Two adaptive kernel selection methods improve the accuracy of Kernelized Diffusion Maps.
problem Selecting an appropriate kernel for Kernelized Diffusion Maps.
method Two complementary approaches: variational outer loop and unsupervised cross-validation.
result Both methods improve the quality and stability of the recovered eigenfunctions.
Conditional diffusion models can approximate target distributions well with Gaussian-mixture reverse kernels.
problem Approximating target distributions in conditional diffusion models.
method Using finite Gaussian mixtures with ReLU-network logits as reverse kernels, reducing the problem to static conditional density approximation.
result The resulting neural reverse-kernel class is dense in conditional KL divergence under exact terminal matching.
This research explores how different discrete diffusion kernels affect graph generation quality.
problem The impact of different discrete diffusion kernels on graph generation quality.
method Developed a family of discrete diffusion kernels that converge to different Bernoulli priors.
result The quality of generated graphs is sensitive to the prior used, challenging previous intuitions.
Diffusion maps are a commonly used kernel-based method for manifold learning, which can reveal intrinsic structures in data and embed them in low dimensions. However, as with most kernel methods, its implementation requires a heavy computational load, reaching up to cubic complexity in the number of data points. This l…
Kernel-smoothed scores improve diffusion models by reducing memorization.
problem Diffusion models can memorize training data, leading to biased samples.
method Interpret empirical score as noisy version of true score, kernel-smoothed.
result Kernel-smoothing reduces variance and improves generalization.
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.
New method circumvents curse of dimensionality in Laplacian estimation.
problem High-dimensional data challenges spectral clustering and diffusion maps.
method Kernelized Laplacian estimation via reproducing kernel Hilbert space.
result Non-asymptotic statistical rates show improved performance in high dimensions.
SJDs unify masked, continuous, and hybrid diffusion models.
problem Unified modeling of diffusion processes.
method Continuous-time Markov processes with token embeddings and hazard rates.
result Unified model recovers masked, continuous, and hybrid diffusion as limits.
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.
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…
Diffusion Maps framework is a kernel based method for manifold learning and data analysis that defines diffusion similarities by imposing a Markovian process on the given dataset. Analysis by this process uncovers the intrinsic geometric structures in the data. Recently, it was suggested to replace the standard kernel …
New method reduces computational cost for learning stationary diffusions.
problem Learning parameters of stationary diffusions efficiently.
method Stein-type discrepancy (SKDS) for estimating generator expectations.
result SKDS guarantees alignment with target stationary distribution.
HyBO optimizes hybrid structures using diffusion kernels.
problem Optimizing complex interactions between discrete and continuous variables.
method HyBO uses diffusion kernels over hybrid spaces with additive kernel formulation.
result HyBO significantly outperforms state-of-the-art methods on real-world benchmarks.
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.
The paper introduces new methods for Asian option pricing using Laguerre quadrature.
problem Developing accurate pricing models for Asian options.
method Utilizes Laguerre quadrature and diffusion kernel approach.
result Demonstrates new techniques to solve complex Asian option pricing equations.
The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.
problem Analyzing Riemannian Gaussian distributions on symmetric spaces.
method Analytical computation of marginals using orthogonal and skew orthogonal polynomials, and diffusion kernels.
result Riemannian Gaussian distributions are random matrix types, and their probability density functions can be computed analytically.
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.
EntroPath learns manifold geometry from diffusion paths.
problem Learning geodesic geometry from data graphs with spurious shortcuts.
method Maximum Entropy Path Ensemble Embedding (MERW) with k-step diffusion paths.
result EntroPath converges to squared geodesic distance in the short-time limit.
Improved image generation quality using closed-form discriminator guidance in diffusion models.
problem Enhancing the quality of images generated by diffusion models.
method Theoretical framework to analyze GAN discriminator's effect on Langevin sampling, proposing IPM-GAN optimization as smoothed score-matching.
result Closed-form kernel-based discriminator guidance improves metrics like CLIP-FID and KID.
