The NNGP kernel's predictions closely match those of the Matern kernel under certain conditions.
problem Comparing NNGP kernels to Matern kernels in practical applications.
method Demonstrated the necessity of normalization for NNGP kernels, explored numerical challenges, and compared predictions and performance.
result NNGP kernel predictions closely match Matern kernel predictions under specific circumstances.
Graph convolutional deep kernel machine learns representations for graph tasks.
problem Limited representation learning in infinite-width neural networks.
method Developed a graph convolutional deep kernel machine as an infinite-width limit.
result Representation learning improves performance for heterophilous node classification tasks.
A new method combines deep kernels with Gaussian processes to avoid overfitting.
problem Losing Bayesian benefits in deep kernel learning due to kernel optimization.
method Using Infinite-width neural networks and Neural Network Gaussian Process (NNGP) as a guide for DKL optimization.
result Robustness to overfitting and good predictive performance on various datasets.
Characterizes neural kernel and NNGP for various activations.
problem Understanding neural kernels and NNGP for non-RELU activations.
method Characterization of RKHS for various activation functions.
result Broad class of non-infinitely smooth activations generate equivalent RKHSs at different depths.
Paper connects neural networks to Gaussian processes for understanding double-descent.
problem Understanding the double-descent phenomenon in neural networks.
method Uses techniques from random matrix theory and Gaussian processes.
result Establishes a connection between NNGP and random matrix theory for neural networks.
This paper explores how kernel methods can explain data effects on neural collapse.
problem Understanding how data affects neural collapse in neural networks.
method Formulating NC1 as a function of kernel, specializing to NNGP and NTK, and exploring a data-aware Gaussian Process kernel.
result The NTK does not represent more collapsed features than the NNGP for Gaussian data, highlighting the limitations of data-independent kernels.
Recent work has established the equivalence between deep neural networks and Gaussian processes (GPs), resulting in so-called neural network Gaussian processes (NNGPs). The behaviour of these models depends on the initialisation of the corresponding network. In this work, we consider the impact of noise regularisation …
Bayesian neural networks explore rare fluctuations for better feature learning.
problem Understanding rare but dominant fluctuations in Bayesian neural networks.
method Large-deviation theory and joint optimization over predictors and internal kernels.
result Posterior rate function optimization reveals data-dependent kernel selection.
Develops a theoretical framework for scalable Gaussian Process regression methods.
problem Limited scalability of Gaussian Process regression for large datasets.
method Introduces and analyzes Nearest Neighbour Gaussian Process (NNGP) and scalable GPnn methods.
result Derives almost sure pointwise limits for predictive criteria and proves risk minimax rates.
Neural-net-induced Gaussian process (NNGP) regression inherits both the high expressivity of deep neural networks (deep NNs) as well as the uncertainty quantification property of Gaussian processes (GPs). We generalize the current NNGP to first include a larger number of hyperparameters and subsequently train the model…
Empirical study compares wide neural networks to kernel methods, resolving open questions.
problem Understanding the relationship between wide neural networks and kernel methods.
method Large-scale empirical study using various neural network architectures and kernel methods.
result Wide neural networks outperform fully-connected finite-width networks in some cases, but underperform convolutional finite-width networks.
New method speeds up neural kernel computations for various activations.
problem Inefficient computation of neural kernels for general activations.
method Fast sketching method using truncated Hermite expansion.
result 106x speedup for approximate CNTK computation on CIFAR-10.
Wide neural networks converge to Gaussian processes, improving generalization.
problem Understanding the generalization of wide neural networks, especially deep equilibrium models.
method Investigation of deep equilibrium models (DEQs) with infinite-depth layers, focusing on their convergence to Gaussian processes as width and depth approach infinity.
result Wide DEQs converge to Gaussian processes, maintaining generalization performance.
Proposes scale mixture of NNGPs for more flexible stochastic processes.
problem Limited focus on broadening the class of stochastic processes from NNGPs.
method Scale mixture of NNGPs with scale priors on last-layer parameters.
result Turns neural networks into a richer class of stochastic processes.
