The purpose of these notes is to explain parts of Gromov's survey of Carnot-Carathedory spaces, in the light of subsequent results of M. Rumin. Among the rich material provided by Gromov, most of which pertains to analysis on metric spaces, we choose to concentrate on the H{ö}lder equivalence problem for Carnot manifol…
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48 results for “weighted Hölder spaces”
Adversarial online nonparametric regression achieves optimal rates with locally adaptive learning.
problem Adversarial online nonparametric regression with general convex losses.
method Parameter-free learning algorithm leveraging chaining trees to compete against H{ö}lder functions, dynamically tracking and adapting to local smoothness variations.
result First computationally efficient algorithm with locally adaptive optimal rates for online regression in an adversarial setting.
Study approximates unknown function levels with queries.
problem Approximating unknown function levels through sequential queries.
method Introduce Bisect and Approximate algorithms to reduce to local function approximation.
result Rate-optimal sample complexity guarantees for H{ö}lder functions.
Toric Pluripotential Theorymath.CV
We study finite energy classes of quasiplurisubharmonic (qpsh) functions in the setting of toric compact K{ä}hler manifolds. We characterize toric qpsh functions and give necessary and sufficient conditions for them to have finite (weighted) energy, both in terms of the associated convex function in R n , and through t…
A variant of Gromov's H{ö}lder-equivalence problem, motivated by a pinching problem in Riemannian geometry, is discussed. A partial result is given. The main tool is a general coarea inequality satisfied by packing energies of maps.
Let be a closed oriented surface of genus at least , and denote by its Teichm{ü}ller space. For any isotopy class of closed curves , we compute the first three derivatives of the length function in the shearing coordinates associated to a maxim…
Analysis of the Risk-Sharing Principal-Agent problem through the Reverse-H{ö}lder inequalityq-fin.RM
In this paper we provide an alternative framework to tackle the first-best Principal-Agent problem under CARA utilities. This framework leads to both a proof of existence and uniqueness of the solution to the Risk-Sharing problem under very general assumptions on the underlying contract space. Our analysis relies on an…
We find a local solution to the Ricci flow equation under a negative lower bound for many known curvature conditions. The flow exists for a uniform amount of time, during which the curvature stays bounded below by a controllable negative number. The curvature conditions we consider include 2-non-negative and weakly $\t…
In the context of stochastic continuum-armed bandits, we present an algorithm that adapts to the unknown smoothness of the objective function. We exhibit and compute a polynomial cost of adaptation to the H{ö}lder regularity for regret minimization. To do this, we first reconsider the recent lower bound of Locatelli an…
Efficient algorithms for contextual bandits with smooth regret in continuous action spaces.
problem Efficient learning in large or continuous action spaces.
method Smooth regret notion and efficient algorithms for general function approximation.
result Statistically and computationally efficient algorithms for contextual bandits with smooth regret.
We consider the problem of online nonparametric regression with arbitrary deterministic sequences. Using ideas from the chaining technique, we design an algorithm that achieves a Dudley-type regret bound similar to the one obtained in a non-constructive fashion by Rakhlin and Sridharan (2014). Our regret bound is expre…
Validates economic scenarios using statistical tests on stochastic processes.
problem Ensuring the accuracy of real-world economic scenario models.
method Applies Chevyrev and Oberhauser's (2022) signature and maximum mean distance test to various stochastic processes.
result Demonstrates the test's effectiveness across different path properties relevant to financial modeling.
New algorithms for interactive learning match minimax bounds efficiently.
problem Interactive learning in the realizable setting with computational efficiency.
method General framework, computationally efficient algorithms, Monte Carlo hit-and-run sampling.
result Sample complexities quantifiable in terms of combinatorial quantities, computationally efficient.
Study shows zero-shot super-resolution in neural operators is impossible in many cases.
problem Understanding the theoretical limits of zero-shot super-resolution in neural operators.
method Systematic theoretical study including information-theoretic and generalization bounds analysis.
result Zero-shot super-resolution is information-theoretically impossible in many settings.
Introduces a space of almost complex structures for complex Lie group bundles.
problem Integrability of almost complex structures on complex Lie group bundles.
method Introduces a space of bundle almost complex structures and studies their properties.
result Locally pseudo-holomorphic sections exist if and only if the obstruction form is zero.
