Paper introduces new neural network models and theories.
problem Understanding neural networks beyond over-parameterized regime.
method Develops two exact models and a novel representor theory.
result Provides insights into neural network training and kernel evolution.
New kernels from neural networks show better performance than traditional methods.
problem Improving neural network performance on small datasets.
method Developed algebraic operations to create compositional kernels from neural network architectures.
result Compositional kernels achieve higher accuracy than neural tangent kernels and neural networks on small datasets.
The neural tangent kernel equivalence theorem fails in practice.
problem Does the neural tangent kernel (NTK) equivalence theorem hold in practical neural network training?
method Rigorously derived NTK and conducted numerical experiments to evaluate the equivalence theorem.
result Adding a layer to a neural network and the corresponding updated NTK do not yield matching changes in predictor error.
Neural networks can learn kernel machines with a data-dependent kernel.
problem Can neural networks in the rich feature learning regime learn a kernel machine?
method Demonstrated silent alignment effect in neural networks, showing they can learn a kernel machine with a data-dependent kernel.
result Neural networks in the rich feature learning regime can learn a kernel machine with a data-dependent kernel due to silent alignment.
Study deep maxout networks and their equivalence to Gaussian processes.
problem Understanding neural networks with infinite width.
method Derive equivalence between deep maxout networks and Gaussian processes, characterize maxout kernel, and provide efficient numerical implementation.
result Bayesian inference based on deep maxout network kernel leads to competitive results compared to finite-width counterparts and deep neural network kernels.
This note explains when neural networks can be seen as Gaussian processes.
problem Understanding the relationship between neural networks and Gaussian processes.
method Formulating a Gaussian process regression based on neural network outputs and analyzing the resulting posterior mean functions.
result The posterior mean functions of neural networks follow a Gaussian process in certain cases, providing an interpretation of reproducing kernel Hilbert spaces.
Paper calculates eigenvalue decay rates for neural network kernels on general domains.
problem Determining eigenvalue decay rates for neural network kernels on arbitrary domains.
method Proved dynamics of wide neural networks approximates NTK on general domains, used minimax optimality and interpolation spaces.
result Provided strategy to calculate eigenvalue decay rates for neural network 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.
Convolutional Neural Networks, as most artificial neural networks, are commonly viewed as methods different in essence from kernel-based methods. We provide a systematic translation of Convolutional Neural Networks (ConvNets) into their kernel-based counterparts, Convolutional Kernel Networks (CKNs), and demonstrate th…
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.
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.
GPs with neural network dual kernels improve reinforcement learning performance.
problem Combining the strengths of DNNs and GPs for reinforcement learning.
method Apply GPs with neural network dual kernels to solve reinforcement learning tasks.
result GPs with neural network dual kernels perform at least as well as conventional methods on the mountain-car problem.
The study explains how neural networks align their kernels to target functions during training.
problem Understanding how neural networks align their kernels to target functions during training.
method Theoretical analysis of kernel evolution in toy models and deep networks.
result Kernel alignment naturally emerges during training to accelerate convergence and improve generalization.
Deep networks are shown to be equivalent to a new type of kernel chain.
problem Identifying an appropriate function space for deep neural networks.
method Extending Reproducing Kernel Banach Spaces (RKBS) to chain RKBS (cRKBS), which composes kernels rather than functions.
result Any deep neural network function is a neural cRKBS function, and conversely, any neural cRKBS function corresponds to a deep neural network.
We applied pre-defined kernels also known as filters or masks developed for image processing to convolution neural network. Instead of letting neural networks find its own kernels, we used 41 different general-purpose kernels of blurring, edge detecting, sharpening, discrete cosine transformation, etc. for the first la…
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.
Adaptive kernels from neural networks improve model performance.
problem Improving neural network performance through adaptive kernels.
method Deriving adaptive kernels from infinite-width neural networks using feature learning and gradient flow training.
result Adaptive kernels achieve lower test loss compared to traditional kernels.
Neural networks outperform kernel methods in classifying high-dimensional Gaussian mixtures.
problem Classifying high-dimensional Gaussian mixtures using kernel methods and neural networks.
method Theoretical analysis and derivation of learning dynamics for 2LNN and comparison with kernel methods.
result 2LNN can achieve near-optimal performance on high-dimensional Gaussian mixture classification tasks, surpassing kernel methods.
