Study spectral sequences in smooth generalized cohomology theories.
problem Systematic study of torsion in differential cohomology.
method Use Atiyah-Hirzebruch spectral sequences with filtrations by Cech resolutions.
result Explicit identification of differentials in spectral sequences for various theories.
S3Attention improves long sequence attention with smoothed skeleton sketching.
problem Quadratic complexity of vanilla Attention makes it unsuitable for long sequence tasks.
method S3Attention uses smoothing and matrix sketching to balance information preservation and computation. result S3Attention significantly outperforms vanilla Attention and other Attention variants. Impute missing events in continuous-time sequences using particle smoothing.
problem Missing events in continuous-time sequences.
method Particle smoothing with trainable bidirectional LSTM proposals.
result Imputed sequences have low Bayes risk compared to ground truth.
RS-Del provides robustness for sequence classifiers against edit distance attacks.
problem Certifying robustness of discrete sequence classifiers against edit distance attacks.
method Randomized deletion (RS-Del) for discrete sequence classifiers, focusing on edit distance-bounded adversaries.
result Achieved a certified accuracy of 91% at an edit distance radius of 128 bytes on malware detection.
Study smooth convergence of metric flows from F-limits.
problem Smooth convergence of F-limit flows. method Extensively studied metric flows and F-limits, showing smooth convergence at regular points. result Each regular point on the limit is a point of smooth convergence.
Transformers handle infinite dimensional inputs effectively by feature extraction and dynamic feature selection.
problem Understanding the approximation and estimation ability of Transformers with infinite dimensional inputs.
method Anisotropic smoothness analysis and feature extraction properties of Transformers.
result Transformers avoid the curse of dimensionality and dynamically select important features.
Conditional COT-GAN predicts sequences using past data and kernel smoothing.
problem Predicting sequences given past data.
method Conditional COT-GAN with kernel smoothing.
result Improved convergence results for sequence prediction.
We calculate the smooth structure set of Sp×Sq, S(p,q), for p,q≥2 and p+q≥5. As a consequence we show that in general S(4j−1,4k) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of forgetful map $F:…
Single-timescale analysis improves convergence in multi-sequence stochastic approximation.
problem Finite-time convergence of nonlinear stochastic approximation with multiple coupled sequences.
method Smoothness property of fixed points and analysis of fine-grained single-timescale SA.
result Improved iteration complexity for achieving ε-accuracy in multi-sequence single-timescale SA.
We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold. This extends the classical notion of asymptotic directions usually defined on smooth submanifolds. We get a simple expression of these cones for polyhedra in E^3, as well as converge…
Smooth conic Kähler metrics with uniform curvature bound.
problem Constructing smooth conic Kähler metrics with bounded curvature.
method Using upper bisectional curvature bound and choosing a new background metric.
result Constructing a smoothing sequence for conic metrics.
Generative Bridging Network improves sequence prediction models by penalizing confidence and smoothing language.
problem Data sparsity and overfitting in sequence prediction tasks.
method Introduces a bridge module to extend ground truth and minimize KL-divergence between bridge distribution and generator.
result Significant improvements over strong baselines in machine translation and text summarization tasks.
The abstract proves every knot type can be parametrized by smooth functions and studies limit knot types.
problem Understanding all knot types and their parametrizations.
method Proving every knot type can be represented by smooth functions and studying limit knot types.
result Every knot type can be parametrized by smooth functions and limit knot types exist.
Study confirms conjecture about Kähler metrics on smooth minimal models.
problem Behavior of constant scalar curvature Kähler metrics on smooth minimal models.
method Analysis of metrics in a neighborhood of the canonical class.
result Convergence to singular Kähler Einstein metric in the canonical class.
We construct a sequence of embedded minimal disks in a ball where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper.
Method approximates Lipschitz domains with smoother shapes.
problem Approximating bounded Lipschitz domains.
method Sequence of smooth, bounded domains with weak curvatures.
result Uniform isocapacitary estimates for approximating sets.
Proves existence of manifolds with Kervaire invariant one in specific dimensions.
problem Existence of smooth framed manifolds with Kervaire invariant one.
method Adams spectral sequence and combination of theorems.
result Smooth framed manifolds with Kervaire invariant one exist in dimensions 2, 6, 14, 30, 62, and 126.
