Efficiently reconstructs jump-diffusion processes from data using neural networks.
problem Reconstructing jump-diffusion processes from data.
method Temporally decoupled squared Wasserstein distance method using parameterized neural networks.
result Enhanced reconstruction of jump-diffusion processes from data.
Error estimates found between SGD with momentum and Langevin diffusion.
problem Quantifying the difference between SGD with momentum and Langevin diffusion.
method Established error estimates using 1-Wasserstein and total variation distances.
result Quantitative error estimates between SGD with momentum and underdamped Langevin diffusion.
Diffusion models achieve nearly optimal distribution estimation in various spaces.
problem Theoretical limitations of diffusion modeling for distribution estimation.
method Analysis of approximation and generalization abilities of diffusion models in Besov spaces.
result Diffusion models achieve nearly minimax optimal estimation rates in total variation and Wasserstein distances.
New network distance based on Laplacian flow captures structure.
problem Measuring similarity between network objects.
method Introducing Laplacian flow to define a new diffusion distance.
result Demonstrated utility and advantage over existing distances.
New analysis improves convergence guarantees for diffusion-based samplers in Wasserstein distance.
problem Improving convergence guarantees for diffusion-based generative models.
method Simple framework to analyze discretization, initialization, and score estimation errors.
result First Wasserstein convergence bound for the Heun sampler and improved results for Euler sampler.
A new method combines EMD and diffusion maps for protein shape analysis.
problem Learning shape spaces of flexible macromolecules.
method Combines Earthmover's distance with diffusion maps for dimensionality reduction.
result EMD-based diffusion maps require fewer samples to recover intrinsic geometry.
In this paper a new dissimilarity measure to identify groups of assets dynamics is proposed. The underlying generating process is assumed to be a diffusion process solution of stochastic differential equations and observed at discrete time. The mesh of observations is not required to shrink to zero. As distance between…
Develops DSD for analyzing multiscale biological networks.
problem Analyzing multiscale structure in biological networks.
method Data-driven diffusion process with multitemporal analysis.
result Parameter-free inference of intrinsic data structure.
Establishes a link between heat diffusion and manifold distances in data.
problem No theoretical link between diffusion-based manifold learning and geodesic distances.
method Formulates heat geodesic embeddings based on Riemannian geometry.
result Method outperforms state-of-the-art in preserving manifold distances and cluster structure.
Solves memorization in diffusion models for manifold data.
problem Memorization effect in diffusion models for manifold data.
method Inertia update at the end of empirical diffusion simulation.
result Approximates true data distribution on a C 2 C^2 C 2 manifold. A new method clusters hyperspectral images using spatially regularized diffusion.
problem Clustering hyperspectral images effectively.
method Spatially regularized random walks and diffusion geometry.
result The method outperforms state-of-the-art algorithms on real data.
DDPMs are robust to noisy score estimates and achieve optimal convergence rates in Wasserstein-2 distance.
problem Evaluating the quality of DDPMs in Wasserstein distance with noisy score estimates.
method Established finite-sample guarantees in Wasserstein-2 distance for DDPMs, considering noisy score estimates.
result Optimal convergence rates in Wasserstein-2 distance for DDPMs, matching Gaussian case.
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.
Diffusion models adapt to low-dimensional data regardless of coefficient choices.
problem Understanding how diffusion models adapt to low-dimensional data structures.
method Analysis of diffusion models with flexible coefficient choices.
result Proven that O ~ ( k / ε ) \widetilde{O}(k/\varepsilon) O ( k / ε ) iterations suffice for accurate sampling in total variation distance. Diffusion models' speed-accuracy relations derived from thermodynamics.
problem Understanding the trade-off between model speed and accuracy.
method Connecting diffusion models to thermodynamics and optimal transport.
result Speed-accuracy relations derived, providing insights into optimal learning protocols.
Paper introduces a new generative learning model using Schrödinger bridge diffusion in latent space.
problem Learning distributions from divergent data distributions.
method Pre-training with large-scale models, Schrödinger bridge diffusion model in latent space.
result Effective control of second-order Wasserstein distance between generated and target distributions.
Paper proves diffusion models work on manifolds.
problem Current diffusion models assume densities are w.r.t. Lebesgue measure, limiting their applicability.
method Introduced convergence results for diffusion models on more general target distributions.
result Quantitative bounds on Wasserstein distance for target and generated distributions.
