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.
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…
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.
In this paper I develop a new computational method for pricing path dependent options. Using the path integral representation of the option price, I show that in general it is possible to perform analytically a partial averaging over the underlying risk-neutral diffusion process. This result greatly eases the computati…
Improves financial instrument pricing using neural networks.
problem Financial instrument pricing within Black-Karasinski model.
method Enhances path-integral approximation with neural networks.
result Demonstrates superior outcomes for multiple calibrations.
GH-PID uses guided harmonic paths for efficient SOT with interpretable diagnostics.
problem Efficiently solving Stochastic Optimal Transport with hard terminal distributions and soft costs.
method Guided Harmonic Path-Integral Diffusion (GH-PID) framework with low-dimensional guidance.
result GH-PID generates geometry-aware, cost-reducing trajectories that match terminal distributions.
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…
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 …
The Black-Scholes theory of option pricing has been considered for many years as an important but very approximate zeroth-order description of actual market behavior. We generalize the functional form of the diffusion of these systems and also consider multi-factor models including stochastic volatility. Daily Eurodoll…
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.
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.
MF-PID uses interacting samples to efficiently transport probability mass.
problem Efficiently transporting probability mass in generative models.
method Introducing Mean-Field Path-Integral Diffusion (MF-PID) where samples become interacting agents.
result MF-PID achieves 19-24% reductions in control energy for demand-response control of energy systems.
New method for sampling from multivariate distributions using optimal control and quantum mechanics.
problem Sampling from continuous multivariate probability distributions efficiently and accurately.
method Harmonic Path Integral Diffusion (H-PID) framework, formulated as a Stochastic Optimal Control problem.
result Efficient sampling algorithms without neural networks, revealing dynamic phase transitions.
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.
Develops a new model-free approach to portfolio theory using rough paths.
problem Handles more general portfolios without probabilistic assumptions.
method Rough path theory for stochastic portfolio theory (SPT).
result Asymptotic growth rates of various portfolios match.
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.
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…
The paper analyzes the stationarity of stochastic Volterra integral equations and introduces fake stationary regimes.
problem Analyzing the stationarity of non-Markovian dynamical systems described by SVIEs.
method Investigates the properties of SVIE solutions, focusing on stationarity over finite and long time horizons, and introduces a deterministic stabilizer to induce a fake stationary regime.
result SVIEs do not exhibit a strong stationary regime unless the kernel is constant or degenerate, but a fake stationary regime can be achieved with a deterministic stabilizer.
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.
New algorithm preserves transport maps for better diffusion model training.
problem Training diffusion models with task-specific optimality structures.
method Generalized Schrödinger Bridge Matching (GSBM), inspired by conditional stochastic optimal control.
result GSBM better preserves transport maps, enabling stable convergence and improved scalability.
Bayesian approach improves rain field reconstruction using CMLs and DMs.
problem Challenges in accurately reconstructing ground-level rainfall from CML path-integrated measurements.
method Bayesian inverse problem with Diffusion Models as priors.
result Improved performance in rainfall estimation compared to existing methods.
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.
We show how effective-potential path-integrals methods, stemming on a simple and nice idea originally due to Feynman and successfully employed in Physics for a variety of quantum thermodynamics applications, can be used to develop an accurate and easy-to-compute semi-analytical approximation of transition probabilities…
Cubature on Wiener space [Lyons, T.; Victoir, N.; Proc. R. Soc. Lond. A 8 January 2004 vol. 460 no. 2041 169-198] provides a powerful alternative to Monte Carlo simulation for the integration of certain functionals on Wiener space. More specifically, and in the language of mathematical finance, cubature allows for fast…
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…
Introduces DF framework for sampling decisions from target distributions.
problem Sampling from target distributions with additional guidance.
method DF framework based on MDP and Path Integral Diffusion.
result DF enhances guided sampling across various applications.
Two semi-supervised manifold alignment methods improve cross-domain classification.
problem Aligning data from multiple sources for better analysis.
method SPUD and MASH methods using graph integration and diffusion.
result SPUD and MASH methods outperform existing methods in cross-domain classification.
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.
Novel framework synthesizes stochastic trajectories with anticipated structural breaks.
problem Synthesizing forward-looking, time-evolving stochastic trajectories with anticipated structural breaks.
method Anticipatory Neural Jump-Diffusion (ANJD) flow, AVNSG for dynamic spectral whitening.
result The framework effectively captures non-commutative moments and high-order stochastic texture.
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.
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…
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…