The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
problem Gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
method Derives Li-Yau and Souplet-Zhang type gradient estimates for the given equations.
result Gradient estimates for the equations on complete noncompact metric measure spaces with compact boundary.
We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: ut=Δφ(up) associated with the Witten Laplacian on Riemannian manifolds. Under the assumption that the m-dimensional Bakry-Emery Ricci curvature is bounded from below, we obtain gradient estimates which…
In this work we derive local gradient and Laplacian estimates of the Aronson-Bénilan and Li-Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discove…
We study the fast diffusion equation (FDE) with a linear forcing term under the Ricci flow on complete manifolds with bounded curvature and nonnegative curvature operator. We prove Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions of the FDE. In addition, we use similar meth…
SA-Solver improves stochastic sampling from DPMs.
problem Efficient sampling from Diffusion Probabilistic Models (DPMs) is time-consuming.
method Proposes SA-Solver, an improved stochastic Adams method for solving diffusion SDE.
result SA-Solver achieves improved or comparable performance compared to SOTA methods for few-step sampling.
Finite-time extinction and smoothing effects in fractional fast diffusion on manifolds.
problem Finite-time extinction and smoothing effects in fractional fast diffusion equations.
method Nonlinear semigroups techniques, weighted Lp spaces, fractional Green function. result Sharp extinction rates and pointwise lower bounds for solutions.
Reflected Diffusion Models improve on score-based models by incorporating data constraints.
problem Numerical error in score-based models leads to unnatural samples.
method Reverses a reflected stochastic differential equation on data support, learning perturbed score function through generalized score matching loss.
result Improves sample quality and fidelity without architectural modifications.
Sig-DEG speeds up diffusion models by distilling them into faster approximations.
problem Computational intensity of diffusion models at inference time.
method Signature-based differential equation generation to summarize Brownian motion.
result Sig-DEG reduces inference steps by an order of magnitude while maintaining generation quality.
We prove the sharp local L^1 - L^\infty smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known L^p - L^\infty estimate for p larger than 1. It also has several applications in geometry, providing …
Study well-posedness of fast diffusion equation on noncompact manifolds.
problem Investigate well-posedness of fast diffusion equation in noncompact Riemannian manifolds.
method Establish existence and uniqueness of solutions for globally integrable initial data.
result Global solutions exist for initial data in Lloc1 on general Riemannian manifolds. We prove that conservation of probability for the free heat semigroup on a Riemannian manifold M (namely stochastic completeness), hence a linear property, is equivalent to uniqueness of positive, bounded solutions to nonlinear evolution equations of fast diffusion type on M of the form ut=Δφ(u), φ being an ar…
New method for fast inference in diffusion models.
problem Intractable probabilistic inference in diffusion models.
method Variational Gaussian Process, exponential family description, convex optimization.
result Improved fast algorithm for learning model parameters.
GF-Net learns Green's functions for linear reaction-diffusion equations.
problem Learning Green's functions for linear reaction-diffusion equations on arbitrary domains.
method GF-Net, a neural network, learns Green's functions in an unsupervised manner using physics-informed approach and symmetry.
result GF-Net efficiently solves linear reaction-diffusion equations under various boundary conditions and sources.
New method infers diffusion equations from sparse data.
problem Statistical inference of diffusion equations from limited data.
method Neural network-based estimators for drift and diffusion tensor.
result Statistical convergence guarantees for Hölder continuous processes.
A fast voice conversion method using diffusion models.
problem One-shot many-to-many voice conversion.
method Diffusion probabilistic modeling with Fast Maximum Likelihood Sampling Scheme.
result Superior quality compared to state-of-the-art approaches.
PRISMA uses PDE residuals for fast, robust, and accurate inference.
problem Slow gradient-based optimization and instability in PDE residual-based methods.
method Integrates PDE residuals directly into the model's architecture via attention mechanisms in the spectral domain.
result Competitive accuracy with significantly lower inference costs and faster speeds.
