Related papers: On cycles and merge trees

In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general…

Forman introduced discrete Morse theory as a tool for studying CW complexes by essentially collapsing them onto smaller, simpler-to-understand complexes of critical cells in [Fo]. Chari reformulated discrete Morse theory for regular cell…

This paper addresses the problem of approximate MAP-MRF inference in general graphical models. Following [36], we consider a family of linear programming relaxations of the problem where each relaxation is specified by a set of nested pairs…

We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give…

Merge trees are a type of topological descriptors that record the connectivity among the sublevel sets of scalar fields. They are among the most widely used topological tools in visualization. In this paper, we are interested in sketching a…

We study some sequences of polynomials that appear when we consider the successive derivatives of the tree function (or Lambert's W function). We show in particular that they are related with a generalization of Cayley trees, called Greg…

Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two…

Weighted model integration (WMI) extends Weighted model counting (WMC) to the integration of functions over mixed discrete-continuous domains. It has shown tremendous promise for solving inference problems in graphical models and…

We prove that a connected simplicial complex is uniquely determined by its complex of discrete Morse functions. This settles a question raised by Chari and Joswig. In the 1-dimensional case, this implies that the complex of rooted forests…

Contour trees have been developed to visualize or encode scalar data in imaging technologies and scientific simulations. Contours are defined on a continuous scalar field. For discrete data, a continuous function is first interpolated,…

We build on recent work of Yeats, Courtiel, and others involving connected chord diagrams. We first derive from a Hopf-algebraic foundation a class of tree-like functional equations and prove that they are solved by weighted generating…

Merge trees are a common topological descriptor for data with a hierarchical component, such as terrains and scalar fields. The interleaving distance, in turn, is a common distance for comparing merge trees. However, the interleaving…

Normalized random measures (NRMs) provide a broad class of discrete random measures that are often used as priors for Bayesian nonparametric models. Dirichlet process is a well-known example of NRMs. Most of posterior inference methods for…

In this paper we offer a new perspective on the well established agglomerative clustering algorithm, focusing on recovery of hierarchical structure. We recommend a simple variant of the standard algorithm, in which clusters are merged by…

Metric trees and metric tree arrangements index cones in the polyhedral fan structure in the Dressian $Dr(2,n)$ and $Dr(3,n)$ respectively. We introduce the notion of generalized metric tree arrangements which parameterize points in…

Merkle trees have become a widely successful cryptographic data structure. Enabling a vast variety of applications from checking for inconsistencies in databases like Dynamo to essential tools like Git to large scale distributed systems…

Reverse search is a convenient method for enumerating structured objects, that can be used both to address theoretical issues and to solve data mining problems. This method has already been successfully developed to handle unordered trees.…

Dynamic trees are mixtures of tree structured belief networks. They solve some of the problems of fixed tree networks at the cost of making exact inference intractable. For this reason approximate methods such as sampling or mean field…

Divided symmetrization of a function $f(x_1,\dots,x_n)$ is symmetrization of the ratio $$DS_G(f)=\frac{f(x_1,\dots,x_n)}{\prod (x_i-x_j)},$$ where the product is taken over the set of edges of some graph $G$. We concentrate on the case when…

We provide a novel framework to compute a discrete vector potential of a given discrete vector field on arbitrary polyhedral meshes. The framework exploits the concept of acyclic matching, a combinatorial tool at the core of discrete Morse…