In this paper we solve support vector machines in reproducing kernel Banach spaces with reproducing kernels defined on nonsymmetric domains instead of the traditional methods in reproducing kernel Hilbert spaces. Using the orthogonality of semi-inner-products, we can obtain the explicit representations of the dual (nor…
This paper extends mirror descent to Banach spaces with reproducing kernels.
problem Optimizing in Banach spaces with reproducing kernels.
method Mirror descent algorithm adapted for Banach spaces with reproducing kernels.
result Mirror descent achieves linear convergence in certain conditions and standard convergence in a constrained setting.
The paper uses Banach spaces to analyze neural networks.
problem Understanding the function spaces of neural networks.
method Theory of reproducing kernel Banach spaces.
result Representer theorem for wide class of Banach spaces.
Extends Gaussian process theory to Banach spaces.
problem Extending Gaussian process theory to Banach spaces.
method Investigates the connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces.
result Characterizes positive definite functions that arise from covariance operators in Banach space setting.
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.
Paper characterizes embeddability of function spaces into Lp-type RKBS via metric entropy.
problem Characterizing embeddability of function spaces into Lp-type RKBS. method Establishes a connection between metric entropy growth and embeddability.
result A bound on metric entropy growth allows embedding into Lp-type RKBS. Motivated by multi-task machine learning with Banach spaces, we propose the notion of vector-valued reproducing kernel Banach spaces (RKBS). Basic properties of the spaces and the associated reproducing kernels are investigated. We also present feature map constructions and several concrete examples of vector-valued RK…
The paper defines a hypothesis space for deep learning using DNNs.
problem Developing a mathematical framework for deep learning.
method Introducing a Banach space of functions of input variables based on DNNs, proving it's a RKBS, and establishing representer theorems for learning models.
result Solutions to learning problems can be expressed as finite sums of kernel expansions based on training data.
A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error and the regularization error. Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding t…
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.
Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the properties that (1) B possesses an l1 norm in the sense that it is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous lin…
Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation and functional analysis. Existing constructions include the reflexive RKBS via a bilinear form, the semi-inner-p…
Transformers are explained as infinite-dimensional kernel machines.
problem Understanding the mechanics of Transformers in AI.
method Characterized Transformers' attention mechanism as a kernel learning method on Banach spaces.
result Transformer's kernel has infinite feature dimension and can learn any binary non-Mercer reproducing kernel Banach space pair.
New neural architectures with multivariate nonlinearities are optimal in function space.
problem Optimality of neural architectures with multivariate nonlinearities.
method Construction of Banach spaces via k-plane transform and sparsity-promoting norm, proving representer theorem. result Neural architectures with multivariate nonlinearities are optimal in function space.
This study approximates neural network features for modeling relations and attention mechanisms.
problem Approximating neural network features for modeling relations and attention mechanisms.
method Analyzes inner products of multi-layer perceptrons for universal approximation of symmetric and asymmetric relation functions.
result Universal approximation of relation functions and attention mechanisms using inner products of neural networks.
New analysis shows a gap between Gaussian RKHS and neural networks on unbounded domains.
problem Understanding the function space bias of neural networks compared to Gaussian RKHS.
method Infinite-center asymptotic analysis of neural network Banach space and Gaussian RKHS on unbounded domains.
result Certain functions in Gaussian RKHS have infinite norm in neural network Banach space on unbounded domains.
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.
Gradient descent in neural networks analyzed using RKBS for broader applicability.
problem Analyzing neural network training in the over-parametrized limit.
method Constructing an exact power-series representation of neural networks in RKBS, proving replicability of gradient descent sequences.
result Gradient descent sequences can be exactly replicated by regularized sequential learning in RKBS, providing new theoretical insights.
We propose a systematic construction of native Banach spaces for general spline-admissible operators L. In short, the native space for L and the (dual) norm ∥⋅∥X′ is the largest space of functions f:Rd→R such that ∥Lf∥X′<∞, subj…
Poor approximators found in neural networks and random feature models.
problem Understanding why certain neural networks and models perform poorly in approximating functions.
method Established a scale separation of Kolmogorov width type and applied it to neural networks and random feature models.
result Reproducing kernel Hilbert spaces and two-layer neural networks are poor L2-approximators for certain functions. There has been growing recent interest in probabilistic interpretations of kernel-based methods as well as learning in Banach spaces. The absence of a useful Lebesgue measure on an infinite-dimensional reproducing kernel Hilbert space is a serious obstacle for such stochastic models. We propose an estimation model for …
We obtain a Bernstein-type inequality for sums of Banach-valued random variables satisfying a weak dependence assumption of general type and under certain smoothness assumptions of the underlying Banach norm. We use this inequality in order to investigate in the asymptotical regime the error upper bounds for the broad …
This paper introduces a new Barron space for graph signals and proves its properties for GCNNs.
problem Understanding and optimizing the performance of GCNNs on graph signals.
method Introducing a Barron space on graph signals, proving its properties, and showing the approximation and learning capabilities of GCNNs within this space.
result GCNN outputs are contained in the Barron space and can be well approximated by functions in this space.