General lower bounds on neural network approximation in L^p norm.
problem Fundamental limits of neural network expressivity.
method General lower bound proof on approximation in L^p norm, applied to feed-forward neural networks.
result Neural networks can't approximate certain functions as well as previously thought.
The Weighted Ambient Metricmath.DG
Proves existence and uniqueness of weighted metrics for smooth spaces.
problem Existence and uniqueness of weighted metrics for smooth metric measure spaces.
method Proves existence and uniqueness using weighted ambient metrics and Poincaré metrics.
result Existence and uniqueness of weighted metrics for smooth metric measure spaces.
Study on deformation of weighted scalar curvature, proving geometric results and stability.
problem Deformation of weighted scalar curvature and related geometric properties.
method Linearization of weighted scalar curvature, studying kernel of formal adjoint.
result Definition and study of weighted vacuum static spaces, stability results on flat spaces.
We study the inverse spectral problem for weighted projective spaces using wave-trace methods. We show that in many cases one can "hear" the weights of a weighted projective space.
Study classifies Einstein spaces and warped products in weighted geometry.
problem Characterizing geometric structures of weighted Einstein spaces.
method Complete local classification of weighted Einstein spaces with harmonic Weyl tensor.
result Spaces decompose into Einstein or specific warped products.
Study proves no minimal surfaces can be contained in certain half-spaces or cones.
problem Prohibiting minimal surfaces from certain geometric configurations.
method Analyzes weighted minimal surfaces in with height-dependent weights.
result No proper surfaces can be contained in specific half-spaces or cones.
This paper selects features in deep neural networks with theoretical guarantees.
problem Feature selection in deep neural networks with unknown nonlinear functions.
method Reformulate neural networks as index models, estimate feature sets using Stein's formula, and apply screening-and-selection mechanism.
result Consistent feature selection with theoretical guarantees, even in high-dimensional settings.
Paper solves CR positive mass and Yamabe problems on weighted spaces.
problem CR positive mass and Yamabe problems on weighted spaces.
method Analyzes sub-Laplacian on Folland-Stein spaces.
result CR positive mass and Yamabe problems resolved.
Functional input neural networks approximate continuous functions on weighted spaces.
problem Approximating continuous functions on infinite-dimensional weighted spaces.
method Additive family mapping, non-linear activation, linear readouts, Stone-Weierstrass theorem.
result Global universal approximation of continuous functions on weighted spaces.
In this paper, we define locally convex vector spaces of weighted vector fields and use them as model spaces for Lie groups of weighted diffeomorphisms on Riemannian manifolds. We prove an easy condition on the weights that ensures that these groups contain the compactly supported diffeomorphisms. We finally show that …
New neural network rates for unbounded domains with weighted Sobolev spaces.
problem Improving neural network approximation rates for unbounded domains.
method Embedding results for weighted Fourier-Lebesgue spaces in weighted Sobolev spaces, followed by asymptotic approximation rates.
result Asymptotic approximation rates for shallow neural networks without curse of dimensionality for unbounded domains and Muckenhoupt weights.
Novel metric space magnitude and weighting vectors improve machine learning tasks.
problem Improving machine learning algorithms using novel metric space concepts.
method Metric space magnitude and weighting vectors for better machine learning.
result The weighting vector effectively detects boundaries and improves classic machine learning tasks.
The moduli space of tropical -weighted stable curves of volume is naturally identified with the dual complex of the divisor of singular curves in Hassett's spaces of -weighted stable curves. If at least two of the weights are , we prove that is homotopic to a wedge sum of spheres, possi…
Classifies smooth metric measure spaces with two weighted Einstein representatives.
problem Classifying smooth metric measure spaces with specific weighted Einstein properties.
method Local and global classification using Einstein and quasi-Einstein warped products.
result Global classification result for complete manifolds, showing specific types of manifolds.
An orbifold version of the Hitchin-Thorpe inequality is used to prove that certain weighted projective spaces do not admit orbifold Einstein metrics. Also, several estimates for the orbifold Yamabe invariants of weighted projective spaces are proved.
Study on convergence rate of weighted Yamabe flow.
problem Weighted Yamabe problem on smooth metric measure spaces.
method Weighted Yamabe flow and its convergence rate analysis.
result Study and analysis of convergence rate of the weighted Yamabe flow.
