Smooth manifolds from locally homogeneous spaces.
problem Understanding smoothness in -Riemannian manifolds.
method Demonstrated locally homogeneous -Riemannian manifolds are smooth.
result Locally homogeneous -Riemannian manifolds are smooth.
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48 results for “local smoothness”
Characterizes when almost smooth spaces become RCD spaces.
problem Understanding conditions for almost smooth spaces to be RCD spaces.
method Characterizations via local volume doubling and Poincaré inequality.
result Characterizes Einstein 4-orbifolds.
Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.
problem Embedding 3-manifolds smoothly in 5-manifolds.
method Homotopy and small homotopy to achieve smooth embeddings.
result Locally flat embeddings are homotopic to smooth ones.
In this paper we aim for a generalisation of the Steenrod Approximation Theorem from, concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalisation is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a c…
Localizes smooth spaces to study their homotopy properties.
problem Understanding the homotopy theory of smooth spaces.
method Model category localization, Quillen equivalences, fibrant replacement.
result Localisation of smooth spaces agrees with motivic-style -localisation.
Adapts Hölder smoothness with normalized gradients.
problem Improving smoothness adaptation methods.
method Black-box adaptation of Levy's method using normalized gradients.
result Bound depends on local Hölder smoothness.
Global and local blowups of manifolds are proven equivalent.
problem Equivalence of global and local blowups in differential topology.
method Proof of equivalence between global and local constructions of blowups.
result Global and local constructions of blowups are shown to be equivalent.
Ricci flow smooths locally collapsing manifolds with controlled curvature.
problem Locally collapsing manifolds with controlled Ricci curvature.
method Ricci flow for a definite period of time, detecting collapsing infranil fiber bundles.
result Topological conditions detect collapsing infranil fiber bundles.
Solves local minima problems on smooth manifolds.
problem Local minima issues on smooth manifolds.
method Introducing valley functions and applying Morse's lemma.
result Eliminates critical points and reduces to 1D.
Local smoothing of metrics with small curvature, removing Ricci curvature condition.
problem Establishing local smoothing of metrics with curvature concentration.
method Local mollification, removing Ricci curvature condition, Sobolev constants and volume growth.
result Compactness of manifolds with small curvature concentration under Ahlfors regularity and Sobolev constant.
Formula proves invariant matches for smooth and orbifold test configurations.
problem Proving equivalence of Donaldson-Futaki invariant and Futaki invariant.
method Equivariant localization formula.
result Invariant matches for smooth and orbifold test configurations.
Locally homogeneous RCD spaces are shown to be smooth manifolds.
problem Understanding the structure of RCD spaces.
method Adapting existing results to new spaces.
result Locally homogeneous RCD spaces are isometric to smooth manifolds.
Study on diffeologies on locally convex spaces and smooth multiplication of distributions.
problem Geometric characterization and smoothness of distribution multiplication.
method Investigation of canonical and -diffeologies on locally convex spaces, proving geometric characterizations, and comparing diffeologies.
result Established a framework for nonlinear distribution theory beyond manifolds, realizing microlocally multipliable distributions as a diffeological colimit.
Motivated by the definition of the smooth manifold structure on a suitable mapping space, we consider the general problem of how to transfer local properties from a smooth space to an associated mapping space. This leads to the notion of smoothly local properties. In realising the definition of a local property at a pa…
Smooth surfaces can always be locally described by Hessians.
problem Locally describing nondegenerate surface metrics as Hessians.
method Analyzing smooth surfaces and their metrics in local coordinates.
result Smooth surfaces can always be locally described by Hessians.
Local minimizers are convex and close to Wulff shapes.
problem Finding local minimizers in anisotropic isoperimetric problems.
method Showed local minimizers are geodesically convex and small smooth perturbations of tangent Wulff shapes.
result Local minimizers are quantitatively close to Wulff shapes.
