Related papers: The Directed Spanning Forest in the Hyperbolic spa…
It is known that any $n$-point set in the $d$-dimensional Euclidean space $\mathbb{R}^d$, for $d = O(1)$, admits: 1) a $(1+\epsilon)$-spanner with maximum degree $\tilde{O}(\epsilon^{-d+1})$ and with lightness $\tilde{O}(\epsilon^{-d})$; 2)…
Decision trees (DTs) and their random forest (RF) extensions are workhorses of classification and regression in Euclidean spaces. However, algorithms for learning in non-Euclidean spaces are still limited. We extend DT and RF algorithms to…
6-dimensional superconformal field theories are exotic and fascinating. They emerge from compactifications of F-theory on Calabi-Yau elliptic fibrations, which grants them a rich array of dualities with various other formulations of string…
We prove the rather counterintuitive result that there exist finite transitive graphs H and integers k such that the Free Uniform Spanning Forest in the direct product of the k-regular tree and H has infinitely many trees almost surely.…
We continue the study of the geometry of infinite geodesics in first passage percolation (FPP) on Gromov-hyperbolic groups G, initiated by Benjamini-Tessera and developed further by the authors. It was shown earlier by the authors that,…
New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in…
In proper hyperbolic geodetic spaces we construct rooted $\mathbb R$-trees with the following properties. On the one hand, every ray starting at the root is quasi-geodetic; so these $\mathbb R$-trees represent the space itself well. At the…
This paper starts with an observation that two infinite series of simplicial complexes, which a priori do not seem to have anything to do with each other, have the same homotopy type. One series consists of the complexes of directed forests…
We study the topological structure of random geometric forests $G$ in the Euclidean plane under mild assumptions: non-crossing edges, stationarity, and finite edge intensity. The framework covers a broad range of constructions, including…
Learning representations according to the underlying geometry is of vital importance for non-Euclidean data. Studies have revealed that the hyperbolic space can effectively embed hierarchical or tree-like data. In particular, the few past…
The weighted shifts are long known and important class of operators. One of known generalisation of this class are weighted shifts on directed trees, where we replace the linear order of coordinates in $\ell^2$ with a possibly more…
Despite the latest prevailing success of deep neural networks (DNNs), several concerns have been raised against their usage, including the lack of intepretability the gap between DNNs and other well-established machine learning models, and…
We study the structure of infinite geodesics in the Planar Stochastic Hyperbolic Triangulations $\mathbb{T}_{\lambda}$, which are the hyperbolic analogs of the UIPT. We prove that these geodesics form a supercritical Galton--Watson tree…
In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy…
The seminal papers of Edmonds \cite{Egy}, Nash-Williams \cite{NW} and Tutte \cite{Tu} have laid the foundations of the theories of packing arborescences and packing trees. The directed version has been extensively investigated, resulting in…
Within the Kardar-Parisi-Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work arXiv:1812.00309 of Dauvergne, Ortmann, and Vir\'ag, this…
Finding meaningful distances between high-dimensional data samples is an important scientific task. To this end, we propose a new tree-Wasserstein distance (TWD) for high-dimensional data with two key aspects. First, our TWD is specifically…
We characterise digraphs of directed treewidth one in terms of forbidden butterfly minors. Moreover, we show that there is a linear relation between the hypertree-width of the dual of the cycle hypergraph of D, i. e. the hypergraph with…
Hierarchical data is common in many domains like life sciences and e-commerce, and its embeddings often play a critical role. While hyperbolic embeddings offer a theoretically grounded approach to representing hierarchies in low-dimensional…
Random forest is widely exploited as an ensemble learning method. In many practical applications, however, there is still a significant challenge to learn from imbalanced data. To alleviate this limitation, we propose a deep dynamic boosted…