This work studies nonnegativity-preserving kernels for stochastic equations and their applications.
problem Nonnegativity preservation in stochastic Volterra equations and related processes.
method Characterization and application of completely monotone kernels; approximation schemes for weak error.
result Positive linear combinations of decaying exponentials can be used for second-order approximation schemes.
Paper introduces a new multi-kernel algorithm for better gradient approximation.
problem Improving gradient approximation in high-dimensional problems.
method Develops a multi-kernel passive stochastic gradient algorithm with variance reduction.
result The multi-kernel algorithm performs better in high-dimensional problems.
Deep kernel learning combines the non-parametric flexibility of kernel methods with the inductive biases of deep learning architectures. We propose a novel deep kernel learning model and stochastic variational inference procedure which generalizes deep kernel learning approaches to enable classification, multi-task lea…
A new method estimates SDEs using occupation kernels.
problem Learning multivariate stochastic differential equations (SDEs).
method Two-step procedure: estimate drift, then diffusion. Occupation kernels used in RKHS.
result Validated on simulated and real-world data.
Quantum stochastic flow computes heat kernel traces for Ricci flat manifolds.
problem Computing heat kernel traces for Ricci flat manifolds.
method Quantum stochastic differential equation (qsde) on Fock space over L2 differential 1-forms, adapted flow construction. result Trace of the connection Laplacian heat kernel can be computed over any compact Ricci-flat Riemannian manifold.
The book explores stochastic areas and heat kernels on manifolds.
problem Understanding stochastic area functionals and heat kernels on manifolds.
method Study of Brownian motions and heat kernels on Lie groups and Riemannian manifolds.
result Rich interactions between stochastic calculus, geometry, and random matrices.
Signature kernel handles sequential data with theoretical and practical advantages.
problem Handling sequential data efficiently and accurately.
method Positive definite kernel based on stochastic analysis with efficient computation.
result Strong empirical performance and theoretical guarantees.
The paper introduces new KMEs to capture stochastic process filtrations.
problem Missing filtration information in stochastic processes.
method Higher order kernel mean embeddings (KMEs) conditioned on filtrations.
result Consistent estimators and tests for filtration-sensitive information.
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.
Pathwise uniqueness shown for specific stochastic equations.
problem Stochastic Volterra equations with singular kernels and Hölder coefficients.
method Established pathwise uniqueness through Hölder continuity of coefficients.
result Pathwise uniqueness and existence of unique strong solutions.
Unified kernel framework extends to stochastic systems, improving numerical stability.
problem Extending kernel methods to stochastic dynamical systems with diffusion.
method Unified kernel framework, Feynman-Kac path-integral representations, collocation-based computational framework.
result Kernel equivalence under uniform ellipticity assumptions and improved numerical stability with moderate diffusion.
Improved deep kernel machine achieves 94.5% test accuracy on CIFAR-10.
problem Achieving high accuracy on complex tasks like image classification.
method Stochastic kernel regularisation to add noise to learned Gram matrices.
result 94.5% test accuracy on CIFAR-10.
Kernel ridgeless regression with random features shows good generalization without explicit regularization.
problem Generalization of kernel ridgeless regression without explicit regularization.
method Investigation of ridgeless regression with random features and stochastic gradient descent, exploring the effect of random features error and spectral density optimization.
result Random features error exhibits the double-descent curve, leading to improved generalization.
Unified quadrature framework for large-scale kernel machines.
problem Efficiently approximating kernel functions for large-scale machine learning.
method Deterministic and randomized interpolatory rules for numerical integration of kernel functions.
result The proposed method reduces the number of nodes needed for accurate kernel approximation.
This paper presents a stochastic behavior analysis of a kernel-based stochastic restricted-gradient descent method. The restricted gradient gives a steepest ascent direction within the so-called dictionary subspace. The analysis provides the transient and steady state performance in the mean squared error criterion. It…
The study analyzes prediction errors in systems with memory kernels, providing bounds and stability results.
problem Prediction errors in stochastic dynamical systems with memory kernels.
method Analysis of generalized Langevin equations (GLEs) with Volterra equations, integrating synchronized noise coupling and weighted norms.
result Prediction discrepancies decay at a rate determined by the memory kernel's decay, quantitatively bounded by kernel estimation errors.