Non-parametric estimation of a multivariate density estimation is tackled via a method which combines traditional local smoothing with a form of global smoothing but without imposing a rigid structure. Simulation work delivers encouraging indications on the effectiveness of the method. An application to density-based c…
We study diffeologies on locally convex spaces and their application to smooth multiplication of distributions.
problem Constructing smooth multiplication of distributions on locally convex spaces.
method Using diffeological colimits and wavefront-set criterion.
result Proving smooth multiplication of microlocally multipliable distributions.
Stochastic gradient methods are dominant in nonconvex optimization especially for deep models but have low asymptotical convergence due to the fixed smoothness. To address this problem, we propose a simple yet effective method for improving stochastic gradient methods named predictive local smoothness (PLS). First, we …
Study smooth mappings between manifolds and their properties.
problem Understanding mappings between manifolds with various dimensions and boundaries.
method Analyzes mappings in terms of iterated directional derivatives and smooth structures.
result Establishes a canonical smooth manifold structure for mappings under certain conditions.
In this work we prove that every locally symmetric smooth submanifold gives rise to a naturally defined smooth submanifold of the space of symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to the locally symmetric manifold. We also present an explicit f…
Book on infinite-dimensional Lie groups, covering basics and various classes.
problem Understanding Lie groups in infinite-dimensional spaces.
method Develops smooth manifolds and Lie groups in locally convex spaces, discussing various classes.
result Detailed exploration of infinite-dimensional Lie groups and their properties.
In this work we prove the fact that, for a short time, it is possible to construct a smooth parametrized family of isometric embeddings of an arbitrary smooth parametrized family of Riemannian metrics on a smooth closed manifold into an Euclidean space. In order to prove this statement we work out stability estimates w…
Local Linear Forestsstat.ML
Random forests are a powerful method for non-parametric regression, but are limited in their ability to fit smooth signals, and can show poor predictive performance in the presence of strong, smooth effects. Taking the perspective of random forests as an adaptive kernel method, we pair the forest kernel with a local li…
Extends curve theory to non-smooth data with finite curvature and torsion.
problem Applying classical curve theory to non-smooth data.
method Using distributional derivative measures of functions of bounded variation.
result Essentially unique non-smooth curve solution with finite total curvature and torsion.
New cohomology theory reveals in group homology.
problem Understanding torsion in group homology of diffeomorphism groups.
method Introduced configured group cohomology, yielding explicit -valued 3-cocycles.
result Found a subgroup isomorphic to $\Q/\Z$ in the third group homology of certain diffeomorphism groups.
Proves local bi-integrability of bi-Hamiltonian systems on real smooth manifolds.
problem Proving local bi-integrability of bi-Hamiltonian systems on real smooth manifolds.
method Proves bi-integrability by constructing a complete set of functions in bi-involution and showing differentials can realize any bi-Lagrangian subspace.
result Bi-Hamiltonian systems are locally bi-integrable on real smooth manifolds.
This paper analyzes saddle points and minimax points in non-convex smooth games.
problem Understanding local optimal points in non-convex smooth games.
method Comprehensive analysis of local minimax points, including their optimality conditions and stability.
result Local saddle points are uniformly local minimax points under mild continuity assumptions.
In this note we prove that the level-set flow of the topologist's sine curve is a smooth closed curve. In previous work it was shown by the second author that under level-set flow, a locally-connected set in the plane evolves to be smooth, either as a curve or as a positive area region bounded by smooth curves. Here we…
FedProx algorithm improved for non-smooth and heterogeneous data.
problem Theoretical understanding of FedProx for non-convex federated optimization.
method Local dissimilarity invariant convergence theory through algorithmic stability.
result Convergence guarantees for non-smooth FL problems and minibatch size.
The paper proves weaker conditions for global smoothings of special Lagrangian submanifolds with conical singularities.
problem Conditions for global smoothings of special Lagrangian submanifolds with isolated conical singularities.
method Proof of weaker conditions for global smoothings.
result Global smoothings are possible under weaker hypotheses than previously known.
