Uniformly branching trees are equivalent to certain metric spaces.
problem Characterizing metric spaces equivalent to uniformly branching trees.
method Proving equivalence between trivalent quasiconformal trees and uniformly branching trees.
result Any two uniformly branching trees are quasisymmetrically equivalent.
Novel algorithm optimizes decision trees for nonlinear metrics.
problem Optimizing decision trees for nonlinear metrics like F1-score.
method Bi-objective optimisation approach to find optimal trees on Pareto frontier.
result The optimal tree for nonlinear metrics lies on the Pareto frontier.
Existence and uniqueness of discrete Einstein metrics on trees proven.
problem Existence and uniqueness of discrete Einstein metrics on trees.
method Using Perron-Frobenius theory and Lin-Lu-Yau Ricci curvature.
result Existence and uniqueness of discrete Einstein metrics on trees established.
Differentiable optimization bridges arbitrary metrics to tree metrics.
problem Designing algorithms to convert arbitrary metrics to tree metrics with guarantees.
method DeltaZero framework, leveraging differentiable Gromov hyperbolicity.
result DeltaZero consistently achieves state-of-the-art distortion on synthetic and real-world datasets.
Paper presents a new method for learning hyperbolic representations using tree structures.
problem Learning faithful low-dimensional hyperbolic embeddings of data.
method Metric-first approach to learn tree structure, then embed into hyperbolic manifold.
result Novel fast algorithm TreeRep learns tree approximating original metric.
New metric learning approach for tree data reduces computation cost.
problem Efficiently computing distances between ordered labeled trees.
method Introduced pq-grams and a differentiable weighted pq-gram distance, combined with LMNN for optimization.
result Significantly reduces computation time for tree classification problems.
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…
The paper explores metrics on tree moduli spaces and a new topological group.
problem Continuous metrics on tree moduli spaces.
method Proposes and analyzes continuous piecewise-smooth metrics.
result Observes a new abelian topological group structure.
Optimal transport for measures on noisy tree metrics is solved with robust approach.
problem Optimal transport problem for measures on noisy tree metrics.
method Max-min robust optimal transport approach considering uncertainty sets of tree metrics.
result Robust optimal transport admits a closed-form expression for fast computation.
A new supervised tree-Wasserstein distance improves document classification.
problem Measuring document similarity efficiently and accurately.
method Rewriting Wasserstein distance on tree metric, using contrastive loss for optimization.
result The Supervised Tree-Wasserstein (STW) distance improves document classification accuracy.
Conditions for reducing quasi-actions to tree actions and group properties.
problem Conditions for reducing quasi-actions to tree actions.
method Reduction to cobounded isometric actions on trees.
result Groups with quasi-orbits quasi-isometric to trees are virtually free.
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…
Study of Ricci flow on trees, focusing on edge weights and curvatures.
problem Understanding the evolution of metrics on trees under Ricci flow.
method Continuous-time Ricci flow based on Lin-Lu-Yau Ollivier Ricci curvature.
result Ricci flow converges to zero curvature on edge weights of positive normalized values in caterpillar trees.
Efficiently computes tree-Wasserstein barycenter for large-scale multilevel clustering and scalable Bayes.
problem Large-scale multilevel clustering and scalable Bayes problems.
method Proposes an efficient algorithm for tree-Wasserstein barycenter and variants.
result Significantly improves efficiency in computation and memory usage for large-scale applications.
Completing segments of a real tree doesn't yield a complete space.
problem Completing segments of a real tree.
method Analyzing the field of real Puiseux series and the tree defined by Brumfiel.
result Completing all segments of the tree does not result in a complete metric space.
Positive-curvature metrics on trees identified for specific configurations.
problem Classifying trees with positive-curvature discrete Einstein metrics.
method Spectral characterization and eigenvalue analysis of the Ricci matrix.
result Positive-curvature metrics found for specific tree configurations.
