The paper defines conditions for Gaussian process sample path regularity.
problem Lack of understanding of Gaussian process sample path regularity.
method Analyzes covariance kernels to determine sample path regularity.
result Necessary and sufficient conditions for Hölder regularity are provided.
This paper optimizes paths for generative models using kinetic energy.
problem Improving generative model performance and sample quality.
method Investigating and optimizing Gaussian probability paths with kinetic energy.
result Kinetic optimal Gaussian paths simplify particle trajectories and improve model performance.
Develops a machine learning framework for computing most probable paths in stochastic systems.
problem Computing the most probable paths in stochastic dynamical systems.
method Reformulates the boundary value problem of Hamiltonian systems and uses a neural network to solve the Euler-Lagrange equation for the Onsager-Machlup action functional.
result Demonstrates the efficacy and accuracy of the machine learning approach in computing most probable paths for stochastic systems with various types of noise.
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.
We introduce time-inhomogeneous stochastic volatility models, in which the volatility is described by a nonnegative function of a Volterra type continuous Gaussian process that may have very rough sample paths. The main results obtained in the paper are sample path and small-noise large deviation principles for the log…
FFM generates functions between Gaussian and data distributions.
problem Generating functions between Gaussian and data distributions.
method Define a path of measures, learn a vector field to generate this path.
result FFM outperforms other function-space generative models.
Extends Gaussian process theory to Banach spaces.
problem Extending Gaussian process theory to Banach spaces.
method Investigates the connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces.
result Characterizes positive definite functions that arise from covariance operators in Banach space setting.
The paper proves a regret bound for a sub-Gaussian mixture on unbounded data.
problem Tackles the challenge of achieving regret bounds for sub-Gaussian mixtures on unbounded data.
method Uses path-wise (deterministic) regret bounds and a cumulative variance process to derive the bound.
result Shows that on a specific event, the regret is eventually bounded by ln(ln V_T).
GAGA accelerates 3D molecular generation by replacing long trajectories with Gaussian approximations.
problem High computational cost of long generative trajectories in 3D molecular generation.
method GAGA identifies a characteristic step where molecular data becomes sufficiently Gaussian, replacing the trajectory with a Gaussian approximation.
result Significant improvement in both generation quality and computational efficiency.
Kernel for Lévy rough paths derived from PDE system.
problem Computing similarity measures for Lévy rough paths.
method Developed a PDE system for the expected signature of inhomogeneous Lévy processes.
result Gaussian martingales' expected signature kernel satisfies a Goursat PDE.
A Bayesian framework models dynamic probability predictions over time.
problem Dynamic probability predictions over time in various settings.
method Gaussian latent information martingale (GLIM) framework.
result GLIM outperforms baseline methods in predicting future uncertainties.
Active learning selects optimal measurement times for inferring continuous paths from sparse data.
problem Inferring continuous probability paths from sparse snapshots in high-fidelity domains like single-cell biology.
method Extends active experimentation to the space of measures using Linearized Optimal Transport (LOT) for probabilistic surrogate modeling.
result Empirical results show that the proposed strategy outperforms uncertainty-agnostic baselines.
This study examines biases in flow matching samplers using finite-sample estimation.
problem Biases in flow matching samplers when using finite-sample surrogates.
method Finite-sample plug-in estimation and hierarchy of empirical FM models.
result Exact empirical minimizer and smoothed plug-in regime identified for affine conditional flows.
In this paper, we establish sample path large and moderate deviation principles for log-price processes in Gaussian stochastic volatility models, and study the asymptotic behavior of exit probabilities, call pricing functions, and the implied volatility. In addition, we prove that if the volatility function in an uncor…
Improved flow matching using Gaussian processes for better sample quality.
problem Training continuous normalizing flows with reduced variance and flexibility.
method Extending conditional flow matching to streams modeled with Gaussian processes.
result Improved quality of generated samples with moderate computational cost.
NUTS mixing time scales as d^(1/4) for Gaussian distributions.
problem Improving the efficiency of the No-U-Turn Sampler (NUTS) for Gaussian distributions.
method Coupling argument leveraging geometric structure of Gaussian concentration, uniformity analysis of NUTS transitions.
result The mixing time of NUTS scales as d^(1/4) for Gaussian distributions, up to logarithmic factors.
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.
The path probability of a particle undergoing stochastic motion is studied by the use of functional technique, and the general formula is derived for the path probability distribution functional. The probability of finding paths inside a tube/band, the center of which is stipulated by a given path, is analytically eval…
New probability path model improves flow matching forecasting performance.
problem Impact of probability path model selection on flow matching forecasting performance.
method Proposed a novel probability path model designed to improve forecasting performance.
result Our model achieves faster convergence during training and improved predictive performance compared to existing models.
In this work we present an analytical model, based on the path-integral formalism of Statistical Mechanics, for pricing options using first-passage time problems involving both fixed and deterministically moving absorbing barriers under possible non-gaussian distributions of the underlying object. We adapt to our probl…
We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended…
Develops methods to find most probable paths on complex manifolds.
problem Identifying optimal paths for manifold-valued processes, especially those with non-trivial structures.
method Constructs a general approach to defining and identifying most probable paths by measuring the Onsager-Machlup function on the anti-development of such processes.
result Derives explicit equations for development most probable paths that encompass various manifold-valued processes.
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 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.
CNFs learn on manifolds using PPD, improving likelihood and sample quality.
problem Training CNFs on manifolds efficiently and accurately.
method Minimizing PPD, a novel divergence, to train CNFs on manifolds.
result CNFs trained with PPD achieve state-of-the-art results on manifold benchmarks.