New sampling method improves efficiency for diffusion models.
problem Efficient sampling from arbitrary smooth distributions in polynomial time.
method Randomized midpoint method for log-concave sampling.
result Achieves best known dimension dependence ( O ~ ( d 5 / 12 ) \widetilde O(d^{5/12}) O ( d 5/12 ) ) for total variation distance. Study on diffusion in non-complete sub-Riemannian manifolds with specific conditions.
problem Analyzing diffusion in incomplete sub-Riemannian manifolds.
method Identifying conditions for Gaussian-type upper bounds and logarithmic asymptotics of heat kernels.
result Optimal constant in exponent for Gaussian-type upper bounds and concentration of diffusion bridge measures.
Starting from a sequence of independent Wright-Fisher diffusion processes on [ 0 , 1 ] [0,1] [ 0 , 1 ] , we construct a class of reversible infinite dimensional diffusion processes on $\DD_\infty:= \{{\bf x}\in Let $ M b e a c o m p l e t e R i e m n n i a n m a n i f o l d a n d be a complete Riemnnian manifold and b e a co m pl e t e R i e mnnianmani f o l d an d μ t h e d i s t r i b u t i o n o f t h e d i f f u s i o n p r o c e s s g e n e r a t e d b y the distribution of the diffusion process generated by t h e d i s t r ib u t i o n o f t h e d i f f u s i o n p r ocess g e n er a t e d b y \ff 1 2\DD+Z w h e r e where w h er e Z$…
LightSBB-M improves generative diffusion modeling with lower 2-Wasserstein distances.
problem Improving generative diffusion models using Schrödinger Bridge and Bass methods.
method Optimizes SBB transport plan with dual representation and tunable beta parameter.
result Achieves up to 32% improvement in 2-Wasserstein distance on synthetic datasets.
A new clustering method for non-linear data on manifolds using diffusion distances.
problem Clustering non-linear data on manifolds with non-Euclidean geometry.
method Diffusion K K K -means clustering on manifolds with polynomial-time convex relaxations via SDP. result Exact recovery of SDPs for diffusion K K K -means under suitable geometric conditions. DIG visualizes complex time series data.
problem Insufficient visualization of high-dimensional dynamical processes.
method DIG (Dynamical Information Geometry) using diffusion framework.
result Reveals structure in multivariate time series data.
This paper analyzes discrete diffusion models, deriving convergence bounds for their generated samples.
problem Theoretical guarantees for discrete-state diffusion models remain under-explored.
method Continuous Time Markov Chain (CTMC) framework and discrete-time sampling algorithm.
result Convergence bounds for KL divergence and TV distance are derived, showing linear dependence on dimension.
A new clustering method uses diffusion processes to reveal hidden structures in data.
problem Clustering data with multimodal, nonlinear densities.
method Combines graph-based diffusion geometry with density estimation techniques.
result Proves sufficient conditions for the accuracy of the LUND procedure.
A novel deep bootstrap framework for nonparametric regression using conditional diffusion models.
problem Nonparametric regression with efficient sampling and accurate estimation.
method Conditional diffusion model for learning conditional distributions, integrating sampling and regression into a unified generative framework.
result Established optimal convergence rates in the Wasserstein distance and convergence guarantees for the bootstrap procedure.
LACD uses unlabeled data to improve conditional diffusion models.
problem Costly and time-consuming acquisition of labeled data.
method Label-augmented conditional diffusion (LACD) with joint denoising score matching.
result LACD converges faster in total variation and Wasserstein-1 distances with sufficient unlabeled data.
Improved KL bounds and Wasserstein guarantees for diffusion flow matching under minimal conditions.
problem Theoretical convergence properties of Brownian motion based diffusion flow matching.
method Refined analysis under Kullback-Leibler and 2-Wasserstein distances.
result State-of-the-art scaling in KL convergence bounds under minimal conditions.
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.
Diffusion models adapt to low-dimensional structures for nonparametric density estimation.
problem High-dimensional statistical inference challenges.
method Viewing diffusion models as implicit density estimators and exploiting their low-dimensional structure.
result Achieves minimax optimal rate for total variation distance with factorizable density.
New theory improves diffusion model convergence for generating data.
problem Improving convergence of diffusion models for data generation.
method Developed a non-asymptotic convergence theory for probability flow ODEs.
result Proves d / ε d/\varepsilon d / ε iterations suffice for approximating target distributions. KIPLMC methods improve statistical inference in latent variable models.
problem Statistical inference in latent variable models.
method Joint diffusion process in parameter and latent variable spaces, with two explicit discretizations.
result KIPLMC methods achieve accelerated convergence rates in Wasserstein-2 distance.
