Optimal kernel improves estimation accuracy in modal statistical methods.
problem Estimation accuracy of kernel-based modal statistical methods depends on the kernel used.
method The study theoretically shows an optimal kernel that minimizes asymptotic error criterion.
result An optimal kernel minimizes the error criterion when using an optimal bandwidth.
Optimal kernel in KR can be data-dependent, improving model performance.
problem Fixed kernel in KR limits model performance.
method Considered data-dependent kernels for KR, using posterior covariance.
result Data-dependent kernel choice leads to optimal performance.
The success of kernel methods has initiated the design of novel positive semidefinite functions, in particular for structured data. A leading design paradigm for this is the convolution kernel, which decomposes structured objects into their parts and sums over all pairs of parts. Assignment kernels, in contrast, are ob…
Efficiently searches through Gaussian process kernels using symbolic representation and Bayesian optimization.
problem Manual selection of kernels in Gaussian processes is complex and computationally expensive.
method Proposes a novel method using symbolic representation and Bayesian optimization to search through a structured kernel space.
result Empirically shows a computationally more efficient way of searching through a discrete kernel space.
Kernel methods linked to feature subspaces and maximal correlation kernels.
problem Understanding kernel methods and their relationship to feature extraction.
method Established a correspondence between feature subspaces and kernels, introduced maximal correlation kernels, and demonstrated their optimality.
result Kernel SVM on maximal correlation kernel achieves minimum prediction error.
NOT learns optimal transport plans, kernel costs improve performance.
problem NOT algorithm learns non-optimal plans with weak quadratic costs.
method Introduced kernel weak quadratic costs to improve NOT's performance.
result Kernel costs provide improved theoretical and practical guarantees.
Unified framework for optimal kernel tests across MMD, HSIC, and KSD.
problem Optimal testing in kernel-based hypothesis testing frameworks.
method Unified derivation of minimax rates, adaptive kernel selection methods.
result Unified power results across MMD, HSIC, and KSD.
We focus on kernel methods for set-valued inputs and their application to Bayesian set optimization, notably combinatorial optimization. We investigate two classes of set kernels that both rely on Reproducing Kernel Hilbert Space embeddings, namely the ``Double Sum'' (DS) kernels recently considered in Bayesian set opt…
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.
The paper constructs optimal confidence bands for kernel gradient flow estimators.
problem Estimating generalization error and constructing confidence bands for kernel gradient flows.
method Established convergence rates and constructed optimal confidence bands under capacity-source condition.
result Optimal confidence bands for kernel gradient flows have shrinkage rates close to minimax optimal rates.
New algorithm optimizes tessellated kernels for larger datasets and improved performance.
problem Limited accuracy and complexity in machine learning algorithms based on kernel optimization.
method 2-step algorithm for optimizing tessellated kernels, scaling to 10,000 data points and extending to regression.
result Significant improvement in performance over Neural Nets and SimpleMKL with similar computation time.
We analyzed optimism in linear and kernel regression models.
problem Understanding predictive complexity in regression models.
method Derived closed-form asymptotic optimism for linear and kernel regression models.
result Scaled optimism is a useful measure for model complexity.
This note optimizes distributions using kernel mean embeddings with a new parameterization.
problem Optimizing distributions using kernel mean embeddings is challenging due to the difficulty of characterizing probability distribution vectors.
method Proposes a new parameterization of positive functions using kernel sums-of-squares to fit distributions in the MMD geometry.
result Distributions with kernel sum-of-squares densities are dense in the MMD geometry, allowing optimization in the finite-sample setting.
Characterizes kernel interpolation in large dimensions, revealing optimal and sub-optimal regions.
problem Understanding the phase diagram of kernel interpolation in large dimensions.
method Characterization of variance and bias under various source conditions.
result Determined the (s,γ)-phase diagram of large-dimensional kernel interpolation. Automates GPU kernel optimization for diverse applications.
problem Lack of systematic evaluation for multi-scenario GPU kernel optimization.
method Introduces MSKernelBench and CUDAMaster for automated optimization.
result Significant speedups across various operators, outperforming existing tools.
