determine efficiently whether a given 2-parameter persistence module belongs to this class? We provide positive answers to both questions, and our class of interest is that of rectangle-decomposable modules. Our contributions include: (a) a proof that the rank invariant is complete on rectangle-decomposable modules, together with an inclusion-exclusion formula for counting the multiplicities of the summands; (b) algorithms to check whether a module induced in homology by a bifiltration is rectangle-decomposable, and to decompose it in the affirmative, with a better complexity than state-of-the-art decomposition methods for general 2-parameter persistence modules. Our algorithms are backed up by a new structure theorem, whereby a 2-parameter persistence module is rectangle-decomposable if, and only if, its restrictions to squares are. This local condition is key to the efficiency of our algorithms, and it generalizes previous conditions from the class of block-decomposable modules to the larger one of rectangle-decomposable modules. It also admits an algebraic formulation that turns out to be a weaker version of the one for block-decomposability. Our analysis focuses on the case of modules indexed over finite grids, the more general cases are left as future work
Computing the interleaving distance is NP-hard
We show that computing the interleaving distance between two multi-graded
persistence modules is NP-hard. More precisely, we show that deciding whether
two modules are -interleaved is NP-complete, already for bigraded, interval
decomposable modules. Our proof is based on previous work showing that a
constrained matrix invertibility problem can be reduced to the interleaving
distance computation of a special type of persistence modules. We show that
this matrix invertibility problem is NP-complete. We also give a slight
improvement of the above reduction, showing that also the approximation of the
interleaving distance is NP-hard for any approximation factor smaller than .
Additionally, we obtain corresponding hardness results for the case that the
modules are indecomposable, and in the setting of one-sided stability.
Furthermore, we show that checking for injections (resp. surjections) between
persistence modules is NP-hard. In conjunction with earlier results from
computational algebra this gives a complete characterization of the
computational complexity of one-sided stability. Lastly, we show that it is in
general NP-hard to approximate distances induced by noise systems within a
factor of 2.Comment: 25 pages. Several expository improvements and minor corrections. Also
added a section on noise system
LIPIcs
Given a locally finite X ⊆ ℝd and a radius r ≥ 0, the k-fold cover of X and r consists of all points in ℝd that have k or more points of X within distance r. We consider two filtrations - one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k - and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in ℝd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module from Delaunay mosaics that is isomorphic to the persistence module of the multi-covers
Rigidity and flexibility of biological networks
The network approach became a widely used tool to understand the behaviour of
complex systems in the last decade. We start from a short description of
structural rigidity theory. A detailed account on the combinatorial rigidity
analysis of protein structures, as well as local flexibility measures of
proteins and their applications in explaining allostery and thermostability is
given. We also briefly discuss the network aspects of cytoskeletal tensegrity.
Finally, we show the importance of the balance between functional flexibility
and rigidity in protein-protein interaction, metabolic, gene regulatory and
neuronal networks. Our summary raises the possibility that the concepts of
flexibility and rigidity can be generalized to all networks.Comment: 21 pages, 4 figures, 1 tabl
Generic Two-Parameter Persistence Modules are Nearly Indecomposable
A fundamental property of one-parameter persistence modules is that the
supports of their indecomposable summands form a stable descriptor of the
module. This is far from true of two-parameter persistence modules: we show
that, in the interleaving distance, any finitely presentable two-parameter
persistence module can be approximated arbitrarily well by an indecomposable,
and that, generically, two-parameter persistence modules are nearly
indecomposable, in the following sense. For every there
exists a dense and open set consisting of modules that decompose as a direct
sum of an indecomposable and an -trivial module. These results
provide further motivation for approaches to multi-parameter persistence that
do not rely on decomposing modules by arbitrary indecomposables.Comment: 17 pages, 2 figure
Towards Stratification Learning through Homology Inference
A topological approach to stratification learning is developed for point
cloud data drawn from a stratified space. Given such data, our objective is to
infer which points belong to the same strata. First we define a multi-scale
notion of a stratified space, giving a stratification for each radius level. We
then use methods derived from kernel and cokernel persistent homology to
cluster the data points into different strata, and we prove a result which
guarantees the correctness of our clustering, given certain topological
conditions; some geometric intuition for these topological conditions is also
provided. Our correctness result is then given a probabilistic flavor: we give
bounds on the minimum number of sample points required to infer, with
probability, which points belong to the same strata. Finally, we give an
explicit algorithm for the clustering, prove its correctness, and apply it to
some simulated data.Comment: 48 page
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