We prove the local existence of unique smooth solutions of the Donaldson geometric flow on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. It is a semiflow on the Besov space for . The Donaldson geometric flow was introduced by Simon Dona…
Given a sequence of properly embedded minimal surfaces in a -manifold with local bounds on area and genus, we prove subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with multiplicity, and we analyze what happens when one blows up the surfaces near a point where…
New subharmonicity concept proves conjecture on Riemannian manifolds.
problem Proving positivity of solutions to a specific PDE on Riemannian manifolds.
method Introducing local -shift defectivity and studying it on locally smoothing spaces.
result Proof of Braverman, Milatovic, Shubin conjecture on positivity of solutions.
Novel method for shape optimization of non-smooth PDEs.
problem Optimizing shapes governed by non-smooth PDEs.
method Functional variational approach and sensitivity analysis.
result Necessary conditions for locally optimal shapes.
StoSOO optimistically maximizes noisy, locally smooth functions.
problem Global maximization of noisy, locally smooth functions with unknown semi-metric.
method StoSOO uses optimistic upper confidence bounds to iteratively decide on the next evaluation point.
result StoSOO performs almost as well as the best tuned algorithms, even without knowing the semi-metric.
We construct series of examples of exotic smooth structures on compact locally symmetric spaces of noncompact type. In particular, we obtain higher rank examples, which do not support Riemannian metric of nonpositive curvature. The examples are obtained by taking the connected sum with an exotic sphere. To detect the c…
We classify smooth locally free actions of the real affine group on closed orientable three-dimensional manifolds up to smooth conjugacy. As a corollary, there exists a non-homogeneous action when the manifold is the unit tangent bundle of a closed surface with a hyperbolic metric.
New method improves counterfactual distribution learning for high-dimensional outcomes.
problem Counterfactual distribution learning for high-dimensional outcomes with concentrated structure.
method Geometry-adaptive diffusion-guided smoothing estimators combining causal nuisance adjustment and local outcome geometry.
result Geometry-adaptive methods show steeper error decay in semi-synthetic experiments.
We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the {\it Riemann-flat condition}. The Riemann-flat condition determines whether or not the es…
The thesis clarifies when local updates outperform centralized methods in heterogeneous data environments.
problem Understanding when local updates are more effective than centralized or mini-batch methods in distributed optimization.
method Fine-grained consensus-error-based analysis framework, focusing on bounded second-order heterogeneity and third-order smoothness.
result Local updates outperform centralized or mini-batch methods under realistic models of data heterogeneity.
This note is a continuation of the author's paper \cite{Li}. We prove that if the metric of a 4-manifold has bounded Ricci curvature and the curvature has no local concentration everywhere, then it can be smoothed to a metric with bounded sectional curvature. Here we don't assume the bound for local Sobolev constan…
On Local Heat Kernelmath-ph
The paper studies local heat kernel properties on smooth manifolds.
problem Understanding heat kernel properties in open convex sets of smooth Riemannian manifolds.
method Utilizes path integral formulation to investigate properties like uniqueness, symmetry, and asymptotics.
result Uniqueness and symmetry of Seeley-DeWitt coefficients are established.
PF-LaCG removes the need for knowing smoothness and strong convexity parameters for locally accelerated CG.
problem Locally accelerated CG requires knowledge of smoothness and strong convexity parameters.
method Parameter-Free Locally Accelerated CG (PF-LaCG) algorithm.
result PF-LaCG achieves local acceleration without requiring knowledge of smoothness and strong convexity parameters.
Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f: O -> P between smooth orbifolds O and P. We show that Sard's theorem holds and that the inverse image of a regular value is a smooth full suborbifold of O. We also study some constraints that the existence o…
Surrogate-based analysis of interactions via local effect smooths
problem Detecting and characterizing feature interactions in machine learning models
method Surrogate-based analysis using generalized additive models
result Empirical validation of effectiveness for pairwise interactions
This is a survey on known results and open problems about Smooth and PL-Rigidity Problem for negatively curved locally symmetric spaces. We also review some developments about studying the basic topological properties of the space of negatively curved Riemannian metrics and the Teichmuller space of negatively curved me…
We identify the 2-groupoid of deformations of a gerbe on a smooth manifold with the Deligne 2-groupoid of a corresponding twist of the DGLA of local Hochschild cochains on infinite jets of smooth functions.