Deep neural-kernel models combine neural networks and kernel machines for scalable large datasets.
problem Combining neural networks and kernel machines for efficient large-scale learning.
method Hybrid neural-kernel architecture using explicit feature mapping and pooling layers.
result The deep neural-kernel models are effective and scalable on benchmark datasets.
Uniform bounds for neural networks' generalization error in overparameterized settings.
problem Generalization error in overparameterized neural networks.
method Neural Tangent kernel theory and Mercer decomposition of the NT kernel in spherical harmonics.
result Uniform generalization bounds for overparameterized neural networks in RKHS.
New kernels from ELU and GELU networks reveal non-trivial fixed points.
problem Understanding fixed-point dynamics in deep neural networks with ELU and GELU activations.
method Deriving covariance functions and analyzing fixed-point dynamics of ELU and GELU networks.
result ELU and GELU networks exhibit non-trivial fixed-point dynamics, explaining implicit regularization in overparameterized models.
The study examines Fisher information matrices and neural tangent kernels for simple ReLU networks with random weights.
problem Understanding the relationship between Fisher information matrices and neural tangent kernels for 2-layer ReLU networks.
method Analyzes Fisher information matrices and neural tangent kernels for 2-layer ReLU networks with random hidden weights, focusing on spectral decomposition and eigenfunctions.
result Obtained an approximation formula for functions represented by 2-layer neural networks.
Study on deep neural networks using branching processes and Mehler's formula.
problem Understanding the mathematical role of activation functions in compositional neural networks.
method Connection between compositional kernels and branching processes via Mehler's formula; new random features algorithm.
result Explicit formulas for eigenvalues of compositional kernels quantify complexity.
Enhances neural networks with prior knowledge through a composite kernel.
problem Lack of effective methods to incorporate prior knowledge into neural networks.
method Integrates a composite kernel combining a neural network kernel and a GP kernel for modeling known properties.
result Demonstrates superior performance and flexibility of the Implicit Composite Kernel (ICK) on synthetic and real-world data.
Generalizes leverage score sampling for neural networks, accelerating kernel methods and deep learning.
problem Accelerating kernel methods and deep learning training.
method Generalizes leverage score sampling to neural networks and proves equivalence to neural tangent kernel ridge regression.
result Equivalence between regularized neural network and neural tangent kernel ridge regression under leverage score sampling initialization.
RNNs are reinterpreted as kernel methods using neural ODEs.
problem Improving generalization and stability of RNNs.
method Connecting RNNs to neural ODEs and reproducing kernel Hilbert spaces.
result RNNs can be viewed as linear functions of a specific feature set.
This study examines the practical equivalence of Laplace and neural tangent kernels.
problem Understanding the practical equivalence of Laplace and neural tangent kernels.
method The study matches the kernels exactly and by matching posteriors of a Gaussian process. It also analyzes the kernels in R^d and experiments with them in regression tasks.
result The Laplace and neural tangent kernels are practically equivalent.
A new multi-kernel RBFNN design improves performance and speed.
problem Improving the performance and speed of RBFNNs.
method Proposes a novel multi-kernel RBFNN where each base kernel has its own weight.
result Better performance gains including faster convergence, better local minima, and resilience against poor local minima.
Develops vector-valued RKBS for neural networks and operators.
problem Understanding function spaces of Rd-valued neural networks and neural operators. method Defines and constructs vector-valued RKBS (vv-RKBS) without restrictive assumptions.
result Establishes Representer Theorem for neural architectures.
Study on SGD for overparameterized neural networks, focusing on convergence rates.
problem Understanding convergence rates of SGD in overparameterized two-layer neural networks.
method Combines NTK approximation with RKHS analysis to explore SGD dynamics.
result Established sharp convergence rates for SGD in overparameterized two-layer neural networks.
This paper explains why ResNets generalize better than FFNets using neural tangent kernels.
problem Understanding why deep ResNets generalize better than deep FFNets.
method Using neural tangent kernels to compare the learnability of functions induced by the kernels of ResNets and FFNets.
result The kernel of ResNets does not exhibit degeneracy as depth increases, unlike FFNets.
The paper shows how multi-task learning in neural networks is similar to kernel regression and Hilbert spaces.
problem Understanding the solutions to multi-task shallow ReLU neural network learning problems.
method Analyzing the properties of solutions to multi-task shallow ReLU neural network learning problems, proving uniqueness and equivalence to minimum-norm interpolation problems in Hilbert spaces.
result The solutions to multi-task neural network interpolation problems are almost always unique and coincide with the solution to a minimum-norm interpolation problem in a Sobolev (Reproducing Kernel) Hilbert Space.
