Characterizes continuity of monotone functionals in mixed topology.
problem Continuity of monotone functionals in mixed topology.
method Characterization through lower semicontinuity and dual representations.
result Continuity in mixed topology is equivalent to dual representation in terms of countably additive measures.
Machine learning classifies topological phases in leaky photonic lattices.
problem Classifying topological phases in leaky photonic lattices using limited data.
method A fully connected neural network trained on bulk intensity measurements.
result Accurate determination of topological properties from intensity distributions.
Paper introduces TVaRD, a new topological risk measure for financial portfolios.
problem Traditional risk measures like VaR and CVaR are insufficient for complex market conditions.
method Topological data analysis (TDA) using cohomology groups on financial time series data.
result TVaRD reveals significant changes in financial time series during stress conditions.
Study on random surfaces in hyperbolic 3-manifolds, focusing on geometric and topological properties.
problem Distribution of nearly geodesic surfaces in hyperbolic 3-manifolds.
method Invariant measures on the Grassmann bundle G(M) derived from limits of random minimal surfaces.
result Topological limiting measures are totally scarring if M contains a totally geodesic subsurface, while geometrical limiting measures are not.
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 R, but does not topologically split. The second space satisfies…
We prove that the bijective correspondence between the space of bounded measured laminations MLb(H) and the universal Teichmüller space T(H) given by λ↦Eλ∣S1 is a homeomorphism for the Fréchet topology on MLb(H) and the Teichmüller topology on T(H), where $E^λ…
Study improves systemic risk assessment by considering local network environments.
problem Identifying systemic financial institutions using network metrics.
method Two-step procedure: 1) recover network communities, 2) regress vulnerability on topological measures at global, local, and aggregated levels.
result Local network metrics predict distress better than global metrics during financial crises.
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.
problem Non-constant topological dimension in metric measure spaces.
method Refines Ketterer and Rajala's example to satisfy CD(0,∞) condition.
result Shows non-constancy of topological dimension for CD spaces.
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 (ε,λ)--topology and the locally L0-- convex topolo…
The paper studies strong topologies for complex Monge-Ampère equations on Kähler manifolds.
problem Analyzing strong topologies for complex Monge-Ampère equations on Kähler manifolds.
method Proving the Monge-Ampère operator is a homeomorphism between finite energy potentials and energy measures with their strong topologies.
result The Monge-Ampère operator produces an homeomorphism between sets of finite energy potentials and measures 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.
problem Understanding structural phase transitions in entangled polymer systems.
method Braided circuit topology framework for multiple-chain systems.
result Circuit topological motif fractions are effective order parameters for structural transitions.
New stable minimal hypersurfaces found in 4-manifolds, proving topology results.
problem Finding stable minimal hypersurfaces with specific topologies in 4-manifolds.
method Geometric measure theory and 4-manifold topology techniques.
result Existence of stable minimal hypersurfaces diffeomorphic to S3 or S2imesS1. Study counts ergodic measures in surface lamination strata.
problem Counting ergodic measures in surface lamination strata.
method Determined through analysis of geodesic laminations.
result Number of ergodic measures identified in each stratum.
Two-dimensional collapsed spaces with lower Ricci bounds are topological surfaces.
problem Topology of collapsed spaces with lower Ricci bounds
method Prove that collapsed spaces are topological surfaces
result Collapsed spaces are topological surfaces
Paper infers intrinsic dimension from quasi-convex measurements.
problem Inferring intrinsic dimension from measurements by quasi-convex functions.
method Developed a method using filtration of Dowker complexes based on discrete data of point orderings.
result Correct intrinsic dimension can be inferred in the limit of large data under generic assumptions.
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.
problem Bounding and predicting the generalization gap of neural networks.
method Empirical evaluation of fractal dimension and persistent homology as generalization measures.
result Fractal dimension and persistent homology fail to predict generalization of models trained from poor initializations.
The paper tackles binary classification with measure data using topological descriptors.
problem Binary classification with measure data.
method Develops classifiers for measure data using topological descriptors (persistence diagrams).
result Upper and lower bounds on the Rademacher complexity of classifiers on measures.