A new sampling method estimates scores without training or nested MCMC.
problem Efficient sampling from complex, unnormalised distributions.
method Multiscale averaging in SDEs for score estimation.
result Empirical results show competitive accuracy and efficiency.
In this paper we discuss the basket options valuation for a jump-diffusion model. The underlying asset prices follow some correlated local volatility diffusion processes with systematic jumps. We derive a forward partial integral differential equation (PIDE) for general stochastic processes and use the asymptotic expan…
DPM-Solver speeds up DPM sampling to 10-20 function evaluations.
problem Slow sampling from Diffusion Probabilistic Models (DPMs).
method Exact formulation of diffusion ODE solutions, using change-of-variable and exponentially weighted integral.
result Generates high-quality samples in 10-20 function evaluations.
The paper examines the sampling dynamics of diffusion models using ODEs.
problem Understanding the sampling dynamics of diffusion models.
method Careful inspection of ODE-based sampling of SDEs, revealing structures and relationships.
result Established a theoretical relationship between optimal ODE-based sampling and mean-shift algorithm.
DBIMs speed up DDBMs and improve image translation.
problem Efficiently sampling from DDBMs for image translation.
method Generalized diffusion bridges and booting noise.
result DBIMs are up to 25imes faster and maintain generation diversity. The Backpropagation algorithm relies on the abstraction of using a neural model that gets rid of the notion of time, since the input is mapped instantaneously to the output. In this paper, we claim that this abstraction of ignoring time, along with the abrupt input changes that occur when feeding the training set, are …
We give a new proof of the fact that the value function of the finite time horizon American put option for a jump diffusion, when the jumps are from a compound Poisson process, is the classical solution of a free boundary equation. We also show that the value function is C1 across the optimal stopping boundary. Our …
Study finds conditions for global minimizers on curved manifolds with fast diffusion and nonlocal interactions.
problem Existence of global minimizers for a free energy functional on negatively curved manifolds.
method Investigation of Carlson-Levin type inequalities for Cartan-Hadamard manifolds.
result Establishes necessary and sufficient conditions for the existence of global energy minimizers.
New method learns discrete graph diffusion via free-energy gradient flows.
problem Challenges in translating continuous diffusion models to discrete spaces.
method Proposes a novel computational approach using a specific metric on the simplex.
result Recover the underlying functional for various graph classes.
A new method speeds up sampling in diffusion models.
problem Slow sample generation in diffusion models.
method Proposed Splitting Integrators for fast stochastic sampling.
result Achieved FID score of 2.36 in 100 NFE, significantly faster than baselines.
We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres $(\Sn, [g_{\Sn}])$, it converges to the standard sphere up to a Möbius diffeomorphism. This result allows us to obtain extinct…
Develops deep learning for fast, accurate option pricing models.
problem Computational efficiency and accuracy in option pricing models.
method Neural network generators solving backward Kolmogorov equations for TPDFs.
result Ultra-fast, highly accurate option pricing models for various asset models.
Efficiently handles contextual bandits with diffusion models.
problem Challenges in online decision-making with contextual bandits.
method Leverage pre-trained diffusion models as priors to capture action dependencies.
result Developed an algorithm for efficient posterior approximation.
Stein's method for measuring convergence to a continuous target distribution relies on an operator characterizing the target and Stein factor bounds on the solutions of an associated differential equation. While such operators and bounds are readily available for a diversity of univariate targets, few multivariate targ…
Let n>2, 0<m≤(n−2)/n, p>\max(1,(1-m)n/2), and 0≤u0∈Llocp(Rn) satisfy liminfR→∞R−n+1−m2∫∣x∣≤Ru0dx=∞. We prove the existence of unique global classical solution of ut=mn−1Δum, u>0, in Rn×(0,∞), u(x,0)=u_0(x) in Rn. If in addition …
Improved diffusion sampling for inverse problems with faster and more robust inference.
problem High computational cost and lack of robustness in diffusion posterior sampling.
method Amortized variational inference with explicit likelihood guidance.
result Improved trade-off between inference speed and robustness to unseen degradations.
