Continuing our previous work (arXiv:1509.07981v1), we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, i…
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48 results for “global kernel estimates”
In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental soluti…
Distance and Kernel-Based Measures for Global and Local Two-Sample Conditional Distribution Testingstat.ME
Unified framework for global and local two-sample conditional distribution testing.
problem Testing equality of two conditional distributions.
method Distance and kernel methods, conditional U-statistics, local bootstrap.
result Developed reliable global and local tests.
Heat kernel estimates on manifolds with mixed boundary conditions.
problem Estimating heat kernels on manifolds with ends and mixed boundary conditions.
method Global harmonic function construction and -transform technique.
result Two-sided heat kernel estimates for Riemannian manifolds with mixed boundary conditions.
We propose a method for nonparametric density estimation that exhibits robustness to contamination of the training sample. This method achieves robustness by combining a traditional kernel density estimator (KDE) with ideas from classical -estimation. We interpret the KDE based on a radial, positive semi-definite ke…
The most direct way to express arbitrary dependencies in datasets is to estimate the joint distribution and to apply afterwards the argmax-function to obtain the mode of the corresponding conditional distribution. This method is in practice difficult, because it requires a global optimization of a complicated function,…
Sharp heat kernel estimates on manifolds lead to solutions of the Parabolic Anderson model.
problem Well-posedness and intermittency of solutions to the Parabolic Anderson model on Riemannian manifolds.
method Sharp global heat kernel bounds and geodesic comparison geometry.
result Upper and lower moment bounds for solutions of the Parabolic Anderson model on general compact Riemannian manifolds.
Improved Strichartz estimates for Schrödinger equation on manifolds with nonpositive curvature.
problem Improving Strichartz estimates for Schrödinger equation on compact manifolds with nonpositive sectional curvature.
method Improved global kernel estimates for microlocalized operators exploiting geometric assumptions.
result No-loss -estimates on intervals of length for all admissible pairs .
New quantum kernels avoid overfitting by combining local and global components.
problem Exponential concentration in quantum kernels leads to overfitting.
method Local-global quantum kernels combining small subsystem and full-system measurements.
result Demonstrated benign overfitting in local-global quantum kernels.
New method guarantees global convergence in variational inference.
problem Limited convergence to local optima in variational inference.
method Minimizes inclusive KL divergence using neural networks and neural tangent kernel.
result Gradient descent dynamics converge to a unique solution in function space.
We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Hölder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with b…
Paper shows DMS as an EM algorithm with improved convergence.
problem Improving the convergence of DMS algorithm.
method Shows DMS as a generalized EM algorithm and provides new proofs.
result Demonstrates global convergence and linear convergence of DMS.
Scalable Global Alignment Graph Kernel Using Random Features: From Node Embedding to Graph Embeddingcs.LG
Graph kernels are widely used for measuring the similarity between graphs. Many existing graph kernels, which focus on local patterns within graphs rather than their global properties, suffer from significant structure information loss when representing graphs. Some recent global graph kernels, which utilizes the align…
KSOS improves kernel learning for dynamical systems via global optimization.
problem Challenges in selecting optimal kernels and tuning parameters in traditional kernel-based methods.
method Global optimization framework with kernel-based surrogate functions.
result KSOS consistently outperforms gradient descent in predicting dynamical systems.
Unified analysis of kernel-based and locally adaptive bandit optimization methods.
problem Performance of bandit optimization algorithms in RKHS functions.
method Investigates the relationship between kernel regularity and algorithmic performance, characterizing spectral properties of various kernels.
result Unified framework for analyzing kernel-based and locally adaptive bandit algorithms, deriving explicit regret bounds.
Improved local feature attributions using neighbourhood reference distributions.
problem Misleading results from global population in local model behaviour.
method Formulation of neighbourhood reference distributions and self-normalised importance sampling.
result Neighbourhood Shapley values provide meaningful sparse feature attributions.
WS-KDE provides robust confidence bounds for stochastic functions.
problem Optimizing time-consuming black-box functions with stochastic outputs.
method Wilson Score Kernel Density Estimation (WS-KDE) for Bayesian optimization.
result WS-KDE provides reliable confidence bounds for any stochastic function.
New framework analyzes deep learning optimization with finite width networks, revealing generalization gaps and excess risks.
problem Analyzing generalization error of deep learning with finite width networks.
method Formulating neural network training as transportation map estimation and analyzing via infinite dimensional Langevin dynamics.
result Achieves fast learning rate and minimax optimal rates for classification and regression problems.