There has recently been much work on the "wide limit" of neural networks, where Bayesian neural networks (BNNs) are shown to converge to a Gaussian process (GP) as all hidden layers are sent to infinite width. However, these results do not apply to architectures that require one or more of the hidden layers to remain n…
Attention mechanisms in deep learning become Gaussian process-like as the number of heads increases.
problem Understanding the behavior of attention mechanisms in deep learning models.
method Extending the equivalence between wide neural networks and Gaussian processes to attention architectures.
result Multi-head attention architectures behave as Gaussian processes as the number of heads tends to infinity.
Fast variational Bayes methods improve geospatial data analysis speed and accuracy.
problem Inaccurate and slow variational Bayes methods for large geospatial data.
method Combination of calculus of variations, closed-form gradient updates, and linear response corrections.
result Comparable accuracy to spNNGP with reduced computational costs and faster speed.
RFAD uses random features to speed up dataset distillation.
problem Efficiently compress large datasets for reduced storage and computation.
method Random feature approximation of the Neural Network Gaussian Process kernel.
result At least 100-fold speedup over KIP with competitive accuracy.
Convolutional DKMs improve kernel methods on MNIST, CIFAR-10, and CIFAR-100.
problem Improving kernel methods for image classification.
method Developed a novel inter-domain inducing point approximation and introduced various techniques to extend DKMs to convolutional networks.
result Achieved state-of-the-art performance on image classification benchmarks.
Better uncertainty estimates for neural networks using Gaussian process priors.
problem Poor uncertainty estimates in neural networks, especially on out-of-distribution data.
method Characterize the function-space prior of an ensemble of infinitely-wide neural networks as a Gaussian process and use it to build a probabilistic model.
result The approach improves calibration of neural networks, especially under distributional shift.
The hedonic approach based on a regression model has been widely adopted for the prediction of real estate property price and rent. In particular, a spatial regression technique called Kriging, a method of interpolation that was advanced in the field of spatial statistics, are known to enable high accuracy prediction i…
Analyzes DNNs trained with noisy gradients, finding FWCs negligible for large n.
problem Analyzing DNNs trained with noisy gradients.
method Introduced analytical framework to analyze non-Gaussian stochastic process.
result FWCs negligible for large n, improving CNN performance.
New theory explains deep learning's success in transforming inputs.
problem Standard theoretical approaches eliminate representation learning.
method Developed a new infinite width limit for representation learning.
result Deep Gaussian processes (DGPs) have multivariate Gaussian posteriors.
New approach predicts generalization of deep neural networks in proportional-width regime.
problem Predicting generalization of deep neural networks in proportional-width regime.
method Equivalent Wishart Ansatz for hierarchical empirical kernels, renormalized NNGP kernel.
result Renormalized NNGP kernel captures dominant stochastic fluctuations in deep neural networks.
Wide Bayesian neural networks have a simpler weight posterior, leading to faster MCMC sampling.
problem Sampling from the posterior of wide Bayesian neural networks is challenging.
method Introducing repriorisation, a data-dependent reparameterisation that simplifies the posterior distribution.
result The repriorisation map accelerates MCMC sampling, achieving up to 50x higher effective sample size.
Many unsupervised kernel methods rely on the estimation of the kernel covariance operator (kernel CO) or kernel cross-covariance operator (kernel CCO). Both kernel CO and kernel CCO are sensitive to contaminated data, even when bounded positive definite kernels are used. To the best of our knowledge, there are few well…
Survey of kernels, RKHS, and their applications in machine learning.
problem Understanding kernels and their applications in machine learning.
method Review of historical context, mathematical definitions, and practical applications of kernels.
result Comprehensive overview of kernels, RKHS, and their applications.
To the best of our knowledge, there are no general well-founded robust methods for statistical unsupervised learning. Most of the unsupervised methods explicitly or implicitly depend on the kernel covariance operator (kernel CO) or kernel cross-covariance operator (kernel CCO). They are sensitive to contaminated data, …
Deep neural kernels and Laplace kernel have equivalent RKHS on spheres.
problem Comparing RKHS of deep neural tangent and Laplace kernels.
method Proof of RKHS equivalence using sphere restrictions and kernel properties.
result RKHS of deep neural tangent kernel and Laplace kernel are the same on Sd−1. 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.
