相关论文: Trees and Markov convexity

This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space,…

In this paper, we introduce a magneto-spectral invariant for finite graphs. This invariant vanishes on trees and is maximized by complete graphs. We compute this invariant for cycles, complete graphs, wheel graphs, hypercubes, complete…

We introduce a certain class of 2-type Galton-Watson trees with edge lengths. We prove that, after an adequate rescaling, the weighted height function of a forest of such trees converges in law to the reflected Brownian motion. We then use…

We analyze variational inference for highly symmetric graphical models such as those arising from first-order probabilistic models. We first show that for these graphical models, the tree-reweighted variational objective lends itself to a…

We show that the Kuratowski imbedding of a Riemannian manifold in L^\infty, exploited in Gromov's proof of the systolic inequality for essential manifolds, admits an approximation by a (1+C)-bi-Lipschitz (onto its image), finite-dimensional…

We prove the sufficiency of the Linear Superposition Principle for linear trees, which characterizes the spectra achievable by a real symmetric matrix whose underlying graph is a linear tree. The necessity was previously proven in 2014.…

Let $\mathscr T$ be a rooted directed tree with finite branching index $k_{\mathscr T}$ and let $S_{\lambda} \in B(l^2(V))$ be a left-invertible weighted shift on ${\mathscr T}$. We show that $S_{\lambda}$ can be modelled as a…

Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes's example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a…

Optimal transport provides a metric which quantifies the dissimilarity between probability measures. For measures supported in discrete metric spaces, finding the optimal transport distance has cubic time complexity in the size of the…

The tree-metric theorem provides a necessary and sufficient condition for a dissimilarity matrix to be a tree metric, and has served as the foundation for numerous distance-based reconstruction methods in phylogenetics. Our main result is…

We study the existence of algebras of hypercyclic vectors for weighted backward shifts on sequence spaces of directed trees with the coordinatewise product. When $V$ is a rooted directed tree, we show the set of hypercyclic vectors of any…

Metric learning has the aim to improve classification accuracy by learning a distance measure which brings data points from the same class closer together and pushes data points from different classes further apart. Recent research has…

We offer a new proof of uniform convexity inequalities for the Finsler manifold of nonpositive curvature taken on the space of positive-semidefinite matrices with the weighted matrix geometric mean defining the geodesic between two points.…

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash $C^1$ Embedding Theorem. For more general metric spaces the same…

Building trees to represent or to fit distances is a critical component of phylogenetic analysis, metric embeddings, approximation algorithms, geometric graph neural nets, and the analysis of hierarchical data. Much of the previous…

We introduce a new sphericalization mapping for metric spaces that is applicable in very general situations, including totally disconnected fractal type sets. For an unbounded complete metric space which is uniformly perfect at a base point…

We establish a novel bijective encoding that represents permutations as forests of decorated (or enriched) trees. This allows us to prove local convergence of uniform random permutations from substitution-closed classes satisfying a…

We observe that embeddings into random metrics can be fruitfully used to study the $L_1$-embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new…

We continue to develop an obstruction theory for embedding 2-spheres into 4-manifolds in terms of Whitney towers. The proposed intersection invariants take values in certain graded abelian groups generated by labelled trivalent trees, and…

Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…