arXiv research

Characterizes continuity of monotone functionals in mixed topology.

Machine learning classifies topological phases in leaky photonic lattices.

Paper introduces TVaRD, a new topological risk measure for financial portfolios.

Study on random surfaces in hyperbolic 3-manifolds, focusing on geometric and topological properties.

We give two examples of metric measure spaces satisfying the measure contraction property MCP(K,N) but having different topological dimensions at different regions of the space. The first one satisfies MCP(0,3) and contains a subset isometric to $\mathbb{R}$, but does not topologically split. The second space satisfies…

We prove that the bijective correspondence between the space of bounded measured laminations $ML_b(\mathbb{H})$ and the universal Teichmüller space $T(\mathbb{H})$ given by $λ\mapsto E^λ|_{S^1}$ is a homeomorphism for the Fréchet topology on $ML_b(\mathbb{H})$ and the Teichmüller topology on $T(\mathbb{H})$, where $E^λ…

Measure homology was introduced by Thurston in order to compute the simplicial volume of hyperbolic manifolds. Berlanga endowed measure homology with a structure of graded locally convex (possibly non-Hausdorff) topological vector space. In this note we completely characterize Berlanga's topology on measure homology of…

Proposes a new metric space example showing non-constant topological dimension.

To provide a solid analytic foundation for the module approach to conditional risk measures, this paper establishes a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the $(\varepsilon,λ)$--topology and the locally $L^0$-- convex topolo…

In the aftermath of the financial crisis, the growing literature on financial networks has widely documented the predictive power of topological characteristics (e.g. degree centrality measures) to explain the systemic impact or systemic vulnerability of financial institutions. In this work, we show that considering al…

The paper studies strong topologies for complex Monge-Ampère equations on Kähler manifolds.

Measure homology was introduced by Thurston in his notes about the geometry and topology of 3-manifolds, where it was exploited in the computation of the simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently proved that there exists a canonical isomorphism between measure homology and singular hom…

A new approach uses circuit topology to study complex polymer interactions.

New stable minimal hypersurfaces found in 4-manifolds, proving topology results.

Study counts ergodic measures in surface lamination strata.

Two-dimensional collapsed spaces with lower Ricci bounds are topological surfaces.

Paper infers intrinsic dimension from quasi-convex measurements.

Regularization plays a crucial role in supervised learning. Most existing methods enforce a global regularization in a structure agnostic manner. In this paper, we initiate a new direction and propose to enforce the structural simplicity of the classification boundary by regularizing over its topological complexity. In…

Robust topological information commonly comes in the form of a set of persistence diagrams, finite measures that are in nature uneasy to affix to generic machine learning frameworks. We introduce a fast, learnt, unsupervised vectorization method for measures in Euclidean spaces and use it for reflecting underlying chan…

This paper evaluates fractal dimension and persistent homology for neural network generalization.

The paper tackles binary classification with measure data using topological descriptors.

Set risk measures extend traditional risk measures to handle sets of positions.

Method learns topological states from randomized measurements.

Measuring wave sources uniquely identifies manifold properties.

New topological complexity measures for neural networks.

New family of measurable pseudo-Anosov maps on spheres.

Machine learning models for repeated measurements are limited. Using topological data analysis (TDA), we present a classifier for repeated measurements which samples from the data space and builds a network graph based on the data topology. When applying this to two case studies, accuracy exceeds alternative models wit…

SM-netFusion estimates brain network atlas by considering multiple topological measures.

New integration theory on topological spaces, including fractals.

Unified toolkit for comparing neural representations using SRTD and NTS.

Topology-based information retrieval improves query accuracy.

It is known that all but finitely many leaves of a measured foliated 2-complex of thin type are quasi-isometric to an infinite tree with at most two topological ends. We show that if the foliation is cooriented, and the associated R-tree is self-similar, then a typical leaf has exactly one topological end. We also cons…

The learnability of different neural architectures can be characterized directly by computable measures of data complexity. In this paper, we reframe the problem of architecture selection as understanding how data determines the most expressive and generalizable architectures suited to that data, beyond inductive bias.…

Entropy measures geodesic flow complexity.

Method quantifies disentanglement of generative models using manifold topology.

Paper estimates area covered by a line-sweep sensor in robotics.

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus $T^2$ for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic…

We provide a measure based topology for certain unions of C2 rectifiable submanifolds of mixed dimensions in Rn. In this topology lower dimensional sets remain in the limit as measures when higher dimensional sets collapse down to them. For example a decreasing sequence of spheres may have a limit consisting of just a …

Contagions such as the spread of popular news stories, or infectious diseases, propagate in cascades over dynamic networks with unobservable topologies. However, "social signals" such as product purchase time, or blog entry timestamps are measurable, and implicitly depend on the underlying topology, making it possible …

We present sufficient conditions for topological stability of continuous functions $f:\mathbb{R}\to\mathbb{R}$ having finitely many local extrema with respect to averagings by discrete measures with finite supports.

Extends rigidity results to non-homogeneous manifolds.

An algorithm preserves topological features in dimensionality reduction.

Approximates measures on curved spaces using Dirac measures.

The bending map of a hyperbolic 3-manifold maps a convex cocompact hyperbolic metric on a hyperbolic 3-manifold with boundary to its bending measured geodesic lamination. In the present paper we study the extension of this map to the space of geometrically finite hyperbolic metrics. We introduce a relationship on the s…

Let $X$ be an infinite Riemann surface equipped with its conformal hyperbolic metric such that the action of the covering group $π_1(X)$ on $\tilde{X}$ is of the first kind-i.e., the surface $X$ is equal to its convex core. We first prove that any geodesic lamination on $X$ is nowhere dense. Given a fixed geodesic pant…

CantorNet tests geometric and topological complexity in neural networks.

Study uses equivariant topology to measure distances between G metric spaces.

Study on free boundary problems in RCD spaces, proving existence and regularity.