Finite Sample Complexity of Sequential Monte Carlo Estimators on Multimodal Target Distributionsstat.CO
New bounds for SMC show its advantage over MCMC in multimodal distributions.
problem Estimating expectations under multimodal distributions with slow global mixing.
method Proves finite sample complexities for SMC with local mixing times, addressing bias through sequential resampling.
result SMC provides fully polynomial time approximation for multimodal problems.
Study uses multi-kernel Hawkes models to analyze high-frequency price dynamics.
problem Understanding responsive speeds of market participants in high-frequency trading.
method Multi-kernel Hawkes models with conditional Hessian analysis for optimization.
result Existence of multi-kernels (UHF, VHF, HF) in high-frequency price dynamics.
We prove the statistical consistency of kernel Partial Least Squares Regression applied to a bounded regression learning problem on a reproducing kernel Hilbert space. Partial Least Squares stands out of well-known classical approaches as e.g. Ridge Regression or Principal Components Regression, as it is not defined as…
AEN-RBF kernel improves robustness in Bayesian optimization for complex systems.
problem Bayesian optimization struggles with outliers in RBF kernel, leading to poor performance.
method Proposes AEN-RBF kernel function, demonstrating improved robustness and convergence.
result The AEN-RBF kernel function reduces mean squared prediction error and improves convergence.
Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Ratesmath.ST
We establish optimal convergence rates for a decomposition-based scalable approach to kernel ridge regression. The method is simple to describe: it randomly partitions a dataset of size N into m subsets of equal size, computes an independent kernel ridge regression estimator for each subset, then averages the local sol…
Estimates Bergman kernel for Siegel varieties, focusing on geodesic distances.
problem Estimating the Bergman kernel for complex manifolds of Siegel varieties.
method Analyzes geodesic distances and derives estimates for the Bergman kernel.
result Derives estimates of the Bergman kernel for Siegel varieties.
Kernel Density Estimation is a very popular technique of approximating a density function from samples. The accuracy is generally well-understood and depends, roughly speaking, on the kernel decay and local smoothness of the true density. However concrete statements in the literature are often invoked in very specific …
Develop a comprehensive theory for regularized M-estimation in reproducing kernel Hilbert spaces.
problem Regularized M-estimation in reproducing kernel Hilbert spaces
method Existence and measurability of the estimator, sharp rates of convergence
result New rates for tensor product Sobolev spaces
Most state-of-the-art graph kernels only take local graph properties into account, i.e., the kernel is computed with regard to properties of the neighborhood of vertices or other small substructures. On the other hand, kernels that do take global graph propertiesinto account may not scale well to large graph databases.…
We consider multi-agent stochastic optimization problems over reproducing kernel Hilbert spaces (RKHS). In this setting, a network of interconnected agents aims to learn decision functions, i.e., nonlinear statistical models, that are optimal in terms of a global convex functional that aggregates data across the networ…
Gradient Estimates For The CR Heat Equation On Complete noncompact Pseudo-Hermitian Manifoldsmath.DG
Derives gradient estimates for CR heat equation on pseudo-Hermitian manifolds.
problem Estimating solutions to CR heat equation on complex manifolds.
method Local and global Li-Yau type gradient estimates.
result Gradient estimates and Harnack inequality for positive solutions.
Bayesian Complementary Kernelized Learning models complex spatiotemporal data.
problem Modeling complex, nonstationary, and nonseparable spatiotemporal data.
method Integrates kernelized low-rank tensor factorization and short-range spatiotemporal Gaussian Processes.
result BCKL offers superior performance in providing accurate posterior mean and high-quality uncertainty estimates.
Two-layer neural networks learn efficiently using kernel methods in mean-field analysis.
problem Feature learning ability of two-layer neural networks in the mean-field regime.
method Mean-field analysis through kernel methods, focusing on dynamics of the first layer's kernel.
result Two-layer neural networks can learn a union of multiple reproducing kernel Hilbert spaces more efficiently than kernel methods.
New method improves Gaussian process regression on complex, sparse point clouds.
problem Traditional Gaussian processes struggle with restricted domains and point clouds.
method Atlas Gaussian Processes (RC-AGPs) combining heat kernel and RBF kernels.
result RC-AGPs outperform existing methods in regression accuracy.
