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.
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48 results for “Generalized Smoothness”
Smooth distributions on subcartesian spaces can be globally finitely generated.
problem Understanding smooth distributions on subcartesian spaces.
method Embedding in Euclidean space, Whitney Embedding Theorem, and distribution theory.
result Smooth generalized distributions and subbundles on connected subcartesian spaces are globally finitely generated.
Label smoothing improves generalization by controlling generalization loss.
problem Lack of mathematical understanding of label smoothing's effectiveness.
method Proposed a theoretical framework to show how label smoothing controls generalization loss in the label noise setting.
result Predicted an optimal label smoothing point that minimizes generalization loss.
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.
A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show …
Generic smooth boundaries for isoperimetric regions in 8D manifolds.
problem Understanding boundaries of isoperimetric regions in high-dimensional spaces.
method Generic regularity results for isoperimetric regions in closed Riemannian manifolds of dimension eight.
result Smooth nondegenerate boundaries for isoperimetric regions for generic metrics and volumes.
New algorithm tackles nonconvex machine learning problems with adaptive normalization and independent sampling.
problem Nonconvex machine learning problems with generalized-smoothness.
method Adaptive gradient normalization, independent sampling, and gradient clipping.
result Achieves an O(ε^(-4)) sample complexity for fast convergence.
New neural network smoothness constraints improve model performance.
problem Improving model sensitivity to input changes for better generalization and robustness.
method Exploring current smoothness constraints and proposing new flexible definitions.
result Current smoothness constraints lack flexibility and understanding of data, tasks, and learning.
Study shows area-minimizing submanifolds are mostly rough, not smooth.
problem Understanding the smoothness of area-minimizing submanifolds.
method Proved non-smoothness by contradiction and established Hausdorff dimension bounds.
result Area-minimizing submanifolds are not generically smooth, resolving a conjecture.
Smooth even solutions found for a generalized convex geometry problem.
problem Dual Orlicz-Minkowski problem in convex geometry.
method Geometric flow involving Gauss curvature and normal vectors.
result Existence of smooth even solutions for smooth even measures.
New structures allow for self-crossing singularities, leading to new families of stable generalized complex manifolds.
problem Stable generalized complex structures in higher dimensions with self-crossing singularities.
method Extending stable generalized complex structures to include anticanonical sections with normal self-crossings.
result Construction of large families of stable generalized complex manifolds in four dimensions.
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.
We introduce non-smooth symplectic forms on manifolds and describe corresponding Poisson structures on the algebra of Colombeau generalized functions. This is achieved by establishing an extension of the classical map of smooth functions to Hamiltonian vector fields to the setting of non-smooth geometry. For mildly sin…
A new method for generating samples without training, using smoothed score matching.
problem Generating samples efficiently and without training.
method Moment-matched score-smoothed overdamped Langevin dynamics (MM-SOLD).
result The method enables fast, robust, training-free sampling with competitive sample fidelity and diversity.
Pairwise Label Smoothing improves deep model generalization by reducing overconfidence.
problem Improving deep model generalization through regularization.
method PLS smooths labels for pairs of samples, learning distribution mass during training.
result PLS significantly outperforms LS and baseline models, reducing up to 30% classification error.
The study examines differential smoothness in specific Artin-Schelter regular algebras of dimension 5.
problem Investigating the differential smoothness of Artin-Schelter regular algebras of dimension 5.
method Analyzing the relationship between the number of generators and Gelfand-Kirillov dimension to identify structural obstructions.
result Certain two- and four-generator AS-regular algebras of global dimension five fail to admit a differential calculus, while a five-generator graded Clifford algebra provides a positive example.
We prove that Riemannian metrics with a uniform weak norm can be smoothed to having arbitrarily high regularity. This generalizes all previous smoothing results. As a consequence we obtain a generalization of Gromov's almost flat manifold theorem. A uniform Betti number estimate is also obtained.