Method learns topological states from randomized measurements.
problem Detecting topologically ordered two-dimensional states on quantum processors.
method Variational tensor network tomography with randomized measurements.
result Demonstrated ability to learn ground states of surface code and quantum spin liquid states.
Set risk measures extend traditional risk measures to handle sets of positions.
problem Handling sets of positions with a single capital requirement.
method Developed an axiomatic framework for set risk measures, dual representation through topology and measures.
result Characterized worst-case set risk measures and provided examples.
Measuring wave sources uniquely identifies manifold properties.
problem Determining Riemannian manifold structure from wave observations.
method Semilinear wave equation measurements at a single point.
result Topological, differential, and geometric structure can be inferred.
New topological complexity measures for neural networks.
problem Measuring complexity of neural network functions.
method Generalized piecewise-linear Morse theory applied to ReLU networks.
result Local complexity can be arbitrarily high.
New family of measurable pseudo-Anosov maps on spheres.
problem Generalizing pseudo-Anosov maps to measurable ones.
method Continuous family of homeomorphisms on sphere, semi-conjugate to core tent map.
result Measurable pseudo-Anosov maps have invariant dense streamlines with uniform measures.
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.
problem Limited BNA estimation methods that overlook topological measures and lack discriminative power.
method Supervised multi-topology network cross-diffusion framework using degree, closeness, and eigenvector centrality measures.
result SM-netFusion produces more centered and representative templates, and improves classification accuracy.
New integration theory on topological spaces, including fractals.
problem Developing a universal integration theory for arbitrary topological spaces.
method Introducing a new integration framework using unital magma valued functions and measures.
result Integration, differentiation, and orientation defined for arbitrary topological spaces.
Proofs a theorem using basic geometric tools.
problem C2-rectifiability problem
method Elementary geometric measure theory and topology
result Gives a proof of Alberti's Luzin-type theorem
Unified toolkit for comparing neural representations using SRTD and NTS.
problem Heuristic asymmetry and unbounded scores in existing divergences.
method Developed SRTD and NTS to address these issues.
result Unified, robust, and scale-invariant metric for comparing neural representations.
Topology-based information retrieval improves query accuracy.
problem Query accuracy in databases with complex structures.
method Dilation-invariant comparative measures of persistent homology.
result Topology-based retrieval outperforms standard methods.
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.
problem Measuring complexity of geodesic flows on manifolds.
method Introduced barcode entropy to measure exponential growth rate of not-too-short bars in Morse-theoretic barcodes.
result Barcode entropy bounds topological entropy and vice versa.
Method quantifies disentanglement of generative models using manifold topology.
problem Challenging and inconsistent measurement of disentanglement in generative models.
method Measures topological similarity of conditional submanifolds in learned representation.
result Method ranks models similarly to existing methods across multiple datasets.
Paper estimates area covered by a line-sweep sensor in robotics.
problem Accurately estimating the area covered by a line-sweep sensor.
method Relies on coverage measure and topological degree in the plane.
result Guaranteed characterization of the explored area using interval analysis.
In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T2 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:R→R having finitely many local extrema with respect to averagings by discrete measures with finite supports.
Extends rigidity results to non-homogeneous manifolds.
problem Measure and topological rigidity in dynamical systems.
method From homogeneous to general manifolds.
result Measure and topological rigidity results extended.
An algorithm preserves topological features in dimensionality reduction.
problem Preserving topological features in dimensionality reduction.
method Simulated annealing for finding a linear projection preserving persistent homology.
result Measures of topological equivalence between filtrations.
Approximates measures on curved spaces using Dirac measures.
problem Topology of invariant measures on curved manifolds.
method Introducing weakly regular vectors and approximating measures by Dirac measures.
result Ergodicity is a generic property in the space of invariant measures supported on weakly regular vectors.
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 π1(X) on 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.
problem Understanding self-similar patterns in neural networks.
method Inspired by Cantor set, CantorNet introduces novel complexity measures.
result CantorNet's decision boundaries are analytically known and can be arbitrarily ragged.
Study uses equivariant topology to measure distances between G metric spaces.
problem Measuring distances between G metric spaces.
method Equivariant topology methods to derive lower bounds.
result Sharp bounds on Gromov Hausdorff distance between spheres.