Develops multifactor approximations for SVEs with completely monotone kernels.
problem Approximating SVEs with kernels of completely monotone type.
method Multifactor approximation, Euler discretization, L2-estimation, convergence analysis. result New multifactor Euler scheme reduces computational cost and outperforms SVEs for option pricing.
Method learns SDEs from data snapshots.
problem Learning drift and diffusion of SDEs from data.
method Two-step process: learn drift by expected value, learn diffusion by SDP.
result Validated on examples and simulations.
FDSKL algorithm trains vertically partitioned data with kernels securely and efficiently.
problem Training vertically partitioned data with kernels while maintaining privacy.
method FDSKL algorithm using random features and doubly stochastic gradients for federated learning.
result FDSKL achieves sublinear convergence and guarantees data security.
Stochastic gradient descent optimizes Nyström samples for kernel matrix approximation.
problem Optimizing Nyström samples for kernel matrix approximation.
method Stochastic gradient descent applied to multisets of landmark points (Nyström samples) using a surrogate criterion (radial SKD).
result Local minimization of the radial SKD yields improved Nyström approximation accuracy.
The existence of stationary Markov perfect equilibria in stochastic games is shown under a general condition called "(decomposable) coarser transition kernels". This result covers various earlier existence results on correlated equilibria, noisy stochastic games, stochastic games with finite actions and state-independe…
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.
Study approximates rough stochastic volatility models using diffusion processes.
problem High computational cost in simulating rough stochastic volatility models.
method Approximates stochastic Volterra equations with an N-dimensional diffusion process.
result Approximations converge strongly with superpolynomial rate in N.
The paper improves boundary detection and density estimation on noisy data.
problem Detecting boundary points and estimating density on noisy data from compact manifolds.
method Doubly stochastic scaling of the Gaussian heat kernel via Sinkhorn iterations.
result The new estimates of boundary points and density outperform standard methods, especially under noise.
The paper explores arbitrage opportunities in derivative markets under specific conditions.
problem Arbitrage opportunities in derivative markets under different conditions.
method Analyzes the relationship between pricing kernel monotonicity and stochastic arbitrage opportunities.
result Pricing kernel nonmonotonicity is equivalent to stochastic arbitrage opportunities under adequacy.
Study small-time CLTs for stochastic Volterra equations with various kernels.
problem Understanding the behavior of stochastic Volterra equations with different kernels.
method Proved convergence of finite-dimensional distributions, functional CLT, and limit theorems for smooth transformations.
result Derived asymptotic pricing formulae for digital calls in rough volatility models.
Paper studies t-SNE convergence with generalized kernels.
problem Understanding convergence of t-SNE with generalized kernels.
method Concrete formulation of generalized kernels, proving convergence to an equilibrium distribution.
result t-SNE converges to an equilibrium distribution under certain conditions for generalized kernels.
The general perception is that kernel methods are not scalable, and neural nets are the methods of choice for nonlinear learning problems. Or have we simply not tried hard enough for kernel methods? Here we propose an approach that scales up kernel methods using a novel concept called "doubly stochastic functional grad…
New graph kernels capture spatio-temporal interactions.
problem Lack of justified spatio-temporal graph kernels for graph problems.
method Derive graph kernels via SPDEs for spatio-temporal modelling.
result Non-separable spatio-temporal graph kernels outperform existing ones.
New algorithms improve GP inference without approximations, achieving better results.
problem Inexact stochastic optimization methods in Gaussian Processes leading to biased results.
method Exact stochastic inference for GPs with finite dimensional RKHS, extending to infinite dimensions.
result Achieves better experimental results than existing methods in constrained resource settings.
Enhances deep kernel learning with stochastic latent variables for better model regularization.
problem Weak model regularization in deep kernel learning, especially on small datasets.
method Introduces DLVKL model with stochastic latent variables, NSDE for expressive posterior, and hybrid prior.
result DLVKL-NSDE outperforms existing deep GPs on large datasets.
We link SVEs to SPDEs and derive Kolmogorov equations for singular kernels.
problem Solving stochastic Volterra equations with singular kernels.
method Establishing connections between SVEs and SPDEs, using stochastic calculus in Hilbert spaces.
result Solutions of SVEs can be expressed in terms of backward Kolmogorov equations.