This paper analyzes error bounds for biased SMC samplers in conditional sampling.
problem Analyzing error bounds for biased SMC samplers in conditional sampling.
method Develops a non-asymptotic error analysis for SMC samplers with biased mutation kernels.
result Derives the first non-asymptotic error bound for conditional sampling with score-based diffusion models.
Generative model on manifolds reduces divergence computation and improves scalability.
problem Difficulties in modeling data on non-Euclidean spaces due to expensive divergence computation and approximations of heat kernel.
method Riemannian Diffusion Mixture, a principled framework using a mixture of bridge processes.
result Achieves superior performance on diverse manifolds with reduced simulation steps.
One of the major problems in natural language processing (NLP) is the word sense disambiguation (WSD) problem. It is the task of computationally identifying the right sense of a polysemous word based on its context. Resolving the WSD problem boosts the accuracy of many NLP focused algorithms such as text classification…
We solve a Schrödinger bridge with a quadratic state cost, finding a closed-form solution.
problem Optimizing diffusion processes between given distributions.
method Regularized Schrödinger bridge with a quadratic state cost.
result Closed-form solution for the Markov kernel of the regularized Schrödinger bridge.
Algorithm learns interaction kernels for particle systems from data.
problem Understanding and modeling interactions in systems of interacting particles.
method Nonparametric algorithm using least squares with regularization, probabilistic error functional, and reproducing kernel Hilbert space convergence.
result The algorithm converges optimally and accurately learns interaction kernels.
Uniform heat kernel and diffusion bridge asymptotics for sub-Riemannian geometry.
problem Analyzing sub-Riemannian heat kernels and their derivatives on incomplete manifolds.
method Localized asymptotic analysis, focusing on minimizing geodesics and the non-abnormal cut locus.
result Uniform bounds and expansions for heat kernels and their derivatives on compacts, including the diffusion bridge measure.
Efficient diffusion model for symmetric manifolds reduces training and computation costs.
problem Heat kernel computations for manifold diffusion models are computationally expensive and infeasible.
method Spatially-varying covariance diffusion model, efficient objective derived via Ito's Lemma.
result Our model reduces training time and arithmetic operations by orders of magnitude.
Spectral algorithms on manifolds using diffusion kernels improve convergence rates.
problem The limitations of existing spectral algorithms in RKHSs for data on manifolds.
method Integrating manifold structure into spectral algorithms using heat kernel diffusion spaces.
result Spectral algorithms converge to the target function and its derivatives in a strong sense, with rates dependent on manifold intrinsic dimension.
RNE provides a flexible framework for diffusion models, enabling inference-time control and energy-based training.
problem Insufficient knowledge of marginal densities in diffusion models.
method Introduces Radon-Nikodym Estimator (RNE) to reveal the connection between marginal densities and transition kernels.
result RNE delivers strong results in inference-time control and energy-based diffusion training.
We prove strong existence and uniqueness, and Hölder regularity, of a large class of stochastic Volterra equations, with singular kernels and non-Lipschitz diffusion coefficient. Extending Yamada-Watanabe's theorem, our proof relies on an approximation of the process by a sequence of semimartingales with regularised ke…
Unified kernel for prediction markets reduces belief variance forecast error.
problem Lack of standardized tools for quoting and hedging belief risk in prediction markets.
method Logit jump-diffusion model with risk-neutral drift, calibration pipeline, and coherent derivative layer.
result Model reduces forecast error compared to diffusion-only and probability-space baselines.
Solves memorization in diffusion models for manifold data.
problem Memorization effect in diffusion models for manifold data.
method Inertia update at the end of empirical diffusion simulation.
result Approximates true data distribution on a C2 manifold. A computational technique borrowed from the physical sciences is introduced to obtain accurate closed-form approximations for the transition probability of arbitrary diffusion processes. Within the path integral framework the same technique allows one to obtain remarkably good approximations of the pricing kernels of f…
A new graph generator uses heat diffusion on graph Laplacians to create new graph structures.
problem Creating realistic and diverse graph structures for various applications.
method Adapting the Generator Matching paradigm to graph data, using graph Laplacian and heat kernel for diffusion.
result The method effectively generates graphs with structural properties of real and synthetic graphs.