Functional central limit theorem for kernel gradient flow and infinitesimal gradient boosting
problem Fluctuations of boosting processes around their deterministic limit
method Stochastic perturbation analysis of ODEs in Banach spaces
result Rescaled deviations converge to a Gaussian process
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.
New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on Lp spaces and sets of measures. 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.
Represents neural networks as solutions to inverse problems in Banach spaces.
problem Understanding the function learned by neural networks.
method Variational framework, representer theorem, polynomial ridge splines.
result Neural networks are solutions to inverse problems in Banach spaces.
Representation costs in data science: Unifying function-space views of parametric methods
problem Analyzing representation costs of parametric data-fitting methods
method Developing a general framework for analyzing representation costs through parameter-space regularizers
result Proving that many natural results hold in this abstract setting, including representer theorems for parametric methods on their native spaces
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.
In this paper we study the variational problem associated to support vector regression in Banach function spaces. Using the Fenchel-Rockafellar duality theory, we give explicit formulation of the dual problem as well as of the related optimality conditions. Moreover, we provide a new computational framework for solving…
The paper presents a novel approach to direct covariance function learning for Bayesian optimisation, with particular emphasis on experimental design problems where an existing corpus of condensed knowledge is present. The method presented borrows techniques from reproducing kernel Banach space theory (specifically m-k…
Kernel methods outperform neural nets in operator learning tasks.
problem Learning operators between Banach spaces from partial observations.
method Kernel-based framework with a priori error analysis and numerical comparisons.
result Kernel methods are competitive with neural nets in cost-accuracy trade-off.
Representations of C∗-algebras are realized on section spaces of holomorphic homogeneous vector bundles. The corresponding section spaces are investigated by means of a new notion of reproducing kernel, suitable for dealing with involutive diffeomorphisms defined on the base spaces of the bundles. Applications of th…
Kernel methods are studied in a mean field limit for high-dimensional data.
problem Analyzing kernel methods in high-dimensional data with many variables.
method Investigation of kernel methods in the mean field limit of interacting particle systems.
result Rigorous mean field limit of kernels and detailed analysis of the limiting reproducing kernel Hilbert space.
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.
We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with…
Unified view of GP approximations improves efficiency.
problem Disparate variational features limit GP efficiency.
method View GP as a Banach space to unify feature selection.
result Unified understanding of existing and new features.
The paper develops a uniform function estimator in RKHS for regression.
problem Reconstructing functions from noisy data at random locations.
method Using reproducing kernel Hilbert spaces and Gaussian random fields.
result The estimator converges uniformly to the conditional expectation.
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.
The paper develops methods to handle missing data using regularized M-estimation in reproducing kernel Hilbert space.
problem Handling missing data in statistical analysis.
method Kernel ridge regression for imputation and maximum entropy method for propensity score estimation.
result The proposed methods achieve statistical consistency and asymptotic equivalence.
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 proposes a method for early stopping in regression using reproducing kernels.
problem Early stopping for iterative learning algorithms in nonparametric regression.
method Data-driven rule based on minimum discrepancy principle, validated by fixed-point analysis of localized Rademacher complexities.
result The proposed rule is minimax-optimal and performs comparably to cross-validation.
Study on Banach half-Lie groups and their properties.
problem Characterizing and understanding the properties of half-Lie groups in infinite dimensions.
method Investigation of Banach half-Lie groups, their extensions, and right invariant strong Riemannian metrics.
result The full Hopf--Rinow theorem holds for Banach half-Lie groups, a surprising result.
Study of regularized least squares in RKKS with indefinite kernels.
problem Asymptotic properties of regularized least squares with indefinite kernels in RKKS.
method Introducing a bounded hyper-sphere constraint, theoretical demonstration of globally optimal solution, modified error decomposition techniques, matrix perturbation theory.
result Derivation of learning rates in RKKS, same as RKHS under certain conditions.
We consider a general regularised interpolation problem for learning a parameter vector from data. The well known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the data points. This is the core of kernel methods in machine learning as it makes th…
Random feature models approximate functions in Banach spaces efficiently.
problem Approximating functions in Banach spaces efficiently.
method Randomly initialized feature maps and linear readout training.
result Universal approximation in Bochner spaces for Banach space-valued models.
We consider a general regularised interpolation problem for learning a parameter vector from data. The well known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the data points. This is at the core of kernel methods in machine learning as it makes…