Flow Matching enables robust training of CNFs with various probability paths.
problem Training Continuous Normalizing Flows (CNFs) at large scales.
method Flow Matching (FM) is a simulation-free approach for training CNFs by regressing vector fields of conditional probability paths.
result Flow Matching with diffusion paths yields more robust and stable training compared to diffusion-based methods.
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.
In this paper we outline methodology to efficiently simulate (jump) diffusion bridge sample paths without discretisation error. We achieve this by considering the simulation of conditioned (jump) diffusion bridge sample paths in light of recent work developing a mathematical framework for simulating finite dimensional …
New MCMC method improves sampling from multimodal distributions.
problem Sampling from multimodal distributions is challenging for classical MCMC methods.
method Interpolating along the diffusion path, preserving mode weights and mixing properties.
result MAD-Path sampler improves global exploration and mode-weight estimation.
AdaPID optimizes diffusion-based samplers by dynamically adjusting schedules.
problem Optimizing the intermediate-time dynamics in diffusion-based samplers.
method Develops a time-varying stiffness schedule using Piece-Wise-Constant (PWC) parametrizations and a hierarchical refinement approach.
result QoS-driven PWC schedules consistently improve sampling fidelity and accuracy.
Develops diffusion samplers for target distributions with efficient score and density estimates.
problem Estimating scores and densities for time-varying distributions.
method Sequential Monte Carlo with diffusion paths and control variates.
result Effective samplers for time-varying distributions with theoretical guarantees and practical applications.
New AI method generates SDE paths without explicit coefficients.
problem Simulating unknown Markovian SDEs with limited data.
method Uses conditional diffusion models on sample paths.
result Consistently outperforms alternative methods in KL divergence.
DALMC provides non-asymptotic error bounds for generative models.
problem Efficiently generating samples from complex data distributions.
method Analysis of diffusion paths and Langevin Monte Carlo.
result Theoretical guarantees for a class of generative models.
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.
Optimizes diffusion processes for target distributions.
problem Efficiently generating target distributions from point masses.
method Stochastic interpolant framework with conditional expectation drift.
result Optimal diffusion coefficient minimizes path-space KL divergence.
Let x denote a diffusion process defined on a closed compact manifold. In an earlier article, the author introduced a new approach to constructing admissible vector fields on the associated space of paths, under the assumption of ellipticity of x. In this article, this method is extended to yield similar results fo…
Study large deviations for hypoelliptic diffusion on sub-Riemannian manifolds.
problem Large deviations for hypoelliptic diffusion measures on sub-Riemannian manifolds.
method Rough path theory and manifold-valued Malliavin calculus.
result Proved a large deviation principle for pinned hypoelliptic diffusion measures.
A new method uses string method to explore diffusion models.
problem Understanding the geometry of learned distributions in diffusion models.
method String method to compute continuous paths between samples.
result The string method identifies realistic morphing sequences and transition pathways.
We consider small-time asymptotics for diffusion processes conditioned by their initial and final positions, under the assumption that the diffusivity has a sub-Riemannian structure, not necessarily of constant rank. We show that, if the endpoints are joined by a unique path of minimal energy, and lie outside the sub-R…
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.
Representations based on random walks can exploit discrete data distributions for clustering and classification. We extend such representations from discrete to continuous distributions. Transition probabilities are now calculated using a diffusion equation with a diffusion coefficient that inversely depends on the dat…
Single model detects abnormal samples across diverse tasks.
problem Detecting abnormal samples in machine learning.
method Introduced Diffusion Paths (DiffPath) using a single unconditional diffusion model.
result Single model performs OOD detection across diverse tasks.
Neural Diffusion Intensity Models simplify Cox processes inference.
problem Intractable nonparametric estimation and posterior inference of latent stochastic intensity in Cox processes.
method Variational framework using neural SDEs, with theoretical guarantee of ELBO maximization coinciding with maximum likelihood estimation.
result Accurate recovery of latent intensity dynamics and posterior paths with significant speedup.