Kernels for structured data are commonly obtained by decomposing objects into their parts and adding up the similarities between all pairs of parts measured by a base kernel. Assignment kernels are based on an optimal bijection between the parts and have proven to be an effective alternative to the established convolut…
New algorithm reduces kernel optimization complexity.
problem Efficiently learn semiseparable kernels for robust machine learning.
method Proposed a SVD-QCQP primal-dual algorithm for minimax optimization.
result Significant improvement in accuracy over non-convex approaches.
A new kernel for probability measures based on optimal transport.
problem Efficiently comparing and modeling distributions.
method Kernel over probability measures using regularized optimal transport and Hilbertian embedding.
result The proposed kernel enables Gaussian process modeling on distributions with theoretical and computational advantages.
A new method optimizes MMD test power by dynamically selecting kernels, overcoming traditional trade-offs.
problem Fixed kernels fail to distinguish certain distributions, leading to overfitting and variance collapse.
method Complexity-Penalized MMD (CP-MMD) criterion, derived from concentration inequality, optimizes kernel selection.
result CP-MMD maximizes true test power while ensuring unconditional Type-I validity, matching or exceeding state-of-the-art performance.
Kernel semi-implicit variational inference improves variational inference without additional optimization.
problem Intractability of hierarchical semi-implicit distributions in variational inference.
method Kernel semi-implicit variational inference (KSIVI) using kernel methods to eliminate lower-level optimization.
result KSIVI reduces variational inference to kernel Stein discrepancy (KSD) optimization, improving expressiveness and tractability.
Bayesian optimization gains efficiency by leveraging symmetries through a modified max kernel.
problem Improving Bayesian optimization efficiency for functions with group symmetries.
method Developed a PSD projection of the max kernel to exploit symmetries without violating kernel properties.
result The modified max kernel achieves lower regret compared to existing invariant and non-invariant kernels.
New method for spot volatility estimation with reduced microstructure noise.
problem Estimating spot volatility from noisy high-frequency data.
method Pre-averaging/kernel estimator to handle microstructure noise.
result Optimal bandwidth selection and kernel functions for minimal variance.
Paper studies distributed kernel regression with imperfect kernels, achieving optimal rates.
problem Optimal rates of distributed regression with imperfect kernels.
method Divide and conquer approach, response weighted base algorithms, leave one out analysis, bias correction.
result Achieves capacity independent optimal rates for distributed kernel regression with imperfect kernels.
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.
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.
Unified analysis for nonlinear parametric models in Bayesian optimization.
problem Limited theoretical guarantees for nonlinear parametric models in Bayesian optimization.
method Kernel-based framework for analyzing regularized nonlinear parametric models trained on adaptively collected data.
result Unified convergence guarantees for nonlinear acquisition and surrogate models.
New method uses multiple kernels to improve SVGD performance.
problem Sub-optimal performance of single kernel in SVGD.
method Combines multiple kernels to approximate optimal kernel, using Kernelized Stein Discrepancy (KSD) and constructing Multiple Kernel SVGD (MK-SVGD).
result Consistently matches or outperforms competing methods in experiments.
This paper uses MIO to select features for kernel SVM classification.
problem Feature selection for kernel SVM classification.
method Mixed-integer optimization (MIO) for feature subset selection.
result The MIO approach can often outperform linear-SVM-based methods in prediction performance.
Kernel DRO uses RKHS to optimize under distributional uncertainty.
problem Optimizing under distributional uncertainty with limited knowledge.
method Kernel DRO using RKHS ambiguity sets and duality theory.
result Unified approach to robust and stochastic optimization.
This work addresses two main issues of the standard Kernel Entropy Component Analysis (KECA) algorithm: the optimization of the kernel decomposition and the optimization of the Gaussian kernel parameter. KECA roughly reduces to a sorting of the importance of kernel eigenvectors by entropy instead of by variance as in K…
This paper studies the optimality of kernel methods in high-dimensional data clustering. Recent works have studied the large sample performance of kernel clustering in the high-dimensional regime, where Euclidean distance becomes less informative. However, it is unknown whether popular methods, such as kernel k-means, …
Optimizes kernel discrepancies by selecting subsets efficiently.
problem Improving kernel discrepancies for QMC methods.
method Introduces a novel subset selection algorithm for kernel discrepancies.
result Efficiently generates low-discrepancy samples from various distributions.
