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 R3 with height-dependent weights. result No proper surfaces can be contained in specific half-spaces or cones.
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 Δg,w of tropical w-weighted stable curves of volume 1 is naturally identified with the dual complex of the divisor of singular curves in Hassett's spaces of w-weighted stable curves. If at least two of the weights are 1, we prove that Δ0,w 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 K-Space. result New variational method to solve weighted Fermat-Frechet problem.
We propose a definition of the weighted σk-curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted σk-curvature prescription problem is governed by a fully nonlinear second order elliptic PDE which is variational when k=1,2 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 C1 and W2,∞ settings, with convex weight functions. result Quantitative stability estimates for weighted inequalities in Rn+1 and Hn+1. 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 Ck-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.
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.
Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.
problem Classifying functions V for bounded Schrödinger operator Δ−V. method Investigates weighted L2-boundedness of Hodge projector. result Characterizes function V for Schrödinger operator boundedness. 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 +1,−1 has an analog of the Fubini-Study form. We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact …
We propose regularizing the empirical loss for semi-supervised learning by acting on both the input (data) space, and the weight (parameter) space. We show that the two are not equivalent, and in fact are complementary, one affecting the minimality of the resulting representation, the other insensitivity to nuisance va…
We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized b…
Study solves Yamabe problems on metric measure spaces with or without boundary.
problem Yamabe-type problems on compact metric measure spaces with or without boundary.
method Analyzes uniqueness, characterization, and existence of minimizers.
result Characterizes weighted Yamabe solitons and existence of positive minimizers.
Study on Bayesian transformers finds issues with weight-space inference and prior specification.
problem Challenges in obtaining meaningful uncertainty estimates for transformer models.
method Proposed a novel method based on implicit reparameterization of the Dirichlet distribution for variational inference on attention weights.
result Proposed method performs competitively with baselines in estimating predictive uncertainty.
We consider a conjecture made by Ge, Wang and Wu regarding weighted Alexandrov-Fenchel inequalities for horospherically convex hypersurfaces in hyperbolic space (a bound, for some physically motivated weight function, of the weighted integral of the kth mean curvature in terms of the area of the hypersurf…
Proves Laplacian and Lichnerowicz Laplacian are sectorial in weighted Hölder spaces.
problem Analyzing sectoriality of Laplacian and Lichnerowicz Laplacian on asymptotically hyperbolic spaces.
method Proves sectoriality in weighted Hölder spaces using asymptotically hyperbolic metrics.
result Analytic semigroups apply, yielding well-posedness results for parabolic evolution equations.
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.
New cyclicity measures defined in weighted Besov spaces, with stability and geometric analysis.
problem Characterizing cyclicity in weighted Besov spaces.
method Defining cyclicity indices based on potential theory and capacity, studying stability under perturbations, and linking zero set structure to cyclicity.
result Novel invariants and conditions for cyclicity in various function spaces.
Unified positive mass theorem and Dirac operator study on weighted manifolds.
problem Establishing a unified positive mass theorem for weighted manifolds and smooth metric measure spaces.
method Analyzing Dirac operators on warped product manifolds and applying results to the positive mass theorem.
result Equivalence of weighted positive mass theorem to usual positive mass theorem.
In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold (M,ω), we construct a weak symplectic structure on each leaf Iw of a foli…
MLDS dataset reveals hidden model behavior via weight-space analysis.
problem Neural networks' opacity makes them hard to evaluate.
method Presented MLDS dataset of trained neural networks.
result Weight-space analysis reveals meaningful divergence with small changes in training data.
Study connects Gaussian processes and regularization for sequence-function mappings.
problem Understanding and interpreting sequence-function maps in biology.
method Relates Gaussian process priors, regularization, and gauge fixing in overparameterized weight space.
result Established the relationship between regularized regression and Gaussian processes in function space.
The study connects minimal and maximal surfaces in 3D and 3-L space.
problem Describing correspondences between minimal and maximal surfaces in different spaces.
method Weierstrass representation and asymptotic analysis.
result Established criteria for singularity types and moduli spaces.
Probabilistic neural networks are typically modeled with independent weight priors, which do not capture weight correlations in the prior and do not provide a parsimonious interface to express properties in function space. A desirable class of priors would represent weights compactly, capture correlations between weigh…
The study explores weightings on submanifolds and their geometric properties.
problem Understanding weightings on submanifolds and their geometric implications.
method Detailed exploration of weighted normal bundles, weighted deformation spaces, and weighted blow-ups.
result A description of weightings in terms of subbundles of higher tangent bundles, leading to new concepts for Lie algebroids and groupoids.
The study examines the independence of GKM manifolds and symmetric spaces.
problem Understanding the independence of isotropy weights in GKM manifolds.
method Using weighted graphs and properties of symmetric spaces, the study analyzes the independence of isotropy weights.
result The maximal independence of G/H is 2, 3, or n=dimT, corresponding to symmetric spaces of rank >2.