URGE improves diffusion model quality without gradients or Hessian.
problem Improving sample quality in diffusion models without gradient evaluations.
method Path-wise importance reweighting via Girsanov change of measure.
result URGE achieves better generation quality than existing methods.
The author has previously constructed a class of admissible vector fields on the path space of an elliptic diffusion process x taking values in a closed compact manifold. In this Note the existence of flows for this class of vector fields is established and it is shown that the law of x is quasi-invariant under the…
Improved diffusion models solve inverse problems more accurately by correcting sample paths off the data manifold.
problem Current diffusion models for inverse problems often produce suboptimal results due to sample paths deviating from the data manifold.
method Proposed an additional correction term inspired by manifold constraints to make iterations closer to the data manifold.
result The proposed method boosts performance by a large margin, producing promising results in various applications.
Study of most probable paths for anisotropic Brownian motions on manifolds.
problem Characterizing paths of Brownian motions with anisotropic diffusion on manifolds.
method Using stochastic development and fiber bundle of linear frames, the study provides a comprehensive characterization of most probable paths.
result Explicit equations and integration methods for most probable paths on different geometries, including constant curvature surfaces.
Parameter inference for stochastic differential equations is challenging due to the presence of a latent diffusion process. Working with an Euler-Maruyama discretisation for the diffusion, we use variational inference to jointly learn the parameters and the diffusion paths. We use a standard mean-field variational appr…
We investigate the extension of the multilevel Monte Carlo path simulation method to jump-diffusion SDEs. We consider models with finite rate activity, using a jump-adapted discretisation in which the jump times are computed and added to the standard uniform dis- cretisation times. The key component in multilevel analy…
New method for estimating diffusion model densities without solving flows.
problem Estimating log densities from diffusion models efficiently.
method Monte Carlo path integral estimation, avoiding flow solving.
result Significantly more scalable and efficient density estimation.
This paper optimizes diffusion schedules for better sampling from data distributions.
problem Choosing an optimal discretization schedule for denoising diffusion models.
method Adaptive algorithm that selects an optimal schedule based on a work cost measure.
result The learned schedule recovers and outperforms manually tuned schedules.
There are many real-world knowledge based networked systems with multi-type interacting entities that can be regarded as heterogeneous networks including human connections and biological evolutions. One of the main issues in such networks is to predict information diffusion such as shape, growth and size of social even…
A discretization scheme for nonnegative diffusion processes is proposed and the convergence of the corresponding sequence of approximate processes is proved using the martingale problem framework. Motivations for this scheme come typically from finance, especially for path-dependent option pricing. The scheme is simple…
Conservation laws improve diffusion model training by optimizing likelihood.
problem Training diffusion models with denoising objectives.
method Developed conservation laws based on GEXIT functions for memoryless noise processes.
result Unified characterization of diffusion model likelihood, reducing training to learning marginal posteriors.
PDNS tackles multimodal sampling challenges using proximal point method.
problem Multimodal distributions with significant barriers between modes.
method Proximal point method on path measures, decomposing into simpler subproblems.
result PDNS effectively promotes thorough exploration across modes.
Starting from a sequence of independent Wright-Fisher diffusion processes on [0,1], we construct a class of reversible infinite dimensional diffusion processes on $\DD_\infty:= \{{\bf x}\in Let $MbeacompleteRiemnnianmanifoldandμthedistributionofthediffusionprocessgeneratedby\ff 1 2\DD+ZwhereZ$…
Paper develops approximation and statistical theory for signature-based path regression.
problem Understanding how fast signatures approximate continuous path functionals.
method Develops \(L^2\) approximation rate for smooth functionals of Itô diffusions and establishes consistency of statistical learning procedures.
result Signature-based methods improve prediction over handcrafted features in various real-data applications.
Proposes a new method for sampling from unknown distributions.
problem Challenges in sampling from distributions without direct sampling.
method Uses a dilation path to estimate score vectors in closed-form, guiding Langevin dynamics.
result Demonstrates improved sampling performance compared to classical methods.
