Related papers: Interval Decomposition of Persistence Modules over…

We address the problem of estimating multiple modes of a multivariate density using persistent homology, a central tool in Topological Data Analysis. We introduce a method based on the preliminary estimation of the $H_0$-persistence diagram…

One of the main objectives of topological data analysis is the study of discrete invariants for persistence modules, in particular when dealing with multiparameter persistence modules. In many cases, the invariants studied for these…

Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a…

Let $R=k[x_1,\dots,x_n]$ be a ring of polynomials over a field $k$ of characteristic $p>0$. There is an algorithm due to Lyubeznik for deciding the vanishing of local cohomology modules $H^i_I(R)$ where $I\subset R$ is an ideal. This…

We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of…

Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically -- as long as the algorithm does not…

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 $1$-interleaved is NP-complete, already for bigraded, interval…

Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under…

For many years, image over-segmentation into superpixels has been essential to computer vision pipelines, by creating homogeneous and identifiable regions of similar sizes. Such constrained segmentation problem would require a clear…

Zigzag persistent homology is a powerful generalisation of persistent homology that allows one not only to compute persistence diagrams with less noise and using less memory, but also to use persistence in new fields of application.…

In the theory of persistent homology, a well known duality relates the barcodes of the absolute homology and relative cohomology of a one-parameter simplicial filtration. Motivated by the problem of computing free presentations of the…

Let $K$ be a field and $P=K[x_1,\dots,x_n]$. The technique of elimination by substitution is based on discovering a coherently $Z=(z_1,\dots,z_s)$-separating tuple of polynomials $(f_1,\dots,f_s)$ in an ideal $I$, i.e., on finding…

We show that if a graded submodule of a Noetherian module cannot be written as a proper intersection of graded submodules, then it cannot be written as a proper intersection of submodules at all. More generally, we show that a natural…

The purpose of this research is to describe an efficient iterative method suitable for obtaining high accuracy solutions to high frequency time-harmonic scattering problems. The method allows for both refinement of local polynomial degree…

We prove rigidity type results on the vanishing of stable (co)homology for modules of finite complete intersection dimension, results which generalize and improve upon known results. We also introduce a notion of pre-rigidity, which…

Let $R$ be a finite ring and let $M, N$ be two finite left $R$-modules. We present two distinct deterministic algorithms that decide in polynomial time whether or not $M$ and $N$ are isomorphic, and if they are, exhibit an isomorphism. As…

Let $G$ be a locally semisimple ind-group, $P$ be a parabolic subgroup, and $E$ be a finite-dimensional $P$-module. We show that, under a certain condition on $E$, the nonzero cohomologies of the homogeneous vector bundle…

We study decomposable N^d-indexed persistence modules via higher dimensional partitions. Their barcodes are defined in terms of the extended interior of the corresponding Young diagrams. For two decomposable N^d-indexed persistence modules,…

Primary decomposition is a very important tool of commutative algebra and geometry. In this paper we generalized some of the existing algorithms of primary decomposition developed by Eisenbud et al. (cf. [EHV]) for free modules and also…

The exact computation of the matching distance for multi-parameter persistence modules is an active area of research in computational topology. Achieving an easily obtainable exact computation of this distance would permit multi-parameter…