Deep learning with noisy gradient descent outperforms linear estimators in high dimensions.
problem Theoretical explanation of deep learning's superiority over linear methods.
method Theoretical analysis of excess risk of a deep learning estimator trained by noisy gradient descent.
result Deep learning achieves a faster learning rate than linear estimators, especially in high dimensions.
GLSKF improves tensor completion by capturing both global and local variations.
problem Tensor completion with missing entries, especially in data with spatial or temporal side information.
method Integrates smoothness-constrained low-rank factorization with a locally correlated residual process.
result GLSKF achieves superior performance and scalability on real-world datasets.
Paper proposes a new GPR-HS framework for accurate VCV estimation in global equity indices.
problem Accurate forecasting of Volatility-Covariance Matrix (VCV) for regulatory processes.
method Hybrid Gaussian Process Regression-Historical Simulation (GPR-HS) framework.
result GPR-HS framework achieves regulatory compliance and outperforms static VaR benchmarks.
Estimates Poisson kernel on negatively curved Hadamard manifolds.
problem Estimating the Poisson kernel on Hadamard manifolds with negative curvature.
method Using techniques from Anderson-Schoen for estimating positive harmonic functions in cones.
result Global upper and lower bounds for the Poisson kernel are derived.
Modal regression is aimed at estimating the global mode (i.e., global maximum) of the conditional density function of the output variable given input variables, and has led to regression methods robust against heavy-tailed or skewed noises. The conditional mode is often estimated through maximization of the modal regre…
Deep learning predicts drug prescriptions across global health records.
problem Predicting drug prescriptions in chronic disease patients.
method Adaptive cross-global attention graph kernel network with support vector machine.
result Model outperforms current methods in accuracy and interpretability.
New method finds global minima using function evaluations and kernel approximations.
problem Finding global minima of smooth functions with limited evaluations.
method Approximates the function using infinite sums of square smooth functions and solves the optimization problem with polynomial time complexity.
result Achieves optimal number of function evaluations with theoretical guarantees and nearly optimal convergence rate.
Two-layer ReLU networks outperform kernel methods in teacher-student settings.
problem Understanding the excess risk of two-layer ReLU neural networks in teacher-student models.
method Investigated a two-phase training process for a student network, comparing it to kernel methods.
result The student network reaches near-global optimality and outperforms kernel methods in minimax optimal rate.
New models exploit invariance to reduce model complexity.
problem Reducing model complexity for tasks with inherent invariances.
method Invariant random features and kernel methods.
result Exploiting invariance saves a factor in model complexity.
Let be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer , we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle , where is the holomorphic cotangent bundle of . Our first main result estimates the corresponding B…
Estimators of information theoretic measures such as entropy and mutual information are a basic workhorse for many downstream applications in modern data science. State of the art approaches have been either geometric (nearest neighbor (NN) based) or kernel based (with a globally chosen bandwidth). In this paper, we co…
Study shows how discrete graph curvature relates to manifold curvature.
problem Relating discrete graph curvature to intrinsic manifold curvature.
method Continuum limits of Ollivier's Ricci curvature on data clouds.
result Random geometric graphs inherit global curvature properties of manifolds.
Extends Mahalanobis distance to Banach spaces for anomaly detection.
problem Anomaly detection in infinite-dimensional spaces.
method Generalizes Mahalanobis distance to Banach spaces via Cameron-Martin norm and variance norm.
result Kernelized nearest-neighbour Mahalanobis distance outperforms traditional methods for time series novelty detection.
Analysis of large-scale sequential data has been one of the most crucial tasks in areas such as bioinformatics, text, and audio mining. Existing string kernels, however, either (i) rely on local features of short substructures in the string, which hardly capture long discriminative patterns, (ii) sum over too many subs…
New variational flows improve Monte Carlo and normalization tasks.
problem Intractable global optimum in expressive variational families.
method Constructing asymptotically exact variational flows from involutive MCMC kernels.
result Provable total variation convergence of new variational families.
Unified framework combines trace-induced quantum kernels for improved machine learning models.
problem Improving performance of quantum machine learning models using trace-induced kernels.
method Developed a unified framework combining various trace-induced quantum kernels, including global fidelity and local projected kernels, as Lego kernels.
result Local projected kernels can achieve comparable performance to global fidelity kernels with fewer quantum resources.