Hyperbolic Ordinal Embedding

Given ordinal relations such as the object i is more similar to j than k is to l, ordinal embedding is to embed these objects into a low-dimensional space with all ordinal constraints preserved. Although existing approaches have preserved ordinal relations in Euclidean space, whether Euclidean space is compatible with true data structure is largely ignored, although it is essential to effective embedding. Since real data often exhibit hierarchical structure, it is hard for Euclidean space approaches to achieve effective embeddings in low dimensionality, which incurs high computational complexity or overfitting. In this paper we propose a novel hyperbolic ordinal embedding (HOE) method to embed objects in hyperbolic space. Due to the hierarchy-friendly property of hyperbolic space, HOE can effectively capture the hierarchy to achieve embeddings in an extremely low-dimensional space. We have not only theoretically proved the superiority of hyperbolic space and the limitations of Euclidean space for embedding hierarchical data, but also experimentally demonstrated that HOE significantly outperforms Euclidean-based methods

Regular finite decomposition complexity

We introduce the notion of regular finite decomposition complexity of a metric family. This generalizes Gromov's finite asymptotic dimension and is motivated by the concept of finite decomposition complexity (FDC) due to Guentner, Tessera and Yu. Regular finite decomposition complexity implies FDC and has all the permanence properties that are known for FDC, as well as a new one called Finite Quotient Permanence. We show that for a collection containing all metric families with finite asymptotic dimension all other permanence properties follow from Fibering Permanence

A new metric invariant for Banach spaces

We show that if the Szlenk index of a Banach space $X$ is larger than the first infinite ordinal $\omega$ or if the Szlenk index of its dual is larger than $\omega$, then the tree of all finite sequences of integers equipped with the hyperbolic distance metrically embeds into $X$. We show that the converse is true when $X$ is assumed to be reflexive. As an application, we exhibit new classes of Banach spaces that are stable under coarse-Lipschitz embeddings and therefore under uniform homeomorphisms.Comment: 22 page

Conical limit sets and continued fractions

Inspired by questions of convergence in continued fraction theory, Erdős, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of Möbius maps acting on the Riemann sphere, S2. By identifying S2 with the boundary of three-dimensional hyperbolic space, H3, we show that these sets of divergence are precisely the sets that arise as conical limit sets of subsets of H3. Using hyperbolic geometry, we give simple geometric proofs of the theorems of Erdős, Piranian and Thron that generalise to arbitrary dimensions. New results are also obtained about the class of conical limit sets; for example, it is closed under locally quasisymmetric homeomorphisms. Applications are given to continued fractions

A notion of geometric complexity and its application to topological rigidity

We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is homotopy equivalent to M, then M x R^n is homeomorphic to N x R^n, for n large enough. This statement is known as the stable Borel conjecture. On the other hand, we show that the class of FDC groups includes all countable subgroups of GL(n,K), for any field K, all elementary amenable groups, and is closed under taking subgroups, extensions, free amalgamated products, HNN extensions, and direct unions.Comment: 58 pages, 5 figure