Study smoothings of singular intersections of ellipsoids.
problem Smooth singular intersections of ellipsoids.
method Generic singularities of coaxial intersections of ellipsoids studied.
result Special attention to 3D case.
The paper smooths out equations on 4-manifolds to show smooth moduli spaces.
problem Understanding the structure of solutions to Vafa-Witten equations on 4-manifolds.
method Constructing perturbation terms to ensure transversality and showing moduli spaces are smooth.
result For generic perturbation, the general part of the moduli space is a smooth 0-dimensional manifold.
Proves a generalized vanishing theorem for quasi-smooth stacks, with applications in K-theory and birational geometry.
problem Vanishing theorems for quasi-coherent sheaves on derived blow-ups of quasi-smooth stacks.
method Derived blow-ups, intrinsic blow-up theory, Kiem-Li-Savvas blow-up theory, virtual localization theorem, desingularization theorem, resolution of diagonal.
result Generalized vanishing theorem for quasi-coherent sheaves on derived blow-ups of quasi-smooth stacks.
We consider spectral sequences in smooth generalized cohomology theories, including differential generalized cohomology theories. The main differential spectral sequences will be of the Atiyah-Hirzebruch (AHSS) type, where we provide a filtration by the Cech resolution of smooth manifolds. This allows for systematic st…
Computes invariant for smooth h-cobordisms families, proving duality and vanishing theorems.
problem Computing invariants for smooth h-cobordisms families.
method Using Dwyer, Weiss, and Williams work, fiberwise generalized Morse function, fiberwise Poincaré--Hopf theory.
result Duality theorem for smooth structure class, vanishing theorem for Rigidity Conjecture.
Study of filtering and smoothing in submanifolds of Euclidean space.
problem Filtering and smoothing in continuous-discrete time on submanifolds.
method Formal expressions and projection approach for prediction and smoothing.
result Agreement with classical results for prediction, differences for smoothing.
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…
In statistical learning theory, convex surrogates of the 0-1 loss are highly preferred because of the computational and theoretical virtues that convexity brings in. This is of more importance if we consider smooth surrogates as witnessed by the fact that the smoothness is further beneficial both computationally- by at…
Minimal surfaces in 8D smooth and nondegenerate.
problem Generic regularity of minimal hypersurfaces in 8D.
method Analysis of -generic metrics.
result All minimal hypersurfaces are smooth and nondegenerate.
SFM generates smooth functional data without exposing real data.
problem Challenges in statistical analysis of functional data.
method Copula framework and smooth flow construction.
result SFM produces high-quality synthetic functional data.
Hypergraph is a general way of representing high-order relations on a set of objects. It is a generalization of graph, in which only pairwise relations can be represented. It finds applications in various domains where relationships of more than two objects are observed. On a hypergraph, as a generalization of graph, o…
Smooth fractal trees via analytic generators, preserving combinatorial and geometric properties.
problem Constructing smooth fractal trees from discrete models.
method Using analytic generator fields to integrate smooth vector fields in an internal state space, generating geometric curves as projections of generator trajectories.
result Analytic generators can represent any discrete tree specification and preserve the asymptotic limit geometry.
The generalization and learning speed of a multi-class neural network can often be significantly improved by using soft targets that are a weighted average of the hard targets and the uniform distribution over labels. Smoothing the labels in this way prevents the network from becoming over-confident and label smoothing…
By means of a general gluing and conformal-deformation construction, we prove that any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannan manifold without boundary. Applications to Sobolev spaces, Nash embedding and loca…
Generalizes smoothness conditions for optimization methods.
problem Optimization under non-uniform smoothness conditions.
method Develops a new analysis technique for bounding gradients.
result Obtains convergence rates for gradient descent and Nesterov's method.
Decomposes smooth manifolds into algebraic submanifolds.
problem Understanding the structure of smooth manifolds induced by continuous selections.
method Generic continuous selection of smooth functions provides stratification of the manifold.
result Stratification leads to local topological structure with nondegenerate critical points.