New method infers and samples point processes from latent diffusion.
problem Modeling point processes with latent diffusion.
method Itô's excursion theory for inference and sampling.
result Proposes a new method to infer and sample point processes.
New SMC sampler improves diffusion model sampling efficiency.
problem Sampling generative diffusion models efficiently.
method Constructs correlated observation paths and designs a sampler.
result Improved statistical efficiency, especially under outlier conditions.
Cosine schedule is optimal for discrete diffusion models.
problem Choosing the best discretization schedule for diffusion models.
method Optimized using Fisher-Rao geometry.
result Cosine schedule is Fisher-Rao optimal.
We use GANs and signatures to approximate conditional laws in filtering and prediction of diffusion processes.
problem Approximating conditional laws for diffusion processes with noisy observations.
method Conditional GANs combined with signatures for approximation.
result Efficient approximation of conditional laws for diffusion processes.
We consider the exact path sampling of the squared Bessel process and some other continuous-time Markov processes, such as the CIR model, constant elasticity of variance diffusion model, and hypergeometric diffusions, which can all be obtained from a squared Bessel process by using a change of variable, time and scale …
The paper predicts cryptocurrency prices using a path-dependent Monte Carlo simulation.
problem Forecasting cryptocurrency prices with volatility and jumps.
method Merton's jump diffusion model with machine learning, traditional, and statistical methods.
result Introduced a path-dependent Monte Carlo simulation for cryptocurrency price prediction.
New method uses neural networks to solve complex PDEs from optimal control theory.
problem Solving high-dimensional Hamilton-Jacobi-Bellman PDEs.
method Iterative diffusion optimization techniques, focusing on path measures and divergences.
result Favourable properties of log-variance divergence for Monte Carlo estimators.
A new slicing method speeds up sliced Wasserstein estimation.
problem Efficiently estimating sliced Wasserstein distance.
method Random-Path Projecting Direction (RPD) for fast sampling.
result RPSW and IWRPSW show favorable performance in training generative models.
New method approximates diffusion process posteriors using moment functions.
problem Approximating posteriors of stochastic differential equations.
method Constructs variational process as controlled prior, approximates posterior with moment functions, uses natural gradient descent.
result Richer variational approximations for state-dependent diffusion terms.
Adjoint sampler targets infinite-dimensional function spaces for efficient sampling.
problem Limited theory and algorithms for sampling infinite-dimensional function spaces.
method Adjoint Sampler for infinite-dimensional function spaces based on stochastic maximum principle.
result FAS achieves superior performance in synthetic and real systems.
Path integral techniques for the pricing of financial options are mostly based on models that can be recast in terms of a Fokker-Planck differential equation and that, consequently, neglect jumps and only describe drift and diffusion. We present a method to adapt formulas for both the path-integral propagators and the …
Combines SMC and diffusion-based samplers for improved sampling performance.
problem Sampling from unnormalized densities efficiently and robustly.
method Viewing SMC and diffusion-based samplers as continuous-time processes, SCLD combines their strengths.
result SCLD achieves improved performance on multiple benchmark problems with less training budget.
This tutorial reviews RL-based methods for optimizing diffusion models to maximize specific metrics.
problem Optimizing diffusion models to generate samples that maximize specific metrics in practical applications.
method Various RL algorithms including PPO, differentiable optimization, reward-weighted MLE, value-weighted sampling, and path consistency learning.
result Exploration of strengths and limitations of RL-based fine-tuning algorithms and their benefits compared to non-RL-based approaches.
The Accardi-Boukas quantum Black-Scholes framework, provides a means by which one can apply the Hudson-Parthasarathy quantum stochastic calculus to problems in finance. Solutions to these equations can be modelled using nonlocal diffusion processes, via a Kramers-Moyal expansion, and this provides useful tools to under…
Develops pathwise analysis for log-optimal portfolios using rough paths theory.
problem Analyzing stability and approximation of log-optimal portfolios.
method Pathwise approach based on càdlàg rough paths theory.
result Establishes pathwise stability and error estimates for log-optimal portfolios.