We investigate the m-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement K-convexity of the m-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the K-convexity of the weig…
SiD distills pretrained diffusion models into a fast one-step generator.
problem Efficiently distilling pretrained diffusion models into a fast generator.
method Reformulates forward diffusion processes as semi-implicit distributions and uses three score-related identities to create a loss mechanism.
result Achieves high FID performance and significantly reduces generation time.
Bayesian inference for biochemical reaction networks using jump-diffusion approximations.
problem Estimating hidden quantities in poorly characterized biochemical processes.
method Developed a Bayesian inference algorithm based on Markov chain Monte Carlo and sequential Monte Carlo methods.
result Numerical evaluation of the algorithm for a partially observed multi-scale birth-death process.
Improved diffusion model generation speed with speculative sampling.
problem Generating samples from computationally expensive diffusion models.
method Extending speculative sampling to diffusion models, using fast draft models for candidate token generation.
result Significant speedup in generation, halving the number of function evaluations.
Efficiently samples complex distributions using tensor train format.
problem Sampling from high-dimensional complex probability densities efficiently.
method Integrates tensor train format with backward stochastic differential equations (BSDEs) for fast, robust, and accurate sampling.
result Improved efficiency in sampling from challenging target distributions.
Smooth convergence shown for curve diffusion flows.
problem Embeddedness and global existence of curves.
method Exponentially fast convergence established.
result Smooth convergence for curve diffusion flows.
Energy-based diffusion models improve molecular sampling and simulation.
problem Inconsistency between diffusion model scores and equilibrium distributions.
method Fokker-Planck regularization to enforce consistency.
result Improved consistency and efficient sampling of biomolecular systems.
A fast method approximates likelihood scores for noisy linear inverse problems.
problem Solving noisy linear inverse problems efficiently.
method Proposes a simple closed-form approximation to the likelihood score for diffusion and flow-based models.
result Significantly faster than baseline methods while maintaining competitive or better reconstruction performances.
ELM combines machine learning and feature engineering for anomalous diffusion detection.
problem Quantitative characterization of anomalous diffusion from single trajectories.
method Extreme Learning Machine (ELM) combined with feature engineering.
result ELM achieves satisfactory performance in AnDi challenge tasks.
Algorithm minimizes risk for multiclass classification of stochastic diffusion paths.
problem Multiclass classification of stochastic diffusion paths with distinct drift functions.
method Empirical risk minimization using L2 risk.
result Achieves fast rates of convergence under margin assumption.
Unified diffusive bounds for non-linear parabolic equations.
problem Proving diffusive upper bounds for parabolic equations.
method Simple exponential deformation argument.
result Unified diffusive upper bounds for a wide class of non-linear parabolic equations.
We consider radial solutions to the fast diffusion equation ut=Δum on the hyperbolic space HN for N≥2, m∈(ms,1), ms=N+2N−2. By radial we mean solutions depending only on the geodesic distance r from a given point o∈HN. We investigate their fine asymptotics near…
New method converts and optimizes sampling schedules for generative models.
problem Optimizing sampling schedules for generative models like flows and diffusions.
method Unified framework for stochastic interpolants, including point mass schedules.
result Demonstrated efficient generation of images with fewer steps.
We study a stochastic equation modeling the lay-down of fibers in the production process of nonwovens. The equation can be formulated as some manifold-valued Stratonovich stochastic differential equation. Especially, we study the long time behaviour of the stochastic process. Demanding mathematical difficulties arising…
CTM improves diffusion model sampling quality with efficient ODE traversal.
problem Lack of natural trade-off between sample quality and speed in consistency models.
method CTM trains a neural network to output scores and traverse ODE trajectories efficiently.
result CTM achieves state-of-the-art FIDs and improves sample quality with increased computational budget.
Study optimal investment and consumption in a stochastic factor model.
problem Optimal investment and consumption decisions in a stochastic factor model.
method Characterization of well-posedness, numerical algorithm, and general theory of sub- and supersolutions for HJB equation.
result Proves existence and provides bounds for the solution to the HJB equation.