Masking diffusion outperforms other discrete diffusion models by incorporating jump times into the model.
problem Improving the performance of discrete diffusion models.
method Conditioning on the jump schedule of discrete Markov processes.
result Schedule-conditioned discrete diffusion (SCUD) models outperform classical and masking diffusion models.
Paper proposes a new method for training diffusion models using Markov operators.
problem Training efficiency and accuracy in diffusion models.
method Operator-informed score matching using spectral decomposition of Markov operators.
result Improved score matching for both low and high-dimensional distributions.
Optimizes control of hybrid systems with multiple switching processes.
problem Optimal control of hybrid systems with multiple Markov switching processes.
method Combines two separate Markov chains into one synthetic chain, derives HJB equations, and solves the portfolio choice problem.
result Derives explicit solutions and value functions for the optimal control problem.
Proposes MLEs for MMJDM with EM-algorithm.
problem Estimating stock prices with varying drift and volatility.
method EM-algorithm for MLEs of MMJDM.
result Validated with simulated data and fitted to Amazon and Netflix stock prices.
This work extends ME-RL using diffusion models to sample optimal policies.
problem Sampling from the optimal policy trajectory distribution in ME-RL.
method Introducing Diffusion-Augmented Markov Decision Processes (DA-MDPs) to minimize reverse KL divergence.
result DA-MDPs enable seamless integration into various ME-RL methods and outperform baselines.
New method simulates sticky boundaries in multidimensional diffusions.
problem Simulating sticky boundaries in multidimensional diffusions.
method Approximate sticky diffusion by a Markov chain, using either finite difference or matching local moments.
result Validates both construction methods for first-order simulation schemes.
New method uses diffusion models to speed up MCMC sampling.
problem Efficiently exploring high-dimensional and multimodal posterior functions.
method Combines Metropolis-Hastings with diffusion models for global sampling.
result Significant reduction in likelihood evaluations for accurate posterior representation.
Paper develops models for better HFT and algorithmic trading.
problem Inaccurate LOB dynamics in financial markets.
method Semi-Markov and Hawkes jump-diffusion models for LOB dynamics.
result Improved trading strategies through precise model application.
Markov chains and diffusion processes are indispensable tools in machine learning and statistics that are used for inference, sampling, and modeling. With the growth of large-scale datasets, the computational cost associated with simulating these stochastic processes can be considerable, and many algorithms have been p…
We develop continuous time Markov chain (CTMC) approximation of one-dimensional diffusions with a lower sticky boundary. Approximate solutions to the action of the Feynman-Kac operator associated with a sticky diffusion and first passage probabilities are obtained using matrix exponentials. We show how to compute matri…
Faster sampling in discrete diffusion models with predetermined transition time.
problem Efficiency in sampling discrete diffusion models.
method Discrete Non-Markov Diffusion Models (DNDM) with predetermined transition time.
result Significantly reduces the number of function evaluations for faster sampling.
Generative Fractional Diffusion Models improve image diversity and quality.
problem Diffusion models struggle with diversity, mode-collapse, and slow convergence.
method Replaces light-tailed BM with fractional Brownian motion (fBM) and its Markov approximation (MA-fBM).
result GFDM achieves greater diversity and quality in image generation.
Discrete diffusion models improve data generation for discrete data like language and graphs.
problem Adapting diffusion models to discrete state spaces for better data generation.
method Formulated as CTMCs, used uniformization of continuous Markov chains for sampling.
result Derive guarantees for sampling from any distribution on a hypercube, aligning with state-of-the-art achievements.
Method calculates Parisian stopping times and option prices using Markov chains.
problem Computing distribution and pricing of Parisian stopping times under Markov processes.
method Continuous-time Markov chain approximation to solve for distribution and convergence analysis.
result Sharp convergence rate and efficient method for diffusion and jump models.
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.
New algorithm learns value and advantage functions for continuous-time Markov processes without structural assumptions.
problem Learning value and advantage functions for continuous-time Markov processes without structural assumptions.
method Proposes Sobolev-prox fitted q-learning algorithm based on Hilbert-space positive definiteness and boundedness properties of Bellman operators. result Identifies ellipticity as a key structural property enabling reinforcement learning for Markov diffusions.
GLASS Flows improves flow and diffusion model performance by optimizing sampling efficiency.
problem Efficiency bottleneck in sampling Markov transitions for flow and diffusion models.
method Introduces GLASS Flows, a new sampling paradigm that simulates a 'flow matching model within a flow matching model' to sample Markov transitions efficiently.
result Eliminates the trade-off between stochastic evolution and efficiency in large-scale text-to-image models.
The paper sets criteria for no arbitrage in complex financial models.
problem Determining conditions for the absence of arbitrage in financial markets.
method Established deterministic conditions for no arbitrage, NUPBR, and NFLVR in diffusion market models.
result Provided criteria in terms of scale function and speed measure.
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…
Paper introduces DMPMs for efficient discrete data generation with sharp convergence bounds.
problem Efficient generation of discrete data with theoretical guarantees.
method Discrete Markov Probabilistic Models (DMPMs) operating in bit space with time-reversal process.
result Sharp convergence bounds established under minimal assumptions, competitive performance in discrete data generation.
A key task in Bayesian statistics is sampling from distributions that are only specified up to a partition function (i.e., constant of proportionality). However, without any assumptions, sampling (even approximately) can be #P-hard, and few works have provided "beyond worst-case" guarantees for such settings. For log-c…
This study bridges discrete and continuous state spaces using the Ehrenfest process and diffusion models.
problem Understanding the relationship between discrete and continuous state spaces in stochastic processes.
method Investigates time-continuous Markov jump processes on discrete state spaces and their correspondence to state-continuous diffusion processes.
result The time-reversal of the Ehrenfest process converges to the time-reversed Ornstein-Uhlenbeck process, bridging discrete and continuous state spaces.