The paper studies geometric properties of quasi-trees and tree approximations.
problem Geometric properties and tree approximations of quasi-trees.
method Construction of a tree approximating quasi-trees, proving quasi-isometric properties.
result Every quasi-tree is (1,C)-quasi-isometric to a simplicial tree. A new metric for comparing measures on tree systems reduces computational burden.
problem Heavy computation in Optimal Transport problems.
method Introducing tree systems and a novel metric (Tree-Sliced Wasserstein distance on Systems of Lines, TSW-SL).
result TSW-SL performs favorably compared to Sliced Wasserstein and its variants.
It is known that PQ-symmetric maps on the boundary characterize the quasi-isometry type of visual hyperbolic spaces, in particular, of geodesically complete \br-trees. We define a map on pairs of PQ-symmetric ultrametric spaces which characterizes the branching of the space. We also show that, when the ultrametric spac…
Reduces conjecture to tree-based Artin groups.
problem Proving K(π,1)-conjecture for all Artin groups. method Actions on Bestvina complexes of Garside groupoids.
result New classes of Artin groups satisfying the conjecture.
A new method for comparing measures in different spaces using flow alignment.
problem Comparing probability measures in different metric spaces with computational efficiency.
method Flow-based Alignment (\FlowAlign) and Depth-based Alignment (\DepthAlign) using tree structures.
result Flow-based Alignment and Depth-based Alignment are pseudo-distances and scalable for large-scale applications.
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…
Paper develops a new method to analyze 3D tree-like objects.
problem Analyzing complex geometrical and topological variations in 3D tree-like objects.
method Extended SRVF representation and new metric for tree-shaped 3D objects.
result Captures full elasticity and topological variations of branches.
Researchers analyze geodesic complexity in robot paths on tree graphs.
problem Understanding optimal paths for robots on tree graphs.
method Examined geodesic complexity in ordered and unordered configuration spaces of graphs in ℓ1 and ℓ2 metrics, finding explicit geodesics and families. result Geodesic complexity matches topological complexity in all cases studied.
Generalizes Rips' result on hyperbolic spaces to metric spaces, showing collapses for tree metrics.
problem Understanding the contractibility of Vietoris-Rips complexes in metric spaces.
method Extending Rips' result using geodesic defect and apparent pairs gradient.
result Vietoris-Rips complexes collapse to subforests for finite tree metrics.
In this paper we prove that if we consider the standard real metric on simplicial rooted trees then the category Tower-Set of inverse sequences can be described by means of the bounded coarse geometry of the naturally associated trees. Using this we give a geometrical characterization of Mittag-Leffler property in inve…
Proves finite step termination of Kähler-Einstein metric singularity formation.
problem Singularity formation of Kähler-Einstein metrics.
method Finite step termination of bubble trees for singularity formation.
result Finite step termination of Kähler-Einstein metric singularity formation proved in non-collapsing situation.
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.
Counting HCMU sphere components using weighted trees.
problem Counting components of moduli space of HCMU spheres.
method Using weighted plane trees to characterize HCMU spheres with a single integral conical angle, and an explicit counting formula is derived.
result An explicit counting formula for the components of the moduli space of HCMU spheres.
Two trees in the boundary of outer space are said to be \emph{primitive-equivalent} whenever their translation length functions are equal in restriction to the set of primitive elements of FN. We give an explicit description of this equivalence relation, showing in particular that it is nontrivial. This question is …
We show that every inner metric space X is the metric quotient of a complete R-tree via a free isometric action, which we call the covering R-tree of X. The quotient mapping is a weak submetry (hence, open) and light. In the case of compact 1-dimensional geodesic space X, the free isometric action is via a subgroup of …
New relation on paths is not transitive.
problem Extending tree-like property to non-Lipschitz paths.
method Analyzing a fractal construction in the plane.
result The resulting relation is not an equivalence relation.
