Related papers: On cycles and merge trees
Metrics for merge trees that are simultaneously stable, informative, and efficiently computable have so far eluded researchers. We show in this work that it is possible to devise such a metric when restricting merge trees to ordered domains…
Dynamic programming is a powerful technique that is, unfortunately, often inherently sequential. That is, there exists no unified method to parallelize algorithms that use dynamic programming. In this paper, we attempt to address this issue…
An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomised algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs…
This paper introduces decorated merge trees (DMTs) as a novel invariant for persistent spaces. DMTs combine both $\pi_0$ and $H_n$ information into a single data structure that distinguishes filtrations that merge trees and persistent…
We prove a new formula for the generating function of multitype Cayley trees counted according to their degree distribution. Using this formula we recover and extend several enumerative results about trees. In particular, we extend some…
Consider the generalized iterated wreath product $\mathbb{Z}_{r_1}\wr \mathbb{Z}_{r_2}\wr \ldots \wr \mathbb{Z}_{r_k}$ where $r_i \in \mathbb{N}$. We prove that the irreducible representations for this class of groups are indexed by a…
In this paper, we revisit implicit regularization from the ground up using notions from dynamical systems and invariant subspaces of Morse functions. The key contributions are a new criterion for implicit regularization---a leading…
We address the problem of building and maintaining distributed spanning trees in highly dynamic networks, in which topological events can occur at any time and any rate, and no stable periods can be assumed. In these harsh environments, we…
Common meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. The inverse of zero is an error term…
We introduce an algorithm that performs a one-directional mesh overset of a parallel forest of octrees with another distributed mesh of unrelated partition. The forest mesh consists of several adaptively refined octrees. Individual smooth…
In scientific visualization, scalar fields are often compared through edit distances between their merge trees. Typical tasks include ensemble analysis, feature tracking and symmetry or periodicity detection. Tree edit distances represent…
We investigate properties of the set of discrete Morse functions on a simplicial complex as defined by Forman. It is not difficult to see that the pairings of discrete Morse functions of a finite simplicial complex again form a simplicial…
If a graph has a non-singular adjacency matrix, then one may use the inverse matrix to define a (labeled) graph that may be considered to be the inverse graph to the original one. It has been known that an adjacency matrix of a tree is…
The hierarchical and recursive expressive capability of rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. On the other hand, such hierarchical…
In this article, we present a binary tree with vertices given by rational functions $p(x)/q(x)$; the root and functional derivation of children are inspired by continued fractions. We prove some special properties of the tree. For example,…
A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general…
Let $F$ be a discrete Morse function on a simplicial complex $L$. We construct a discrete Morse function $\Delta(F)$ on the barycentric subdivision $\Delta(L)$. The constructed function $\Delta(F)$ "behaves the same way" as $F$, i. e. has…
We introduce generalizations of Aldous' Brownian Continuous Random Tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a k-th root…
Random forests on the one hand, and neural networks on the other hand, have met great success in the machine learning community for their predictive performance. Combinations of both have been proposed in the literature, notably leading to…
We present a way to normalize a combinatorial Morse function into an integer-valued canonical representative of the set of discrete Morse functions inducing a given gradient field.