Sparse Kernel Flows learns dynamical systems from data.
problem Learning dynamical systems from limited data.
method Sparse Kernel Flows: trains optimal kernel from a dictionary of kernels.
result Sparse Kernel Flows can learn from 132 chaotic systems.
Extends heat kernel estimates for super Ricci flow.
problem Heat kernel estimates for super Ricci flow.
method Generalizes Bamler-Zhang's geometric analysis to super Ricci flow.
result Obtains Gaussian heat kernel estimates for super Ricci flow.
Paper explores Fisher-Rao gradient flows and their kernel approximations.
problem Understanding and analyzing approximations of Fisher-Rao gradient flows.
method Rigorous investigation of Fisher-Rao and Wasserstein type gradient flows, focusing on kernel approximations.
result Proves evolutionary Γ-convergence for kernel-approximated Fisher-Rao flows, providing theoretical guarantees.
The paper constructs optimal confidence bands for kernel gradient flow estimators.
problem Estimating generalization error and constructing confidence bands for kernel gradient flows.
method Established convergence rates and constructed optimal confidence bands under capacity-source condition.
result Optimal confidence bands for kernel gradient flows have shrinkage rates close to minimax optimal rates.
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.
Proves upper bounds for heat kernels evolving on manifolds.
problem Bounding heat kernels on evolving manifolds.
method Logarithmic Sobolev inequalities and ultracontractivity estimates.
result Gaussian upper bounds for heat kernels are derived.
Identifies a gradient flow to solve kernel learning problems with noise reduction.
problem Kernel learning problem with Gaussian noise.
method Riemannian gradient flow with continuous Lyapunov functionals.
result Flow reduces noise and finds stationary points.
Paper establishes a generalization bound for gradient flow using a data-dependent kernel.
problem Understanding the generalization properties of gradient-based optimization methods.
method Establishes a generalization bound for gradient flow through a data-dependent kernel called the loss path kernel (LPK).
result The LPK captures the entire training trajectory and leads to tighter generalization guarantees.
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.
This paper shows equivalence between SVGD and BBVI using kernel gradient flows.
problem Bayesian inference methods and their equivalence.
method Formalizes equivalence between SVGD and BBVI using kernel gradient flows.
result BBVI corresponds precisely to SVGD when using the neural tangent kernel.
The paper proves existence and growth estimates for inverse mean curvature flow and related p-Laplacian Green kernel decay.
problem Existence and growth estimates for inverse mean curvature flow.
method Proving new decay estimates for the Green kernel of the p-Laplacian. result Existence and optimal growth estimates for the weak inverse mean curvature flow.
New approach to learning kernels from data using AIT principles.
problem Learning kernels from data in machine learning.
method Sparse Kernel Flows method based on AIT principles.
result Sparse Kernel Flows aligns with MDL principle and offers a robust theoretical foundation.
Theory explains why neural nets better learn Calabi-Yau metrics.
problem Learning Calabi-Yau metrics with neural networks.
method Developed a theory of metric flows in neural network space.
result Finite-width neural networks learn Calabi-Yau metrics better than fixed kernel methods.
Proves heat kernel superconvexity in hyperbolic space.
problem Heat kernel superconvexity in hyperbolic space.
method Proves conjecture by Bernstein in all dimensions.
result Analog of Huisken's monotonicity formula for mean curvature flow.
Proves existence and uniqueness of mean curvature flow.
problem Existence and uniqueness of mean curvature flow.
method Heat kernel estimates and contraction mapping principle.
result Continuous dependence of mean curvature flow on initial data.
New kernel improves MMDs with theoretical guarantees for gradient flows.
problem Non-smoothness of negative distance kernel in MMDs.
method Smoothed 1D absolute value function followed by fractional integral transform.
result Improved theoretical guarantees for Wasserstein gradient flows.