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.
In this paper we present an algorithm for pricing barrier options in one-dimensional Markov models. The approach rests on the construction of an approximating continuous-time Markov chain that closely follows the dynamics of the given Markov model. We illustrate the method by implementing it for a range of models, incl…
We show by explicit closed form calculations that a Hurst exponent H that is not 1/2 does not necessarily imply long time correlations like those found in fractional Brownian motion. We construct a large set of scaling solutions of Fokker-Planck partial differential equations where H is not 1/2. Thus Markov processes, …
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.
The paper analyzes sampling and estimation on manifolds using Langevin diffusion.
problem Sampling and estimation on compact Riemannian manifolds.
method Discretization of Langevin diffusion with error bounds derived.
result First-order error bounds for bias and variance in estimators.
New algorithm for continuous-time switching systems using variational inference.
problem Inference in time-series data with continuous-time switching systems.
method Developed a variational inference algorithm combining Gaussian process approximation and posterior inference for Markov jump processes.
result Bayesian latent state estimates and point estimates of unknown parameters for arbitrary points on the real axis.
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…
Deep networks can approximate score functions in high-dimensional graphical models efficiently.
problem Approximation efficiency of score functions by deep neural networks in high-dimensional graphical models like Markov random fields.
method Variational inference denoising algorithms and efficient neural network representation.
result Efficient sample complexity bound for diffusion-based generative modeling when score functions are learned by deep neural networks.
In this paper we propose a semi-Markov modulated model of interest rates. We assume that the switching process is a semi-Markov process with finite state space E and the modulated process is a diffusive process. We derive recursive equations for the higher order moments of the discount factor and we describe a Monte Ca…
RML improves generative modeling of complex distributions.
problem Learning complex distributions in applications.
method RML defines a forward process to a known distribution, then learns a reverse Markov process.
result RML efficiently captures complex distributions in simulations and climate data.
DiGS improves sampling from multi-modal distributions.
problem Inadequate mixing in MCMC methods for multi-modal distributions.
method Integrates diffusion models and Gibbs sampling to create an auxiliary noisy distribution.
result DiGS exhibits better mixing for multi-modal distributions than state-of-the-art methods.
New method learns diffusion transition density for Bayesian inference.
problem Bayesian inference on diffusions with inaccessible boundaries.
method Neural Galerkin framework to solve FP equation with Dirac mass.
result Approximates likelihood function for efficient posterior sampling.
QTD integrates quantization with diffusion for efficient data generation.
problem Challenges in continuous diffusion models, especially long-range transitions and biases.
method Quantized Transition Diffusion (QTD) integrates data quantization with discrete diffusion dynamics.
result QTD achieves efficient data generation with minimal score evaluations.
SGLDiff approximates Bayesian posterior distributions with subsampling error.
problem Approximating Bayesian posterior distributions in large-scale data settings.
method Stochastic Gradient Langevin Diffusion (SGLDiff) with subsampling.
result The Wasserstein distance between the posterior and SGLDiff's limiting distribution is bounded by a fractional power of the mean waiting 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.
Neural network approximates diffusion bridges for efficiency and robustness.
problem Efficient simulation of conditioned diffusion processes, especially rare events and multimodal distributions.
method Trains a neural network to approximate bridge dynamics, eliminating MCMC and score modeling.
result Efficient sampling of conditioned diffusion bridges at comparable cost to unconditioned process.
Develops RL for optimal market-making in non-Markov processes.
problem Optimal market-making in non-Markov price processes.
method Deep reinforcement learning with Soft Actor-Critic (SAC) algorithm.
result Optimal strategy for market-making in semi-Markov and Hawkes Jump-Diffusion dynamics.
Most previous contributions to BSDEs, and the related theories of nonlinear expectation and dynamic risk measures, have been in the framework of continuous time diffusions or jump diffusions. Using solutions of BSDEs on spaces related to finite state, continuous time Markov chains, we develop a theory of nonlinear expe…
Improves sampling quality in model composition using MH-like acceptance rule for score-based diffusion models.
problem Inability to apply MH corrections in score-based diffusion models for model composition.
method Introduces a novel MH-like acceptance rule based on line integration of the score function.
result Relative improvements similar to energy-based models without explicit energy parameterization.
This study shows how DDPM can be represented by the OU process.
problem Designing optimal noise schedules for DDPM.
method Formal equivalence between DDPM and OU process, heuristic designs based on Fisher Information.
result Fisher-Information-motivated schedule corresponds to cosine noise schedule.
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.
A key task in Bayesian machine learning is sampling from distributions that are only specified up to a partition function (i.e., constant of proportionality). One prevalent example of this is sampling posteriors in parametric distributions, such as latent-variable generative models. However sampling (even very approxim…
Schrödinger bridge solved with Weyl calculus for quadratic state cost.
problem Optimal control policy to steer joint state statistics.
method Weyl calculus in quantum mechanics for reaction-diffusion PDEs.
result Explicit Markov kernel for quadratic state cost found.
New theory improves diffusion models' convergence rates.
problem Understanding and optimizing diffusion models for faster data generation.
method Developed non-asymptotic theory for diffusion models with minimal assumptions.
result Established convergence rates for two diffusion models.
Using results from our companion article [arXiv:1112.4824v2] on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and mathematical finance, concerning degenerate diffusion processes. We show that th…
Paper reviews methods for conditional sampling in generative diffusion models.
problem Extending generative diffusion models to sample from conditional distributions.
method Review of existing computational approaches to conditional sampling.
result Highlight key methodologies for constructing conditional generative samplers.