arXiv research

The paper constructs optimal confidence bands for kernel gradient flow estimators.

Deep belief networks can approximate any multivariate density with binary hidden units.

Gaussian process (GP) regression is a powerful interpolation technique due to its flexibility in capturing non-linearity. In this paper, we provide a general framework for understanding the frequentist coverage of point-wise and simultaneous Bayesian credible sets in GP regression. As an intermediate result, we develop…

Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.

Study heat profiles and eigenfunctions using Brownian motion.

Deep learning method proves convergence for solving HJI equations.

New algorithms for efficient return distribution approximation in reinforcement learning.

Quantum neural networks approximate periodic functions more efficiently.

A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and othe…

Proves continuum limits of Lipschitz learning using Γ-convergence.

Sharp lower bound for Hodge Laplacian on Kähler hyperbolic manifolds.

We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it pr…

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincaré metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the p…

We prove that for a so-called sticky process $S$ there exists an equivalent probability $Q$ and a $Q$-martingale $\tilde{S}$ that is arbitrarily close to $S$ in $L^p(Q)$ norm. For continuous $S$, $\tilde{S}$ can be chosen arbitrarily close to $S$ in supremum norm. In the case where $S$ is a local martingale we may choo…

The study analyzes the performance of a nonparametric estimator for dynamical systems.

Let $X$ be an infinite Riemann surface equipped with its conformal hyperbolic metric such that the action of the covering group $π_1(X)$ on $\tilde{X}$ is of the first kind-i.e., the surface $X$ is equal to its convex core. We first prove that any geodesic lamination on $X$ is nowhere dense. Given a fixed geodesic pant…

Study identifies numerical signs of blow-up in hydrodynamic equations.

We consider 2-dimensional orientable self-shrinkers $Σ$ for the Mean Curvature Flow of polynomial volume growth immersed in $\mathbb R^n$. We look at closed one forms minimizing the norm $\int_Σ\eterm |ω|^2$ in their cohomology class. Any closed form satisfying the Euler-Lagrange equation for this minimization will be …

In machine learning, we are given a dataset of the form $\{(\mathbf{x}_j,y_j)\}_{j=1}^M$, drawn as i.i.d. samples from an unknown probability distribution $μ$; the marginal distribution for the $\mathbf{x}_j$'s being $μ^*$. We propose that rather than using a positive kernel such as the Gaussian for estimation of these…

Deep residual networks can approximate any continuous function using control theory.

This paper presents a method to compute the {\it quasi-conformal parameterization} (QCMC) for a multiply-connected 2D domain or surface. QCMC computes a quasi-conformal map from a multiply-connected domain $S$ onto a punctured disk $D_S$ associated with a given Beltrami differential. The Beltrami differential, which me…

New adapted renormalized volume for hyperbolic 3-manifolds with compressible boundary.

The automation of posterior inference in Bayesian data analysis has enabled experts and nonexperts alike to use more sophisticated models, engage in faster exploratory modeling and analysis, and ensure experimental reproducibility. However, standard automated posterior inference algorithms are not tractable at the scal…

New bounds on machine learning model generalization error moments.

GANs learn distributions well from samples, with rates depending on intrinsic dimension.

New bounds tighten the generalization error of Gibbs algorithm.

Study loop corrections in random feature models affecting training and test errors.

Efficient classifier error estimation without re-training.

Study shows how classifiers can approach Bayes error in high-dimensional settings.

Confidence measures for the generalization error are crucial when small training samples are used to construct classifiers. A common approach is to estimate the generalization error by resampling and then assume the resampled estimator follows a known distribution to form a confidence set [Kohavi 1995, Martin 1996,Yang…

Full-batch GD achieves generalization close to any stationary point with fewer assumptions.

The paper analyzes CycleGAN's error components for unpaired data generation.

This research analyzes the error convergence rate of GAN models.

Study shows infoGAN's generalization error bound for two-layer networks.

Paper analyzes Gibbs and Langevin Monte Carlo for interpolation regime, showing generalization from low errors.

SGD reduces test error by decorrelating updates.

Random forest training yields confidence intervals for generalization error.

First order discretizations of Langevin diffusion can achieve better generalization error with additional smoothness assumptions.

The success of deep learning has led to a rising interest in the generalization property of the stochastic gradient descent (SGD) method, and stability is one popular approach to study it. Existing works based on stability have studied nonconvex loss functions, but only considered the generalization error of the SGD in…

Randomly sampled interpolators achieve zero generalization error with enough data.

New bounds study class-specific generalization error in machine learning.

Paper calculates the exact error of LDA models.

Method estimates LLM error rates using Pareto optimization.

Modified training direction reduces generalization error in neural networks.

Unified error analysis for discrete flow models.

A form of generalisation error known as Off Training Set (OTS) error was recently introduced in [Wolpert, 1996b], along with a theorem showing that small training set error does not guarantee small OTS error, unless assumptions are made about the target function. Here it is shown that the applicability of this theorem …

The paper examines prediction and estimation risks of ridgeless least squares under general error assumptions.

Analyzes generalization error in distributed linear regression.