In this paper, we study the behavior of Bergman kernels along the Kähler Ricci flow on Fano manifolds. We show that the Bergman kernels are equivalent along the Kähler Ricci flow under certain condition on the Ricci curvature of the initial metric. Then, using a recent work of Tian and Zhang, we can solve a conjecture …
Proposes a new method for posterior sampling using MMD with negative distance kernel.
problem Posterior sampling and conditional generative modeling.
method Approximates joint distribution using discrete Wasserstein gradient flows of MMD with negative distance kernel.
result Establishes an error bound for posterior distributions and proves the method is a Wasserstein gradient flow.
Paper reconciles different Ricci flow approaches and proves weak solutions.
problem Proving weak solutions for Ricci flows with singularities.
method Introducing a novel hitting estimate for Brownian motion, compensating for lack of lower heat kernel bounds.
result Every noncollapsed limit of Ricci flows and singular Ricci flows are weak solutions.
Regularizes f-divergences with MMD to analyze Wasserstein flows.
problem Limitations of f-divergences in measures' support. method Rewriting MMD regularization as Moreau envelope in RKHS, analyzing gradients.
result Analysis of Wasserstein flows of MMD-regularized f-divergences. We investigate the prescribed Q-curvature flow for GJMS operators with non-trivial kernel on compact manifolds of even dimension. When the total Q-curvature is negative, we identify a conformally invariant condition on the nodal domains of functions in the kernel of the GJMS operator, allowing us to prove the global ex…
In this paper, we first derive a Sobolev inequality along the harmonic-Ricci flow. We then prove a linear parabolic estimate based on the Sobolev inequality and Moser's iteration. As an application, we will obtain an upper bound estimate for the heat kernel under the flow.
New variational flows improve Monte Carlo and normalization tasks.
problem Intractable global optimum in expressive variational families.
method Constructing asymptotically exact variational flows from involutive MCMC kernels.
result Provable total variation convergence of new variational families.
Uniform proof for Ricci flows on complete manifolds.
problem Proving short-time existence, uniqueness, and continuous dependence for Ricci flows.
method Using Koch-Lamm framework and tensor heat kernel estimates, with a new continuous dependence estimate.
result Uniform proof of short-time existence, uniqueness, and continuous dependence for Ricci flows.
Paper improves heat kernel estimates on Ricci shrinkers.
problem Estimates on heat kernels for Ricci shrinkers.
method Improves estimates from previous work and extends recent progress.
result Theory of $\IF$-convergence holds on Ricci flows induced by Ricci shrinkers.
In this paper, we study how to get the Ricci expanders from W+-functional through the heat kernel estimate of the conjugate heat equation to the type III singularity of Ricci flow. The Gaussian upper and lower bounds are established for the related heat kernel in accordance to the interesting work of Cao-Zhang for the …
We characterize the conjugate linearized Ricci flow and the associated backward heat kernel on closed three--manifolds of bounded geometry. We discuss their properties, and introduce the notion of Ricci flow conjugated constraint sets which characterizes a way of Ricci flow averaging metric dependent geometrical data. …
Study geometric bounds on generalized Ricci flow.
problem No specific problem stated; focuses on bounds.
method Analogous geometric quantities and bounds proven.
result Geometric and analytic bounds established.
Note establishes a local maximum principle for Ricci flow under curvature conditions.
problem Preserving nonnegativity of curvature along Ricci flow with unbounded curvature.
method Combining scaling invariant curvature condition with Dirichlet heat kernel estimates.
result Unified and more direct proof of localized maximum principle.
Paper learns optimal kernels for Gaussian process regression in aerodynamics.
problem Approximating complex functions from limited data in aerodynamics.
method Two algorithms: Kernel Flow and Spectral Kernel Ridge Regression.
result Explicit construction of optimal kernels based on target function features.
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
Improved forecasting for irregularly-sampled time series using kernel flows.
problem Forecasting dynamical systems from irregularly-sampled time series data.
method Directly approximating the vector field using time differences in data-adapted kernels.
result Significant improvement in forecasting accuracy compared to classical methods.
Paper improves MMD flow efficiency with Riesz kernels for image generation.
problem High computational costs in MMD flows for large scale computations.
method Introduces Riesz kernels and sliced MMD for efficient computation.
result Efficient computation of MMD gradients in one-dimensional setting.