Improved error estimate for SGLD sampling algorithm.
problem Establishing a precise error bound for SGLD.
method Sharp uniform-in-time error estimate for SGLD under mild assumptions.
result Uniform-in-time O ( η 2 ) O(η^2) O ( η 2 ) bound for KL-divergence between SGLD and Langevin diffusion. Diffusion models learn multi-modal distributions with optimal efficiency.
problem Learning high-dimensional distributions with low-dimensional multi-modal structures.
method Score-based diffusion models, focusing on subgaussian distributions within subspaces.
result Diffusion models require O ~ ( ε − k ∨ 2 ) \widetilde{O}(\varepsilon^{-k \vee 2}) O ( ε − k ∨ 2 ) samples for 1-Wasserstein ε \varepsilon ε error, improving over prior guarantees. Study shows diffusion models adapt to manifold hypothesis without dimensionality issues.
problem Empirical success of diffusion models in high-dimensional data.
method Developed a new framework connecting diffusion models to Gaussian Processes theory.
result Achieves rates independent of ambient dimension in terms of score learning and sampling complexity.
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.
New method aligns diffusion models for inference-time properties without retraining.
problem Aligning pre-trained diffusion models for desired inference-time properties.
method Variationally stable Doob's matching for provable guidance estimation.
result Consistent estimator of guidance with non-asymptotic convergence guarantees.
Conditional diffusion models improve data generation with non-asymptotic convergence bounds.
problem Lack of non-asymptotic properties in conditional diffusion models.
method Integrates a pre-trained model into the diffusion model framework to capture conditional distributions.
result Established upper error bounds for the convergence between original and generated conditional distributions.
The paper analyzes reflected diffusion models on hypercube data.
problem Challenges in modeling bounded domains with low-dimensional data.
method Employed an infinite series expansion of transition densities to bound the score function and its approximation.
result Established convergence rates for generative algorithm adapting to intrinsic dimensionality.
Generative diffusion models improve channel sampling from limited data.
problem Challenges in channel modelling and data collection for wireless systems.
method Diffusion model with U-Net architecture for frequency domain synthesis.
result Stable training and diverse high-fidelity samples generated from true channel distribution.
FKEE estimates expectations without samples, using diffusion bridges and PINNs.
problem Estimating expectations without large sample sizes.
method Diffusion bridge models and Feynman-Kac operator approximation using PINNs.
result Significantly reduces variance and improves efficiency.
High-dimensional curved diffusions show abrupt convergence at a critical time.
problem Understanding abrupt convergence in high-dimensional curved diffusions.
method Functional inequalities and spectral rigidity.
result Abrupt convergence (cutoff) occurs in high dimensions, linked to spectral rigidity.
Unified framework for multi-view diffusion geometries using intertwined diffusion trajectories.
problem Constructing multi-view diffusion geometries with flexible view interaction and fusion.
method Intertwined multi-view diffusion trajectories (MDTs) as a class of inhomogeneous diffusion processes.
result Established theoretical properties and derived diffusion distances and embeddings.
In this paper we will give a new proof of the monotonicity of Wasserstein distances of two diffusions under super Ricci flow. Our proof is based on the coupling method of B.Andrew and J.Clutterbuck. The same method can also be applied to the contractivity of normalized L-Wasserstein distance under backward Ricci flow.
This paper improves non-asymptotic bounds for denoising diffusions, focusing on the Ornstein-Uhlenbeck process.
problem Improving non-asymptotic bounds for denoising diffusions, especially for the Ornstein-Uhlenbeck process.
method Explicit non-asymptotic bounds on forward diffusion error in total variation, considering multi-modal data distributions.
result The Ornstein-Uhlenbeck process cannot be significantly improved in terms of reducing terminal time T T T for multi-modal data distributions. A new method compares unaligned datasets using log-Euclidean signatures of SPD matrices.
problem Efficiently comparing datasets with unknown alignment.
method Diffusion operators, Riemannian geometry, log-Euclidean metric.
result LES distance recovers meaningful structural differences, outperforming existing methods.
Paper establishes fast convergence theory for diffusion models under minimal assumptions.
problem Establish theoretical guarantees for diffusion models under minimal assumptions.
method Developed a convergence theory for denoising diffusion probabilistic models (DDPM) under minimal assumptions.
result Achieved convergence rate of O(d/T) for target distributions with finite first-order moment.
New bounds close the score matching gap for diffusion models.
problem The difference between sample quality and score matching loss in diffusion models.
method Theoretical analysis of score matching gap, developing tighter bounds for KL divergence, reverse KL divergence, and Wasserstein distance.
result The quality of score approximation impacts closing the score matching gap for low noise scales.