Study shows deterministic equivalent for neural network kernel convergence.
problem Understanding convergence of neural network kernels.
method Analyzes empirical spectral distribution of Conjugate Kernel, proving convergence to a deterministic limit.
result Obtains a deterministic equivalent for the Stieltjes transform and resolvent of the Conjugate Kernel.
Proposes a meta-learning method for Bayesian optimization with neural network kernels.
problem Improving BO performance on high-dimensional data with complex structure.
method Meta-learning approach that trains a neural network policy to select the next data point to evaluate, using a reinforcement learning framework.
result The proposed method achieves better BO performance than existing methods on text document datasets.
Wide neural networks become linear, with constant tangent kernel, due to Hessian scaling.
problem Understanding the linearity of large non-linear models and the tangent kernel.
method Analyzing the scaling properties of the Hessian matrix of neural networks as their width increases.
result The constancy of the tangent kernel is due to the scaling properties of the Hessian matrix.
VC dimensions of group CNNs are infinite for certain kernels and groups.
problem Estimating the generalization capacity of group convolutional neural networks.
method Identifying precise VC dimension estimates for simple sets of group CNNs.
result Two-parameter families of convolutional neural networks have an infinite VC dimension for infinite groups and certain kernels.
This paper studies neural network operators and their convergence properties.
problem Understanding the approximation and convergence of neural network operators.
method Proves density results, convergence estimates, and Voronovskaya-type theorems.
result Establishes quantitative convergence estimates and derives Voronovskaya-type theorems.
A recent breakthrough in deep learning theory shows that the training of over-parameterized deep neural networks can be characterized by a kernel function called \textit{neural tangent kernel} (NTK). However, it is known that this type of results does not perfectly match the practice, as NTK-based analysis requires the…
This paper improves neural network generalization by dynamically learning kernel parameters.
problem Improving neural network generalization and adaptability.
method Diagonal adaptive kernel model that learns kernel eigenvalues and output coefficients during training.
result The diagonal adaptive kernel model significantly improves generalization over fixed-kernel methods.
This work challenges the Neural Tangent Kernel's role in overparameterized neural networks, especially with large width and depth.
problem The Neural Tangent Kernel's behavior in overparameterized neural networks with large width and depth is unclear.
method Experimental and theoretical analysis of ReLU networks with large width and depth.
result The aggregate norm of hidden neuron deviations does not vanish in infinitely-wide ReLU networks, indicating non-trivial behavior.
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.
How can neural networks such as ResNet efficiently learn CIFAR-10 with test accuracy more than 96%, while other methods, especially kernel methods, fall relatively behind? Can we more provide theoretical justifications for this gap? Recently, there is an influential line of work relating neural networks to kernels in t…
Deep neural networks define suitable reproducing kernel Banach spaces.
problem Characterizing the function spaces of deep neural networks.
method Reproducing kernel Banach spaces and variational results.
result Deep neural networks define suitable reproducing kernel Banach spaces.
New insights into why neural networks can overfit without interpolating data.
problem Understanding why neural networks can overfit without interpolating data in fixed dimensions.
method Analyzing the smoothness of estimators and their derivatives.
result Benign overfitting is possible with estimators that have large enough derivatives, not just in high dimensions but also in fixed dimensions.
Paper introduces RNTK for recurrent neural networks, improving performance across various datasets.
problem Understanding and optimizing overparametrized recurrent neural networks.
method Developed the Recurrent Neural Tangent Kernel (RNTK) to compare inputs of different lengths.
result RNTK offers significant performance gains over other kernels, including standard NTKs, across multiple datasets.
Substring kernels are classical tools for representing biological sequences or text. However, when large amounts of annotated data are available, models that allow end-to-end training such as neural networks are often preferred. Links between recurrent neural networks (RNNs) and substring kernels have recently been dra…
Empirical study shows standard CNNs deviate from NTK predictions.
problem Understanding how standard finite-width CNNs behave compared to their infinite-width NTK counterparts.
method Empirical analysis of AlexNet and LeNet architectures.
result Standard CNNs deviate significantly from their NTK counterparts, but deviation decreases with wider networks.
Improved sample efficiency with normalized RBF kernels in neural networks.
problem Learning more with less data in deep learning models.
method Two-phase method to train neural networks with normalized RBF kernels as output layer.
result Normalized RBF kernel networks achieve higher sample efficiency, compactness, and separability.