PGF kernels analyze spherical data using generalized RBF kernels.
problem Analysis of spherical data.
method Introduced PGF kernels and a semi-parametric learning algorithm.
result PGF kernels generalize RBF kernels for spherical data.
Adapts manifold structure for better clustering performance.
problem Lack of consideration for local manifold structure in existing multiple kernel k-means methods.
method Adopts manifold adaptive kernel to integrate local manifold structure of kernels.
result Proposed method outperforms state-of-the-art methods.
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.
Quantum kernels can be efficiently embedded into classical feature spaces.
problem Can all quantum kernels be efficiently embedded into classical feature spaces?
method Invoking computational universality and using techniques like random Fourier features, the authors show that certain classes of quantum kernels can be efficiently embedded.
result For shift-invariant and composition kernels, embedding quantum kernels are universal and efficient.
New random feature maps for Laplacian and related kernels.
problem Challenges in approximating the Laplacian kernel and its generalizations.
method Developed random feature maps for Laplacian and related kernels, providing efficient sampling schemes.
result Demonstrated the efficacy of these random feature maps on real datasets.
New method for learning with non-Euclidean data using decomposable kernels.
problem Difficulty in using classical kernels for non-Euclidean data.
method Reproducing kernel Krein space (RKKS) methods for kernels that admit a positive decomposition.
result Invariant kernels can be used for learning in non-Euclidean spaces.
New estimator reduces kernel mean estimation error.
problem Kernel mean estimation in reproducing kernel Hilbert spaces.
method Corrupt data with known distributions and estimate kernel mean under the corrupted distribution.
result The marginalized kernel mean estimator achieves lower estimation error.
We present Random Partition Kernels, a new class of kernels derived by demonstrating a natural connection between random partitions of objects and kernels between those objects. We show how the construction can be used to create kernels from methods that would not normally be viewed as random partitions, such as Random…
In this paper, we compare 5 different nonlinear kernels: min-max, RBF, fRBF (folded RBF), acos, and acos-χ2, on a wide range of publicly available datasets. The proposed fRBF kernel performs very similarly to the RBF kernel. Both RBF and fRBF kernels require an important tuning parameter (γ). Interestingly, for a …
New kernels allow learning from non-separable data.
problem Learning from non-separable data.
method Introducing entangled kernels and a two-step algorithm.
result Efficient algorithm for learning entangled kernels.
Laplace kernel and Neural Tangent Kernels are shown to be nearly identical for normalized data.
problem Understanding the similarity between Laplace and Neural Tangent Kernels.
method Theoretical analysis and experiments on normalized data.
result Laplace kernel and Neural Tangent Kernels have nearly identical eigenfunctions and RKHS for normalized data.
Kernel smoothing on unknown manifolds with bounds and asymptotic normality.
problem Data on unknown manifolds without boundaries.
method Finite sample bounds and asymptotic normality for kernel smoothing and its derivatives.
result Established finite sample bounds and asymptotic normality for kernel smoothing.
In this paper we propose a family of tractable kernels that is dense in the family of bounded positive semi-definite functions (i.e. can approximate any bounded kernel with arbitrary precision). We start by discussing the case of stationary kernels, and propose a family of spectral kernels that extends existing approac…
The success of kernel-based learning methods depend on the choice of kernel. Recently, kernel learning methods have been proposed that use data to select the most appropriate kernel, usually by combining a set of base kernels. We introduce a new algorithm for kernel learning that combines a {\em continuous set of base …
New kernels capture both local and non-local interactions efficiently.
problem Designing kernels that capture both local and non-local interactions while remaining computationally tractable.
method Spectral truncation kernels based on C∗-algebra. result Spectral truncation kernels induce interactions across the data function domain and reduce computational cost.
Quantum kernel machines need to use more complex kernels to fully exploit their potential.
problem Current quantum kernels struggle with complex learning tasks due to limited degrees of freedom.
method Propose using operator-valued kernels and C∗-algebraic representations to enhance quantum kernels. result Quantum operator-valued kernels can reveal structural dependencies that scalar-valued kernels miss.
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…
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.