Paper analyzes regret bounds for unconstrained online optimization.
problem Minimizing regret in dynamic online learning for strongly convex and smooth functions.
method Preconditioned OGD, Online Optimistic Newton (OON), multiple gradient queries.
result Achieves O(C2,T∗) regret bound with one gradient query per round. Modified flow smooths conical Kähler-Ricci problem.
problem Smooth approximation of modified conical Kähler-Ricci flow.
method Constructing a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.
result Long-time solution of the modified conical Kähler-Ricci flow.
This paper is devoted to the study of geometric structures modeled on homogeneous spaces G/P, where G is a real or complex semisimple Lie group and P⊂G is a parabolic subgroup. We use methods from differential geometry and very elementary finite-dimensional representation theory to construct sequences of invar…
Cubic spline smoothing improves interpolation between irregularly sampled data.
problem Interpolation discontinuity in recurrent neural networks for irregularly sampled sequences.
method Cubic spline smoothing compensation module trained end-to-end with ODE-RNN.
result Improves interpolation between irregularly sampled data points.
In the present work we study the behavior of sequences of smooth global isothermic immersions of a given closed surface and having a uniformly bounded total curvature. We prove that, if the conformal class of this sequence is bounded in the Moduli space of the surface, it weakly converges in W^{2,2} away from finitely …
In this paper, we give a lower bound of Bergman kernels for a sequence of almost Kähler-Einstein Fano manifolds, or more general, a sequence of Fano manifolds with almost Kähler-Ricci solitons. This generalizes a result by Donaldson-Sun, Tian for Kähler-Einstein manifolds sequence with positive scalar curvature. As an …
Study finite group actions on exotic aspherical space forms.
problem Classify finite group actions on M#Σ where M is a closed aspherical space form and Σ is an exotic n-sphere. method Combines geometric and topological rigidity results with smoothing theory and spectral sequence computations.
result Classification of free actions of finite groups on M#Σ when M is 7-dimensional. Study proves neck properties for smooth maps with bounded energy.
problem Understanding neck formation in smooth maps with bounded energy.
method Proved energy identity and no neck property for extrinsic polyharmonic maps.
result Established neck properties for maps with bounded total energy.
A regular n-gon inscribing a knot is a sequence of n points on a knot, such that the distances between adjacent points are all the same. It is shown that any smooth knot is inscribed by a regular n-gon for any n.
Given a construction of smooth homotopy class invariants of smooth immersions Mn→Rn+k. The particular case of k=1,n≥1 is a sequence of non-zero integrals, where the n=2 term is the Gauss-Bonnet integral
The study finds counterexamples to curvature estimates for minimizing surfaces.
problem Curvature estimates for minimizing surfaces in metric convergence.
method Constructing sequences of smooth minimizing surfaces in metrics converging to Euclidean.
result Found counterexamples with diverging L2 norm of second fundamental form. Constructs equivariant cohomology models for differentiable stacks.
problem Developing cohomology theory for stacks with group actions.
method Extends classical results for smooth manifolds to differentiable stacks.
result Derives spectral sequences generalizing Bott's spectral sequence.
The paper shows that certain Einstein orbifolds cannot be limits of smooth Einstein metrics.
problem Understanding the limits of smooth Einstein metrics on compact Einstein orbifolds.
method Analyzing sequences of compact Einstein manifolds and their limits, providing an explicit obstruction for certain orbifolds.
result Explicit obstruction for negative Einstein orbifolds appearing as limits of compact Einstein manifolds, which does not vanish for hyperbolic orbifolds.
We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $\barpartial$-Laplacian on compact Kähler manifolds. This situation typically occurs for a sequence of Fano manifolds with anticanonical Kähler class. We apply it to show that, if an almost smooth Fano-Ricci limit space admits a Kähler…
The paper improves smoothed analysis for online problems with adaptive adversaries.
problem Online prediction, discrepancy minimization, and online optimization with adaptive adversaries.
method General technique to prove smoothed guarantees against adaptive adversaries, reducing to simpler oblivious adversaries.
result Strong smoothed guarantees for three online problems, matching or improving previous results.