Following Donaldson's oppenness theorem on deforming a conical Kähler-Einstein metric, we prove a parabolic Schauder-type estimate with respect to conical metrics. As a corollary, we show that the conical Kähler-Ricci Flow exists for short time. The key is to establish the relevant heat kernel estimates, where we use t…
We estimate the heat kernel on a closed Riemannian manifold M, with dim(M)≥3, evolving under the Ricci-harmonic map flow and the result depends on some constants arising from a Sobolev imbedding theorem. In a special case, when the scalar curvature satisfies a certain natural inequality, we obtain, as a corolla…
In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estim…
New method uses Fokker-Planck equation for sampling and inference.
problem Intractability of evaluating probability density in practical applications.
method Reformulates Fokker-Planck equation as a particle flow method, using velocity field.
result Turns intractable density evaluation into an advantage for variational inference, kernel mean embeddings, and sequential Monte Carlo.
The paper establishes bounds for Ricci flows using entropy and heat kernel methods.
problem Analyzing geometric and analytic properties of Ricci flows.
method Entropy and heat kernel bounds, monotonicity formula for variance of conjugate heat kernels.
result Optimal bounds for Ricci flows, including volume, heat kernel, and entropy estimates.
The paper describes flows of MMD functionals with distance kernel and quantile functions.
problem Wasserstein gradient flows of MMD functionals with negative distance kernel.
method Characterization via Cauchy problem on L2(0,1), solution via subdifferential construction. result Flow invariance and smoothing properties on subsets of C(0,1), absolute continuity of initial measures. This work develops a particle system to approximate Fisher-Rao gradient flows in mean-field optimization.
problem Optimizing probability measures in neural network contexts.
method Constructing an interacting particle system approximating Fisher-Rao gradient flows.
result Propagation of chaos for the Fisher-Rao gradient flow in entropic mean-field optimization.
Kernelised flows improve density estimation and generation with fewer parameters.
problem Limited expressiveness of flow-based models due to invertibility constraints.
method Integrates kernels into normalising flows to enhance expressiveness and efficiency.
result Kernelised flows outperform neural network-based flows in parameter efficiency and low-data scenarios.
Paper analyzes convergence rates of mean-field SVGD method.
problem Establishing quantitative rates of convergence for mean-field SVGD.
method Quantitative analysis of mean-field SVGD dynamics on torus.
result Explicit polynomial convergence rates in L2-norm for Riesz-type kernels.
Generative models use kernel smoothing for conditioning on small example sets.
problem Improving generative models' performance with limited conditioning examples.
method Showed that cross-attention conditioning is equivalent to kernel smoothing, specifically a Nadaraya--Watson kernel smoother.
result The approach predicts and confirms three failure regimes for kernel-based conditioning.
In this article we derive Harnack estimates for conjugate heat kernel in an abstract geometric flow. Our calculation involves a correction term D. When D is nonnegative, we are able to obtain a Harnack inequality. Our abstract formulation provides a unified framework for some known results, in particular including corr…
New method uses normalizing flows to improve force fields for coarse-grained molecular dynamics.
problem Lack of reference atomistic forces makes force matching infeasible for MLCG force fields.
method Introduces noise-based kernels adapted to low-data regimes using normalizing flows.
result Flow-based kernels reduce local distortions while preserving global accuracy.
The paper extends Ricci flow theory with Type-I scalar curvature bounds, proving entropy convergence and characterizing singular sets.
problem Extending Ricci flow theory with Type-I scalar curvature bounds.
method Type-I rescaling procedure and entropy analysis of conjugate heat kernels.
result Entropy of Ricci flow solutions converges to soliton entropy, characterizing singular sets.
Paper develops physics-informed, boundary-constrained Gaussian process for fluid flow field reconstruction.
problem Reconstructing fluid flow fields from limited data.
method Physics-informed, boundary-constrained Gaussian process regression.
result Derives physics-informed kernels for simulating incompressible flows.
Study the spectral flow of Dirac operators on spinor bundles.
problem Understanding the asymptotic behavior of spectral flow for Dirac operators.
method Variation of eta invariant and local index theory technique.
result Established a uniform estimate of the eta invariant for large parameter values.