arXiv research

In this paper, we solve a logarithmic $\bar{\partial}$-equation on a compact Kähler manifold associated to a smooth divisor by using the cyclic covering trick. As applications, we discuss the closedness of logarithmic forms, injectivity theorems and obtain a kind of degeneration of spectral sequence at $E_1$, and we al…

We generalize Cartan's logarithmic derivative of a smooth map from a manifold into a Lie group $G$ to smooth maps into a homogeneous space $M=G/H$, and determine the global monodromy obstruction to reconstructing such maps from infinitesimal data. The logarithmic derivative of the embedding of a submanifold $Σ\subset M…

This work improves policy evaluation and selection using logarithmic smoothing for pessimistic off-policy estimation.

Solutions near infinity to special Lagrangian equations are asymptotic to quadratic polynomials with logarithmic terms.

We shall introduce the notion of $C^\infty$ logarithmic symplectic structures on a differentiable manifold which is an analog of the one of logarithmic symplectic structures in the holomorphic category. We show that the generalized complex structure induced by a $C^\infty$ logarithmic symplectic structure has unobstruc…

New bounds for online portfolio selection without smoothness assumptions.

We develop a new theoretical framework, the \emph{envelope complexity}, to analyze the minimax regret with logarithmic loss functions and derive a Bayesian predictor that adaptively achieves the minimax regret over high-dimensional $\ell_1$-balls within a factor of two. The prior is newly derived for achieving the mini…

Fix a smooth projective curve over a field of characteristic zero and a finite set of punctures. Let G be a connected linear algebraic group. We prove that the moduli of G-bundles with logarithmic connections having fixed residue classes at the punctures is an algebraic stack of finite type.

Proves a Baum--Bott formula for foliations by curves with logarithmic terms.

New algorithm for sequential off-policy learning improves performance over batch methods.

New method tackles bilevel optimization with polyhedral constraints.

In this note we study logarithmic transformations in the sense of differential topology on two fibers of the Hopf surface. It is known that such transformations are susceptible to yield exotic smooth structures on four-manifolds. We will show here that this is not the case for the Hopf surface, all integer homology Hop…

A new subdivision scheme for Heisenberg group values with central smoothness loss.

Round handles are affiliated with smooth 4-manifolds in two major ways: 5-dimensional round handles appear extensively as the building blocks in cobordisms between 4-manifolds, whereas 4-dimensional round handles are the building blocks of broken Lefschetz fibrations on them. The purpose of this article is to shed more…

Study Higgs bundles on curves with punctures, extending spectral correspondence.

Logarithmic corrections to Price's law near black hole event horizon.

Analyzes structure of log smooth pairs when equality holds in Bogomolov-Gieseker inequality.

This work removes logarithmic singularities from hyperboloidal initial data without creating new ones.

LMC algorithm converges to target in Chi-squared and Renyi divergence.

In the previous article we derived a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with conic singularities. In this article we investigate how the terms in the expansion reflect the geometry of the manifold. Since the general expansion contains a logarithmi…

The paper challenges the smooth null infinity model by constructing counter-examples and showing non-smoothness of null infinity.

Deep ReLU networks can approximate and learn smooth functions efficiently.

Study local features of decorated representation spaces for spherical surfaces.

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic …

New estimator achieves minimax optimal risk in transfer learning.

New Dynkin condition for manifolds with boundary yields bi-Lipschitz equivalence and spectral properties.

Improved sampling for high-dimensional posteriors with underdamped Langevin.

We prove the sharp local L^1 - L^\infty smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known L^p - L^\infty estimate for p larger than 1. It also has several applications in geometry, providing …

Study on lengths and curvatures of harmonic functions on smooth and singular surfaces.

Improved sampling from high-dimensional Gaussians using smoothed scores.

Given a smooth manifold $M$ equipped with a properly and discontinuous smooth action of a discrete group $G$, the nerve $M_{\bullet}G$ is a simplicial manifold and its vector space of differential forms $\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right)$ carry a $C_{\infty}$-algebra structure $m_{\bullet}$. We sh…

Unified analysis of online optimization with self-concordant barriers, improving regret bounds.

Study on optimal rates for sequential probability assignment using smoothed analysis.

Let $(X, D)$ be a logarithmic pair, and let $h$ be a singular metric on the tangent bundle, smooth on the open part of $X$. We give sufficient conditions on the curvature of $h$ for the logarithmic and the standard cotangent bundles to be big. As an application, we give a metric proof of the bigness of logarithmic cota…

We develop a general theory of log spaces, in which one can make sense of the basic notions of logarithmic geometry, in the sense of Fontaine-Illusie-Kato. Many of our general constructions with log spaces are new, even in the algebraic setting. In the differentiable setting, our theory yields a framework for treating …

Modified ReLU networks improve regression estimation rates.

This work improves the convergence theory of diffusion models for generating samples from complex distributions.

The logarithmic Riemann surface Sigma_{log} is a classical holomorphic 1-manifold. It lives into R^4 and induces a covering space of C - 0 defined by exp. This paper suggests a geometric construction of it, derived as the limit of a sequence of vector fields extending exp suitably to embeddings of C into R^3, which tur…

This paper investigates the nonparametric regression problem using SVMs with anisotropic Gaussian RBF kernels. Under the assumption that the target functions are resided in certain anisotropic Besov spaces, we establish the almost optimal learning rates, more precisely, optimal up to some logarithmic factor, presented …

We use the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formu…

In this paper, we will establish an elliptic local Li-Yau gradient estimate for weak solutions of the heat equation on metric measure spaces with generalized Ricci curvature bounded from below. One of its main applications is a sharp gradient estimate for the logarithm of heat kernels. These results seem new even for s…

The article calculates the $\mathbb{F}$-convergence rate for Ricci flows with closed and smooth tangent flows.

Parallel-in-time solver reduces ODE simulation time from linear to logarithmic.

Bayesian methods estimate regression functions on submanifolds using graph Laplacian eigenbasis.

Adaptive smooth non-stationary bandits achieve optimal regret rates without knowing parameters.

The paper extends entropy formulas to super Ricci flows on metric measure spaces.

The paper provides new gradient estimates for solutions to a nonlinear elliptic equation on smooth metric measure spaces.

The paper studies a 1D diffusion equation with nonlinear Robin boundary conditions and finds conditions for global and finite time blow-up or blow-down.