Let M be a differentiable manifold. We say that a tensor field g defined on M is non-regular if g is in some local Lp space or if g is continuous. In this work we define a mollifier smoothing g_t of g that has the following feature: If g is a Riemannian metric of class C2, then the Levi-Civita connection and the Rieman…
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48 results for “mollifier smoothing”
A -Finsler structure is a continuous function defined on the tangent bundle of a differentiable manifold such that its restriction to each tangent space is an asymmetric norm. We use the convolution of with the standard mollifier in order to construct a mollifier smoothing of …
New optimization method for sampling from unknown density measures.
problem Sampling from measures with unknown normalization constants.
method Mollified Interaction Energy Descent (MIED) method.
result Gradient flow of MIE converges to chi-square divergence.
Study smooths Finsler structures on Lie groups, proving extremal convergence.
problem Smooth left-invariant strongly convex -Finsler structures on Lie groups.
method Introduce mollifier smoothing, study extremals using Pontryagin maximum principle.
result Pontryagin extremals on smoothed Finsler structures converge uniformly to those on original structure.
Study well-poses Dirac operator problem with APS boundary conditions.
problem Well-posedness of Cauchy problem for Dirac operator on Lorentzian manifolds.
method Derived energy estimates, established uniqueness and existence of weak solutions, introduced mollifier operators.
result Well-posedness of Cauchy problem for Dirac operator with APS boundary conditions.
We consider the problem of pricing basket options in a multivariate Black Scholes or Variance Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical i…
Let be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various -related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the reso…
We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in . If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the surface. We establish longtime-existence of the flow and investigate the projection of…
This study analyzes theoretical guarantees for VI with fixed-variance Gaussian mixtures.
problem Theoretical analysis of variational inference with non-Gaussian distributions.
method Formulates variational inference as minimizing a mollified relative entropy, solving it through gradient descent on particle positions.
result Establishes descent lemma and approximation error bounds for optimization of variational inference.
Gaussian Process Hydrodynamicsphysics.flu-dyn
Gaussian Process Hydrodynamics approximates fluid flow equations using probabilistic kernels.
problem Approximating fluid flow equations with fewer particles and uncertainty estimates.
method Lagrangian particle-based approach with Gaussian Process (GP) prior and physics-informed kernels.
result GPH requires fewer particles and provides uncertainty estimates.
Random concave functionsmath.PR
Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave functions on the unit simplex measure the concentration of capital, and their gr…
New uncertainty principle for Schrödinger equations on hyperbolic manifolds.
problem Uncertainty principle for Schrödinger equations on hyperbolic manifolds.
method General strategy of Escauriaza-Kenig-Ponce-Vega, new Carleman estimates, logarithmic convexity, new mollifier and weight function.
result Similar rigidity phenomenon as in Euclidean space persists in hyperbolic geometry.
Novel neural network layer improves long-range interactions in point clouds.
problem Efficiently treating long-range interactions in point clouds.
method Long-range convolutional (LRC)-layer that incorporates Fourier transforms.
result Global all-to-all convolution operation performed in nearly-linear time.
Smooth and F--smooth systemsmath.DG
Abstract reviews geometric theories of smooth and F-smooth systems.
problem Geometric theories of smooth and F-smooth systems.
method Reviews geometric theories of smooth and F-smooth systems.
result Discusses geometric theories of smooth and F-smooth systems.
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.
Smooth families of biholomorphisms between strongly pseudoconvex domains are shown to be smooth.
problem Smoothness of families of biholomorphisms between strongly pseudoconvex domains.
method Riemannian geometry of Bergman metrics and smoothness of families of isometries.
result Smoothness of families of biholomorphisms between strongly pseudoconvex domains.
We prove that smooth cube manifolds have normal smooth structures.
On a compact, oriented, Riemannian manifold, the Hodge decomposition theorem associates a smooth primitive to any exact smooth form omega. In this paper, we show that given a smooth family of exact smooth forms omega(t), the family of associated primitives is also a smooth family with respect to t.
Smoothness of submetriesmath.DG
Smooth submetries between curved spaces are smooth.
problem Smoothness of submetries between curved spaces.
method Proving smoothness of submetries in a general setting, including Riemannian submersions and isometric actions.
result Smoothness of the base manifold is implied by the smoothness of the total manifold without curvature assumptions.
Estimates time-varying network connections using multi-stage smoothing.
problem Estimating edge probabilities of time-varying networks.
method Multi-stage smoothing: temporal local smoothing followed by node-domain smoothing.
result Captures both smooth temporal evolution and structural patterns in connectivity.
The paper constructs infinitely many -smoothings of a -manifold.
problem Constructing -smoothings of a -manifold.
method Using controlled -cobordisms.
result Infinitely many -smoothings of a -manifold are constructed and are isotopic after taking a product with .
Investigates smoothness of specific algebra structures.
problem Smoothness of bi-quadratic algebras on three generators.
method Analyzes differential smoothness with PBW basis.
result Characterizes conditions for smoothness.
New smoothing techniques for topological surfaces in 4-manifolds.
problem Topological isotopy of surfaces in smooth 4-manifolds.
method Combining Quinn's smoothing theory with Gabai's light bulb theorem and other developments.
result Proves topological = smooth results for certain disks and spheres.
We define a diffeomorphism invariant of smooth 4-manifolds which we can estimate for many smoothings of R^4 and other smooth 4-manifolds. Using this invariant we can show that uncountably many smoothings of R^4 support no Stein structure. (Gompf has constructed uncountably many smoothings of R^4 which do support Stein …
The paper examines smoothness in diffusion algebra.
problem Smoothness in diffusion algebras.
method Not explicitly detailed in the abstract.
result Not explicitly detailed in the abstract.