Researchers develop weighted GJMS operators for smooth metric measure spaces.
problem Developing mathematical tools for smooth metric measure spaces.
method Constructing and proving formal self-adjointness of weighted GJMS operators.
result Formal self-adjointness of weighted GJMS operators proved.
The paper proves new inequalities in hyperbolic space using Euclidean methods.
problem Proving weighted isoperimetric inequalities in hyperbolic space.
method Using isoperimetric inequality with log-convex density in Euclidean space.
result Removed horo-convex assumption and proved new inequalities for star-shaped domains.
We generalize Fulton and MacPherson's configuration space construction to weighted filtered manifolds.
problem Infinitesimal collision data in filtered manifolds with higher-order compatibility.
method Generalizing Fulton and MacPherson's blow-up approach to weighted arrangements of submanifolds.
result Smoothness of the weighted blow-up under reasonable assumptions.
Extends online learning to metric spaces using exponential weights.
problem Online learning in metric spaces.
method Exponentially weighted average forecaster, barycenters, Jensen's inequality, measure contraction property.
result Results in a statistical learning framework.
Paper studies weighted Fermat-Frechet problem for simplex edge lengths.
problem Finding optimal edge lengths for simplex deformations.
method Isometric embedding techniques for -Space.
result New variational method to solve weighted Fermat-Frechet problem.
We propose a definition of the weighted -curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted -curvature prescription problem is governed by a fully nonlinear second order elliptic PDE which is variational when or the smooth metric measure space is lo…
The current paper deals with some new classes of Finsler metrics with reversible geodesics. We construct weighted quasi-metrics associated with these metrics. Further, we investigate some important geometric properties of weighted quasi-metric space. Finally, we discuss the embedding of quasi-metric spaces with general…
Study Bochner formula on metric measure spaces for vanishing Betti numbers.
problem Vanishing Betti numbers on metric measure spaces.
method Introduce weighted curvature conditions.
result Vanishing of all Betti numbers.
The paper extends geometric inequalities for nearly spherical sets in various space forms.
problem Investigating weighted inequalities for nearly spherical sets in space forms.
method Generalizing and extending inequalities for nearly spherical sets in and settings, with convex weight functions.
result Quantitative stability estimates for weighted inequalities in and .
The weighted Yamabe flow converges on smooth metric measure spaces.
problem Analyzing convergence of the weighted Yamabe flow on metric measure spaces.
method Introduced the weighted Yamabe flow and proved its long-time existence and convergence under certain conditions.
result Long-time existence and convergence of the weighted Yamabe flow on smooth metric measure spaces.
In our previous paper, we discussed the hyperbolization of the configuration space of n(> 4) marked points with weights in the projective line up to projective transformations. A variation of the weights induces a deformation. It was shown that this correspondence of the set of the weights to the Teichmüller space when…
Kähler information manifolds for signal filters in weighted Hardy spaces are explored.
problem Developing a geometric framework for signal processing filters in weighted Hardy spaces.
method Introducing weighted Hardy spaces and smooth transformations of transfer functions, demonstrating the Kähler manifold structure.
result The Riemannian geometry of weighted Hardy norms for transfer functions forms a Kähler manifold.
The paper shows neural networks can approximate functions over non-compact domains with non-polynomial activation.
problem Approximating functions over non-compact domains using neural networks.
method Using single-hidden-layer feedforward neural networks with non-polynomial activation functions over non-compact subsets of Euclidean spaces.
result Neural networks can approximate functions in weighted -spaces and weighted Sobolev spaces over unbounded domains.
A new weighted FDA method improves face recognition accuracy.
problem Equal treatment of all class pairs in FDA leads to suboptimal performance.
method Cosine-weighted and automatically weighted FDA methods are proposed.
result Improved face recognition accuracy through weighted FDA.
Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.
problem Classifying functions for bounded Schrödinger operator .
method Investigates weighted -boundedness of Hodge projector.
result Characterizes function for Schrödinger operator boundedness.
We investigate the relation between weighted quasi-metric Spaces and Finsler Spaces. We show that the induced metric of a Randers space with reversible geodesics is a weighted quasi-metric space.
Smooth superspace with special weights has a Fubini-Study form.
problem Describing a new smooth superspace with a special structure.
method Construction of weighted projective superspace and description of its structure.
result Smooth superspace with weights has an analog of the Fubini-Study form.