We prove the existence of a local smooth Levi decomposition for smooth Poisson structures and Lie algebroids near a singular point. In the appendix of this paper, we show an abstract Nash-Moser normal form theorem, which generalizes our Levi decomposition result and which may be helpful in the study of other smooth nor…
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.
Paper tackles over-smoothing in deep GCNs, proposing DropEdge to improve performance.
problem Over-smoothing reduces expressivity in deep GCNs, especially affecting node classification.
method Theoretical analysis of GCN behavior with depth, proposing DropEdge to alleviate over-smoothing.
result DropEdge improves performance on various GCNs, shallow and deep.
This paper is a short version of some joint work with Stefan Haller. It describes the structure of "smooth manifold with corners" on the space of possibly broken instantons and on the completion of unstable manifolds of a generic smooth vector field. The result is stated in Theorem 1.4.
Topologies on algebraic and equational theories are used to define germ determined, near-point determined, and point determined rings of smooth functions, without requiring them to be finitely generated. It is proved, that any commutative algebra morphism (without requiring continuity) between near-point determined rin…
New bounds explain deterministic non-smooth deep nets without large Lipschitz constants.
problem Challenges in explaining generalization of deterministic non-smooth deep nets.
method De-randomized PAC-Bayes margin bounds for deterministic non-convex and non-smooth predictors.
result New bounds avoid large Lipschitz constants, providing generalization guarantees.
Generic smooth minimal hypersurfaces exist in 8D manifolds.
problem Existence of smooth minimal hypersurfaces in high-dimensional manifolds.
method Global perturbation argument and a novel geometric invariant.
result Generic metrics on 8D manifolds admit smooth minimal hypersurfaces.
Linear Microbundlesmath.DG
Extends calculus to topological manifolds using generalized functions.
problem Proving the existence of non-singular generalized tangent vector fields on spheres.
method Develops a theory of generalized functions and applies it to continuous maps between topological spaces.
result Shows coherence between non-existence of smooth vector fields on spheres and existence of generalized ones.
Study on optimal rates for sequential probability assignment using smoothed analysis.
problem Optimal rates for sequential probability assignment under smoothed adversaries.
method General-purpose reduction from minimax rates to transductive learning, development of an efficient algorithm using MLE oracle.
result Optimal (logarithmic) fast rates for parametric and finite VC dimension classes, sublinear regret for general classes.
Proves existence of smooth metrics with specific curvature properties.
problem Existence of smooth metrics with prescribed negative Ricci curvature.
method Formulated and proved for general domains in Euclidean space.
result Existence of smooth complete conformal metrics with prescribed negative Ricci curvature.
Improves safety region certification for smoothed classifiers without changing smoothing scheme.
problem Certified safety regions for smoothed classifiers are often small compared to optimal.
method Generalizes certified radius calculation as nested optimization problem, uses 0th-1st order information, and designs efficient estimators.
result Certified safety regions are significantly larger than current methods, achieving significant improvements on various metrics.
Study smooth loops and loop bundles, relating to -structures.
problem Properties of smooth loops and their applications.
method Analyze smooth loops, introduce loop bundles, define torsion and curvature.
result Showed how loop bundles relate to -structures.
Derandomizing PAC-Bayes bounds for smooth loss functions
problem Derandomizing PAC-Bayes bounds for smooth loss functions
method Exploiting smoothness properties of both the loss and the predictor class
result Bounds for deterministic predictors that involve flatness quantities
Optimal smooth subspaces approximate large data sets efficiently.
problem Approximating large data sets with invariant subspaces.
method Smooth functions under lattice translations or crystallographic groups, with optimal selection of Paley-Wiener space.
result Optimal lattice selection enhances approximation efficiency.
Generalizes Thurston's jiggling lemma for piecewise smooth solutions.
problem Creating piecewise smooth solutions of differential relations without homotopical assumptions.
method Jiggling arbitrary sections of to construct solutions of .
result Generalization of Thurston's lemma for piecewise smooth solutions of differential relations.