Smoothness Adaptivity in Constant-Depth Neural Networks: Optimal Rates via Smooth Activationsstat.ML
Smooth activations enable optimal error rates in neural networks for Sobolev function classes.
problem Achieving optimal approximation and estimation error rates for neural networks in Sobolev function classes.
method Study of neural networks with smooth activations, proving optimal rates via approximation and statistical properties.
result Constant-depth networks with smooth activations achieve optimal rates of approximation and estimation, demonstrating smoothness adaptivity.
Entropy data replaces classical charts for smooth manifolds.
problem Establishing smooth structures on topological manifolds.
method Using entropy data to define admissible coordinate functions and reconstruct smooth atlases.
result Entropy-smooth structures are equivalent to classical smooth structures and stable under perturbations.
Variant of previous work on smooth algebraic functions with compact and non-compact preimages.
problem Constructing smooth algebraic functions with specific preimage properties.
method Explicit construction of smooth real algebraic functions with controlled preimage compactness.
result New results in singularity theory and real algebraic geometry.
Proof outlined for 4D smooth Poincaré conjecture.
problem 4-dimensional smooth Poincaré conjecture.
method Outline of proof.
result Proof of 4D smooth Poincaré conjecture.
Study on smoothness of special algebra types.
problem Differential smoothness of bi-quadratic algebras with PBW basis.
method Investigation of algebra properties.
result Results on differential smoothness.
The study bounds and characterizes surfaces containing smooth conics and twistor fibers in a flag threefold.
problem Bounding and characterizing surfaces containing smooth conics and twistor fibers in a flag threefold.
method Analyzing the family of smooth conics and using algebraic properties to construct surfaces.
result The only smooth cases of surfaces containing infinitely many twistor fibers are of bidegree (1,1).
In this note we introduce the notion of a smooth structure on a conical pseudomanifold in terms of -rings of smooth functions on . For a finitely generated smooth structure we introduce the notion of the Nash tangent bundle, the Zariski tangent bundle, the tangent bundle of , and the …
The paper examines smoothness in graded skew Clifford algebras.
problem Smoothness of graded skew Clifford algebras.
method Investigation of differential smoothness.
result Results on the differential smoothness of graded skew Clifford algebras.
Kontsevich's classes distinguish smooth structures on fiber bundles.
problem Distinguishing smooth structures on fiber bundles.
method Using Kontsevich's characteristic classes and real blow-up construction.
result Kontsevich's classes are determined by the topology of the 2-point configuration space bundle.
Label smoothing improves model performance even with noisy labels.
problem Mitigating label noise in deep learning models.
method Examined label smoothing as a technique to cope with label noise and compared it to loss-correction methods.
result Label smoothing is competitive with loss-correction techniques under label noise and beneficial for distillation from noisy data.
Smooth K-Theorymath.KT
We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth K-orientation of a proper submersion and define the associated push-forward which satisfies functoriality, compatibility with pull-back diagrams, and projection and bordism formulas. We construct a multiplicati…
Theory of smooth relative connections on quiver bundles developed.
problem Existence of smooth relative connections over quiver bundles.
method Developed a theory over on smooth twisted quiver bundles, provided obstructions and necessary/sufficient conditions.
result Established a necessary and sufficient condition for the existence of smooth relative connections on tree-type quiver bundles.
New smooth models for string groups defined in ∞-categories.
problem Defining string group models in smooth spaces.
method Homotopy-theoretic definition using singular complex functor.
result New smooth models for the string group.
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.
We prove that any isometry between the unit spheres of -smooth (more generally, absolutely smooth) smooth Banach spaces extends to a linear isometry of the Banach spaces. This answers the famous Tingley's problem in the class of absolutely smooth -dimensional Banach spaces.
On Transverse Triangulationsmath.DG
We show that every smooth manifold admits a smooth triangulation transverse to a given smooth map. This removes the properness assumption on the smooth map used in an essential way in Scharlemann's construction [5].
Smooth actions of cyclic groups on 3-manifolds are conjugate to smooth ones.
problem Understanding smoothness in group actions on 3-manifolds.
method Finite cyclic group actions by -bilipschitz homeomorphisms on closed 3-manifolds.
result Finite cyclic group actions by -bilipschitz homeomorphisms on closed 3-manifolds are conjugate to smooth actions.
Confirming operator characterization on smooth manifolds.
problem Characterizing an operator on smooth sections of tangent bundles.
method Using algebraic axioms and H^1(M, R) = {0} assumption.
result Operator can be characterized universally for any smooth manifold.
Smooth Schrödinger Bridges improve trajectory inference by smoothing Gaussian processes.
problem Improving trajectory inference in applications like particle tracking.
method Generalizes Schrödinger Bridge problem to smooth Gaussian processes, solving the problem on phase space.
result The method outperforms existing methods on real datasets.
Solves smoothing problem in Chow's connectivity theorem.
problem Smoothing problem in Chow's connectivity theorem.
method Solution provided.
result Solves the smoothing problem in Chow's theorem.
Smoothness analysis of adversarial training reveals constraints cause more non-smoothness.
problem Non-smoothness of adversarial training loss function.
method Analyzed the smoothness of adversarial training loss function using optimal attacks for model parameters.
result The constraint causes more non-smoothness than constraint.
We study a non-parametric multi-armed bandit problem with stochastic covariates, where a key complexity driver is the smoothness of payoff functions with respect to covariates. Previous studies have focused on deriving minimax-optimal algorithms in cases where it is a priori known how smooth the payoff functions are. I…
Smooth contact maps are always smooth in rigid Carnot groups.
problem Smoothness of contact maps in rigid Carnot groups.
method Analyzing -rigid Carnot groups to show -contact maps are smooth.
result Smooth contact maps are always smooth in rigid Carnot groups.