arXiv research

Study on embedding tree products into groups, distinguishing them.

Paper presents a new method for learning hyperbolic representations using tree structures.

New method embeds phylogenetic trees for clustering, recovering evolutionary relationships.

The Farey tree helps embed rational balls and lens spaces into complex projective space.

Spaces of circle embeddings in curved surfaces indexed by trees.

DTE uses tree leaf means to embed data, balancing accuracy and speed.

We show that for each n\ge 2 there is a quasi-isometric embedding of the hyperbolic space H^n in the product T^n=Tx...xT of n copies of a (simplicial) metric tree T. On the other hand, we prove that there is no quasi-isometric embedding H^2 --> TxR^m for any metric tree T and any m\ge 0.

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 demonstrated that metric learning approaches can also be applied to trees, such as m…

We prove the equivalence between a relative bottleneck property and being quasi-isometric to a tree-graded space. As a consequence, we deduce that the quasi-trees of spaces defined axiomatically by Bestvina-Bromberg-Fujiwara are quasi-isometric to tree-graded spaces. Using this we prove that mapping class groups quasi-…

We prove that every visual Gromov hyperbolic space X whose boundary at infinity has the finite capacity dimension n admits a quasi-isometric embedding into (n+1)-fold product of metric trees.

A new method for hierarchical clustering using continuous embeddings and optimization.

Proposes a hierarchical clustering method for positive and negative dissimilarities.

Maximal representations are studied using tree embeddings and geodesic currents.

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 demonstrated that metric learning approaches can also be applied to trees, such as m…

We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of the trees, showing that they are convex-cocompact and asymptotically isometric.…

Study on inflection points of plane curve shadows with fixed embedded shapes.

Sparse oblique decision tree improves security rules for renewable power systems.

Proposes a deep tree-ensemble model for multi-output prediction.

A new method, tree-SNE, solves the scale problem in t-SNE.

Paper develops a new method for game options in local volatility models.

The paper explores metrics on tree moduli spaces and a new topological group.

Study classifies Halin graphs with positive curvature.

Infinite fractal tree solves shortest connection problem.

This paper approximates 1-Wasserstein distance using tree-based embedding.

Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. The Bollobás-Riordan-Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chor…

Tree complex linked to polyhedral shapes like associahedra and cyclohedra.

We tackle the problem of building explainable recommendation systems that are based on a per-user decision tree, with decision rules that are based on single attribute values. We build the trees by applying learned regression functions to obtain the decision rules as well as the values at the leaf nodes. The regression…

A new tree-Wasserstein distance for high-dimensional data with latent feature hierarchy.

We prove that the universal cover of any graph manifold quasi-isometrically embeds into a product of three trees. In particular we show that the Assouad-Nagata dimension of the universal cover of any closed graph manifold is 3, proving a conjecture of Smirnov.

A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary (relatively) hyperbolic groups, rank 1 CAT(0) groups, mapping class groups and Out(Fn). As an application, we show that mapping clas…

New groups constructed from tree automorphisms, proving finiteness.

Study of discrete period matrices on embedded graphs, relating to Riemann surfaces.

We prove that an arbitrary right-angled Artin group $G$ admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree. Consequently, $G$ admits quasi-isometric group embeddings into a pure braid group and into the area-preserving diffeomorphism groups of the 2--disk an…

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relativ…

Protein Thoughts interprets protein interactions with clear reasoning, improving prediction accuracy.

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asy…

We present some explicit constructions of universal R-trees with applications to the asymptotic geometry of hyperbolic spaces. In particular, we show that any asymptotic cone of a complete simply connected manifold of negative curvature is a complete homogeneous R-tree with the valency $2^{\aleph_0}$ at every point. It…

A result about spanning forests for graphs yields a short proof of Krebes's theorem concerning embedded tangles in links.

We show that a relatively hyperbolic group quasi-isometrically embeds in a product of finitely many trees if the peripheral subgroups do, and we provide an estimate on the minimal number of trees needed. Applying our result to the case of 3-manifolds, we show that fundamental groups of closed 3-manifolds have linearly …

Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinat…

Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram $L$, there is an associated ribbon graph whose quasi-trees correspond bijectively to …

TreeDOX predicts chaotic systems without hyperparameter tuning.

Any-gram kernels are a flexible and efficient way to employ bag-of-n-gram features when learning from textual data. They are also compatible with the use of word embeddings so that word similarities can be accounted for. While the original any-gram kernels are implemented on top of tree kernels, we propose a new approa…

Asymptotic subcone of an unbounded metric space is another metric space, capturing the structure of the original space at infinity. In this paper we define a functional metric space S which is an asymptotic subcone of the hyperbolic plane. This space is a real tree branching at every its point. Moreover, it is a homoge…

Gradient boosted decision trees are a popular machine learning technique, in part because of their ability to give good accuracy with small models. We describe two extensions to the standard tree boosting algorithm designed to increase this advantage. The first improvement extends the boosting formalism from scalar-val…

Small neural networks embed arbitrary metric spaces into Gaussian mixtures.

In every dimension $n\ge 3$ we introduce a class of orthogonal graph-manifolds and prove that the fundamental group of any orthogonal graph-manifold quasi-isometrically embeds into a product of $n$ trees. As a consequence, we obtain that asymptotic and linearly-controlled asymptotic dimensions of such group are equal t…

Determinants of theta curves and symmetric graphs are studied.