We present a construction, called the limit of a tree system of spaces (or, less formally, a tree of spaces). The construction is designed to produce compact metric spaces that resemble fractals, out of more regular spaces, such as closed manifolds, compact polyhedra, compact Menger manifolds, etc. Such spaces are pote…
We prove that if X is a complete geodesic metric space with uniformly generated first homology group and f:X→R is metrically proper on the connected components and bornologous, then X is quasi-isometric to a tree. Using this and adapting the definition of hyperbolic approximation we obtain an intrinsic sufficent …
New algorithm solves unbalanced optimal transport on trees in quasi-linear time.
problem Efficiently solving unbalanced optimal transport problems on trees.
method Proposed an algorithm that solves a more general unbalanced optimal transport problem exactly in quasi-linear time on a tree metric.
result Solves unbalanced optimal transport on trees in quasi-linear time (less than one second for a tree with one million nodes).
Proposes a new phylogenetic tree space with biologically principled geometry.
problem Developing a space for statistical analysis of phylogenies with biologically informed assumptions.
method Introduces wald space, a new phylogenetic tree space, and two related geometries based on Fisher information and Gaussian processes.
result Geodesics in wald space are similar to those in the Fisher information geometry, but the two geometries are distinct.
For a finitely generated group G, we introduce an asymmetric pseudometric on projectivized deformation spaces of G-trees, using stretching factors of G-equivariant Lipschitz maps, that generalizes the Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichmüller space. We show that in t…
Optimal transport (\OT) theory defines a powerful set of tools to compare probability distributions. \OT~suffers however from a few drawbacks, computational and statistical, which have encouraged the proposal of several regularized variants of OT in the recent literature, one of the most notable being the \textit{slice…
The paper improves decision tree stability for health care applications.
problem Stability of decision trees in health care applications.
method Introducing a new distance metric to determine tree stability and proposing a novel training methodology.
result On average, a 4.6% decrease in predictive power yields a 38% increase in model stability.
Tree Mover's Distance measures graph attributes and improves GNN performance.
problem Measuring generalization and robustness in graph neural networks.
method Introducing Tree Mover's Distance (TMD) for attributed graphs.
result TMD correlates with GNN performance under distribution shifts.
Explains visual metrics on hyperbolic space boundaries.
problem Understanding the geometry of hyperbolic spaces.
method Construction of visual metrics, quasisymmetries, and invariants.
result Detailed examples and applications of Gromov's round trees.
In this paper, we study the weak compactness of the set of conformal metrics in any Riemann surface without boundary whose Calabi energy and area are uniformly bounded. We prove that for any sequence of such metrics, there alwasy exists a subsequence which converges in H\sp{2,2}_\sb{loc} everywhere except a finite numb…
There is a well-known correspondence between infinite trees and ultrametric spaces which can be interpreted as an equivalence of categories and comes from considering the end space of the tree. In this equivalence, uniformly continuous maps between the end spaces are translated to some classes of coarse maps (or even c…
Geodesic currents on hyperbolic surfaces have dual spaces that are metric trees.
problem Understanding the dual spaces of geodesic currents on hyperbolic surfaces.
method Analyzing the geometric properties of dual spaces, including their hyperbolicity and completeness.
result The dual spaces of geodesic currents are Gromov hyperbolic metric tree-graded spaces.
New method for comparing different mass measures on tree structures using entropy partial transport.
problem Comparing nonnegative measures with different masses on tree structures.
method Entropy Partial Transport (EPT) on extended trees, regularized for fast computation and negative definiteness.
result First closed-form solution for unbalanced OT on tree structures.
Since its inception in the 1980s, ID3 has become one of the most successful and widely used algorithms for learning decision trees. However, its theoretical properties remain poorly understood. In this work, we introduce a novel metric of a decision tree algorithm's performance, called mean iteration statistical consis…
Maximal representations are studied using tree embeddings and geodesic currents.
problem Maximal representations of surface groups in symplectic groups.
method Metric properties, geodesic currents, and tree embeddings.
result Translation length can be computed as intersection with a geodesic current.
Latent tree models are graphical models defined on trees, in which only a subset of variables is observed. They were first discussed by Judea Pearl as tree-decomposable distributions to generalise star-decomposable distributions such as the latent class model. Latent tree models, or their submodels, are widely used in:…