Constructing the adjacency graph is fundamental to graph-based clustering. Graph learning in kernel space has shown impressive performance on a number of benchmark data sets. However, its performance is largely determined by the chosen kernel matrix. To address this issue, the previous multiple kernel learning algorith…
New research optimizes HSIC estimation rate for translation-invariant kernels.
problem Optimizing the rate of HSIC estimation for translation-invariant kernels.
method Proved minimax optimal rate of O(n−1/2) for HSIC estimation. result Optimality of various HSIC estimators proven.
Despite their success, kernel methods suffer from a massive computational cost in practice. In this paper, in lieu of commonly used kernel expansion with respect to N inputs, we develop a novel optimal design maximizing the entropy among kernel features. This procedure results in a kernel expansion with respect to en…
Improves learning of spectral mixture kernels with approximate Bayesian inference.
problem Difficult optimization of large number of SM kernel parameters.
method Approximate Bayesian inference using variational distribution of spectral points and random Fourier features.
result Accelerates convergence and leads to better optimal parameters.
Laplace kernel feature selection offers statistical guarantees for nonparametric models with few samples.
problem Statistical guarantees for kernel-based feature selection in nonconvex optimization problems.
method Sharp characterization of the gradient of the objective function for Laplace kernel feature selection.
result Model-selection consistency for Laplace kernel-based feature selection in nonparametric settings with n∼logp samples. LIBO optimizes repeated bandit tasks without prior knowledge or regret.
problem Optimizing repeated bandit tasks without prior knowledge or regret.
method LIBO sequentially meta-learns a kernel to adapt to the environment and solve tasks with the latest estimate.
result LIBO achieves sublinear lifelong regret, converging to oracle performance as more tasks are solved.
Optimizes differentially private kernel learning with random projection.
problem Privacy-preserving learning algorithms with optimal performance.
method Differentially private kernel ERM algorithm based on random projection in reproducing kernel Hilbert space.
result Achieves minimax-optimal excess risk rates for various loss functions.
OKSVM optimizes RBF kernel hyperparameter for SVMs, improving classification performance.
problem Intrinsic dependence of RBF kernel hyperparameter on SVM performance.
method Gradient descent method for automatic hyperparameter learning and SVM weights adjustment.
result OKSVM outperforms classical SVM regardless of initial RBF hyperparameter values.
Optimizes Gaussian process hyperparameters using Bayesian autoregression.
problem Optimizing hyperparameters for Matérn kernel temporal Gaussian processes.
method Recursive Bayesian estimation for autoregressive parameters.
result Outperforms traditional optimization methods in runtime and accuracy.
A new kernel for ranked data tackles computational challenges.
problem Complex geometric structure and partial rankings make existing algorithms infeasible for real-world applications.
method Derives a graph cut kernel that combines submodular optimization and kernel-based methods.
result The graph cut kernel efficiently handles large-scale ranked data.
Random Fourier features (RFF) represent one of the most popular and wide-spread techniques in machine learning to scale up kernel algorithms. Despite the numerous successful applications of RFFs, unfortunately, quite little is understood theoretically on their optimality and limitations of their performance. Only recen…
Kernel approximation methods create explicit, low-dimensional kernel feature maps to deal with the high computational and memory complexity of standard techniques. This work studies a supervised kernel learning methodology to optimize such mappings. We utilize the Discriminant Information criterion, a measure of class …
Study proves optimal controls for stochastic Volterra equations with singular kernels.
problem Existence of optimal controls for stochastic Volterra equations with singular kernels.
method Sufficient conditions based on integrability and growth hypotheses.
result Existence of optimal relaxed and strict controls under classical convexity assumptions.
Paper improves clustering risk bounds for kernel k-means.
problem Improving clustering risk bounds for kernel k-means.
method Analyzes kernel k-means and Nyström approximation.
result Achieves nearly optimal excess clustering risk bound.
New method speeds up Bayesian optimization in high dimensions.
problem High-dimensional expensive function optimization struggles.
method Structured automatic differentiation for kernel matrices.
result First-order Bayesian optimization scalable to high dimensions.
In practical Bayesian optimization, we must often search over structures with differing numbers of parameters. For instance, we may wish to search over neural network architectures with an unknown number of layers. To relate performance data gathered for different architectures, we define a new kernel for conditional p…