A new smooth edit distance for easier optimization in machine learning.
problem Hard optimization of edit distance for variable-length sequences.
method Soft edit distance (SED) as a differentiable approximation.
result SED can be optimized with gradient methods and used for clustering.
We analyze the limit of the p-form Laplacian under a collapse, with bounded sectional curvature and bounded diameter, to a smooth limit space. As an application, we characterize when the p-form Laplacian has small positive eigenvalues in a collapsing sequence.
Paper optimizes statistical estimation for randomized smoothing to reduce adversarial robustness certification time.
problem Efficiently estimating robustness of points against adversarial attacks.
method Developed estimation procedures using confidence sequences and randomized Clopper-Pearson intervals.
result Achieved optimal sample complexities and stronger certificates with reduced computational burden.
We analyze the signature type of a cascade of periodic orbits associated to period doubling renormalizable maps of the two dimensional disk. The signature is a sequence of rational numbers which describes how periodic orbits turn each other and is invariant by topological conjugacies that preserve orientation. We prove…
Given a sequence of properly embedded minimal surfaces in a 3-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…
If f is a smooth function on a Hodge manifold, we construct a canonical sequence of real algebraic functions that converge to f in the smooth topology. The definition of of the approximants is inspired by Berezin-Toeplitz quantization. The proof follows quickly from known results of Fine, Liu and Ma.
We investigate a generalization of the so-called metric splitting of globally hyperbolic space-times to non-smooth Lorentzian manifolds and show the existence of this metric splitting for a class of wave-type space-times. Our approach is based on smooth approximations of non-smooth space-times by families (or sequences…
In principle, Floer theory can be extended to define homotopy invariants of families of equivalent objects (e.g. Hamiltonian isotopic symplectomorphisms, 3-manifolds, Legendrian knots, etc.) parametrized by a smooth manifold B. The invariant of a family consists of a filtered chain homotopy type, which gives rise to a …
Solves logarithmic ∂-equation on Kähler manifolds with smooth divisors.
problem Closedness of logarithmic forms and injectivity theorems.
method Cyclic covering trick to solve ∂-equation.
result Unobstructed deformations for smooth divisors.
Proves a tropical version of Clemens-Schmid sequence for tropical varieties.
problem Existence of tropical cycles with specific cohomology classes.
method Tropical Hodge theory and Clemens-Schmid sequence analogy.
result Proves tropical Hodge conjecture for rationally triangulable varieties.
In this note, we show that the conical Kähler-Ricci flows introduced in \cite{CYW} exist for all time t∈[0,∞) in the weak sense. As a key ingredient of the proof, we show that a conical Kähler-Ricci flow is actually the limit of a sequence of smooth Kähler-Ricci flows.
Classifies smooth manifolds homotopy equivalent to sphere products
problem Classifying smooth manifolds homotopy equivalent to sphere products
method Using normal-invariant map and explicit families of manifolds
result Classifies smooth manifolds up to almost diffeomorphism
Nontrivial boundary Dehn twist found on K3#K3 manifold.
problem Proving nontriviality of a Dehn twist on a specific 4-manifold.
method Algebraic criterion and equivariant topological K-theory to show non-isotopy.
result Boundary Dehn twist is nontrivial in the smooth mapping class group.
A theory of differential characters is developed for manifolds with boundary. This is done from both the Cheeger-Simons and the deRham-Federer viewpoints. The central result of the paper is the formulation and proof of a Lefschetz-Pontrjagin Duality Theorem, which asserts that the pairing: Ch^k(X,dX) x Ch^{n-k-1}(X) --…
Constructs improperly embedded minimal disks with curvature issues.
problem Properly embedded minimal disks with curvature issues in a cylinder.
method Constructs a sequence of improperly embedded minimal disks with curvature blow-up at a single point.
result The disks are not properly embedded in open subsets of R^3 that exhaust R^3.
Paper proves convergence of warped product manifolds to a nonnegative scalar curvature limit.
problem Proving convergence of sequences of manifolds with nonnegative scalar curvature.
method Warped product manifolds with diverging circular fibers, proving convergence in W1,p sense. result Sequence converges to an extreme limit space with nonnegative scalar curvature.