Doubly-stochastic normalization improves robustness to heteroskedastic noise.
problem Robustness to heteroskedastic noise in affinity matrix construction.
method Doubly-stochastic normalization of the Gaussian kernel.
result Doubly-stochastic normalization converges to clean matrix with rate m−1/2 under heteroskedastic noise. Develops EFT for ResNets, revealing limitations of kernel-only approach.
problem Limitations of kernel-only approach in deep neural networks.
method Collective kernel EFT for pre-activation ResNets based on G-only closure hierarchy. result Numerical findings show V4 equation residual accumulates to an O(1) error. Develops a framework for learning nonlinear operators using Mercer kernels.
problem Learning nonlinear operators between infinite-dimensional spaces.
method Stochastic approximation framework with Mercer operator-valued kernels.
result Establishes dimension-free polynomial convergence rates for nonlinear operator learning.
New method speeds up training of large kernel models.
problem Scaling kernel machines to large datasets and model sizes.
method Delayed projections in Preconditioned Stochastic Gradient Descent (PSGD).
result Significant training speed up over existing methods.
Paper develops a novel approach for optimal control using kernel methods.
problem Optimal control of nonlinear stochastic systems.
method Infinitesimal generator approach in reproducing kernel Hilbert spaces.
result Data-driven solution to optimal control problems.
Averaged SGD achieves optimal convergence rate for neural networks in the NTK regime.
problem Convergence analysis of averaged stochastic gradient descent for neural networks.
method Analyzed convergence of averaged stochastic gradient descent for overparameterized two-layer neural networks.
result Achieved minimax optimal convergence rate with global convergence guarantee.
The sequence of moments of a vector-valued random variable can characterize its law. We study the analogous problem for path-valued random variables, that is stochastic processes, by using so-called robust signature moments. This allows us to derive a metric of maximum mean discrepancy type for laws of stochastic proce…
Unified approach to stochastic Volterra systems' deviations.
problem Large and moderate deviations for stochastic Volterra systems.
method Weak convergence approach by Budhijara, Dupuis and Ellis.
result Unified treatment of deviations for a broad class of stochastic Volterra equations.
The stable under iterated tessellation (STIT) process is a stochastic process that produces a recursive partition of space with cut directions drawn independently from a distribution over the sphere. The case of random axis-aligned cuts is known as the Mondrian process. Random forests and Laplace kernel approximations …
New method transforms complex stochastic equations into simpler ones for efficient simulation.
problem Efficient simulation of complex path-dependent stochastic processes.
method Transforms Volterra-type SDEs into standard diffusion processes using convolution kernels.
result Proposes a numerical simulation scheme with a strong convergence rate of 1/2.
New method interpolates high-dimensional scattered data using kernel theory.
problem Scattered data in high-dimensional spaces defy traditional distributional assumptions.
method Kernel interpolation framework based on integral operator theory.
result Spectra of kernel matrices predict performance of interpolation methods.
Unified derivation of high-dimensional linear models using stochastic gradient descent.
problem Performance analysis of high-dimensional linear models trained with stochastic gradient descent.
method Derivation of a deterministic equivalence for the two-point function of a random matrix resolvent.
result Unified understanding of model performance including previously known and novel results.
This paper presents a new insight into improving the performance of Stochastic Neighbour Embedding (t-SNE) by using Isolation kernel instead of Gaussian kernel. Isolation kernel outperforms Gaussian kernel in two aspects. First, the use of Isolation kernel in t-SNE overcomes the drawback of misrepresenting some structu…
We present a novel variation of online kernel machines in which we exploit a consensus based optimization mechanism to guide the evolution of decision functions drawn from a reproducing kernel Hilbert space, which efficiently models the observed stationary process.
The study extends stochastic completeness to landmark spaces with any number of landmarks.
problem Stochastic completeness for landmark spaces with arbitrary numbers of landmarks.
method Volume growth criterion and eigenvalue bounds for geodesic balls.
result Stochastic completeness for landmark spaces with any number of landmarks is proven.
We provide existence, uniqueness and stability results for affine stochastic Volterra equations with L1-kernels and jumps. Such equations arise as scaling limits of branching processes in population genetics and self-exciting Hawkes processes in mathematical finance. The strategy we adopt for the existence part is b…