A new framework RTK accelerates diffusion inference by breaking down the process into fewer, more efficient subproblems.
problem Efficiently generating data from trained diffusion models using discretized reverse SDEs or ODEs.
method Developed a general RTK framework that decomposes the diffusion process into fewer, more balanced subproblems, using MALA and ULD for sampling.
result The RTK-MALA and RTK-ULD algorithms achieve faster convergence rates and lower error compared to existing methods.
This primer explains diffusion models in general state spaces.
problem Diffusion models in general state spaces are not well-introduced.
method Develops discrete-time and continuous-time views of diffusion models, deriving Fokker-Planck and master equations.
result Unified understanding of diffusion models across continuous and discrete domains.
DMPS uses diffusion maps and LAWGD for efficient generative modeling.
problem Efficiently modeling complex data distributions.
method Diffusion maps for manifold learning and LAWGD for sampling.
result DMPS outperforms other methods on moderate-dimensional data.
Flow Matching improves statistical guarantees through kernel density estimation.
problem Improving statistical guarantees for generative models.
method Connecting Flow Matching to kernel density estimation and verifying optimal rates of convergence.
result Flow Matching achieves optimal rates up to logarithmic factors for large networks and on lower-dimensional manifolds.
Local data coverage governs memorization in diffusion models.
problem Memorization in diffusion models
method Derive a theoretical criterion based on local data coverage
result Predicts memorization based on density of training data in neighborhood and dataset size
G-FuNK learns solutions for nonlinear PDEs on multiple domains and parameters.
problem Predicting time-dependent dynamics of complex systems governed by nonlinear PDEs with varying parameters and domains.
method Graph Fourier Neural Kernels combining domain-adapted and transferable components for non-diffusive and diffusive terms.
result G-FuNK achieves low relative errors on unseen domains and fiber fields, significantly accelerating predictions.
Graph diffusion processes approximate manifold heat semigroups using graph transition matrices.
problem Approximating manifold heat semigroups from graph data under low regularity conditions.
method Iterating graph transition matrix P to approximate Qt=etΔ, bounding error in ∞-norm. result Convergence rates O(N−2/(d+6)) for manifold heat semigroup approximation, valid for in-sample and out-of-sample. DM uses semigroup property to tune diffusion time for better data analysis.
problem Difficulty in tuning diffusion time for optimal data analysis.
method Proposes a semigroup criterion to select diffusion time.
result Effective and robust method for picking diffusion time.
We consider a special family of occupation-time derivatives, namely proportional step options introduced by Linetsky in [Math. Finance, 9, 55--96 (1999)]. We develop new closed-form spectral expansions for pricing such options under a class of nonlinear volatility diffusion processes which includes the constant-elastic…
Researchers calculate entropy of heat kernel on manifolds for very small times.
problem Estimating entropy of heat kernel on compact Riemannian manifolds for small times.
method Asymptotic expansion, polynomial expressions in curvature tensor components.
result First three coefficients of entropy expansion computed and expressed as polynomials.
We introduce {\em vector diffusion maps} (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian…
A new method embeds data using Gaussian processes based on the heat kernel.
problem Embedding high-dimensional data in a low-dimensional space.
method Computing embeddings based on the Karhunen-Loève expansion of the heat kernel.
result The embedding approximates diffusion distances and is robust to outliers.
We develop a maximum penalized quasi-likelihood estimator for estimating in a nonparametric way the diffusion function of a diffusion process, as an alternative to more traditional kernel-based estimators. After developing a numerical scheme for computing the maximizer of the penalized maximum quasi-likelihood function…
One-step diffusion samplers reduce sampling time and computational costs.
problem Efficient sampling from complex distributions.
method One-step diffusion, self-distillation, deterministic flow.
result Achieves competitive sample quality with fewer evaluations.
Score-based diffusion models achieve optimal error bounds under non-parametric assumptions.
problem Improving the minimax optimality of score-based diffusion models.
method Kernel-based score estimation and early stopping strategy.
result Achieves minimax optimal error bounds under sub-Gaussian and Sobolev space assumptions.