This work develops a generic framework, called the bag-of-paths (BoP), for link and network data analysis. The central idea is to assign a probability distribution on the set of all paths in a network. More precisely, a Gibbs-Boltzmann distribution is defined over a bag of paths in a network, that is, on a representati…
We construct a path integral based on the coupling of the Liouville action and the Mabuchi K-energy on a one-dimensional complex manifold. To the best of our knowledge this is the first rigorous construction of such an object and this is done by means of probabilistic tools. Both functionals play an important role resp…
Path-dependent PDEs model VIX and Realised Variance options.
problem Modeling volatility derivatives with path-dependence.
method Continuous stochastic volatility model with Gaussian Volterra process, proving well-posedness of PDEs.
result Formulae for greeks and implied volatility provided, finite-dimensional pricing PDEs obtained in Markovian models.
The paper proposes a method to learn evolving multivariate distributions from sample paths.
problem Learning the temporal evolution of multivariate densities from sample data.
method Normalizing flows to construct time-dependent mappings.
result The method can approximate evolving probability density functions from observed data.
New path-gradient estimator for continuous normalizing flows.
problem Limitation of simple Gaussian variational distributions in complex applications.
method Proposed a path-gradient estimator for continuous normalizing flows.
result Empirical evidence of superior performance of the new estimator.
The aim of this paper is to associate a measure for certain sets of paths in the Euclidean plane R2 with fixed starting and ending points. Then, working on parameterized surfaces with a specific Riemannian metric, we define and calculate the integral of the length over the set of paths obtained as the image…
Temporal aggregation reveals latent default correlation from monthly data.
problem Understanding effective default correlation from monthly default data.
method Temporal coarse-graining of latent default-probability paths.
result Temporal coarse-graining improves identifiability and reduces over-allocation of long-horizon fluctuations.
Temporal coarse-graining of latent default paths explains effective correlation in corporate defaults.
problem Understanding effective default correlation in corporate defaults.
method Temporal coarse-graining of latent default-probability paths, applied to corporate default-count data.
result Temporal coarse-graining provides a scale-consistent baseline that improves identifiability and reduces over-allocation of long-horizon fluctuations.
New control methods improve dynamic measure transport paths.
problem Improving paths for dynamic measure transport.
method Connecting mean-field games to optimization problems for learning paths, advocating for smoothness of velocities.
result Our method recovers more efficient and smooth transport models compared to untilted paths.
BWFlow improves graph generation by smoothly interpolating graph components.
problem Disjoint modeling of graph nodes and edges leads to irregular and non-smooth probability paths.
method Modeling graphs as MRFs and using optimal transport displacement for a smooth probability path.
result BWFlow achieves better training convergence and efficient sampling in graph generation.
We consider the problem of function approximation by two-layer neural nets with random weights that are "nearly Gaussian" in the sense of Kullback-Leibler divergence. Our setting is the mean-field limit, where the finite population of neurons in the hidden layer is replaced by a continuous ensemble. We show that the pr…
This paper introduces a novel, well-founded, betweenness measure, called the Bag-of-Paths (BoP) betweenness, as well as its extension, the BoP group betweenness, to tackle semisupervised classification problems on weighted directed graphs. The objective of semi-supervised classification is to assign a label to unlabele…
Study uses Bayes Hilbert framework to recover probability measure flows from sensors.
problem Recovering probability measure flows from moving sensors in a Hilbert space.
method Bayes Hilbert framework, minimum-energy transport, linearization, variational theory.
result Localized sensors can recover reduced path directions but not full state space.
This paper uses probability tensors for efficient path planning in complex scenarios.
problem Efficient path planning in complex environments with obstacles and multiple goals.
method Probability tensors are used to model agent motion and decision-making, incorporating past and future information.
result The model finds solutions in complex scenarios, demonstrating realistic emergent behaviors.
Bayesian framework for SSP problem learns optimal strategy through interactions.
problem Sequential decision-making in stochastic shortest path problems.
method Develops a Bayesian framework to learn optimal action-value function Q∗ through interactions, avoiding unrealistic assumptions. result Demonstrates data efficiency and uncertainty quantification compared to other methods.
This paper analyzes regret bounds for Gaussian process Thompson sampling.
problem Analyzing the performance of Gaussian process Thompson sampling (GP-TS) in Bayesian optimization.
method The paper derives several regret bounds for GP-TS, including a lower bound, upper bounds on the second moment of cumulative regret, expected lenient regret, and improved cumulative regret.
result The paper provides improved regret upper bounds for GP-TS, showing that it suffers from a polynomial dependence on 1/δ with probability δ. Scalable machine learning with path signatures for time series and graphs.
problem Challenges in real-world time series and graph data.
method Combines rough path theory with probabilistic, deep, and kernel methods.
result Scalable models for time series and graph data.
We describe the pricing and hedging of financial options without the use of probability using rough paths. By encoding the volatility of assets in an enhancement of the price trajectory, we give a pathwise presentation of the replication of European options. The continuity properties of rough-paths allow us to generali…
A generalized bridge is the law of a stochastic process that is conditioned on N linear functionals of its path. We consider two types of representations of such bridges: orthogonal and canonical. The orthogonal representation is constructed from the entire path of the underlying process. Thus, future knowledge of the …
The paper develops divergences for Gaussian processes and RKHS settings.
problem Estimating divergences in infinite-dimensional spaces.
method Formulations of Alpha Log-Det divergences, continuity in norm, laws of large numbers, consistent estimation from finite samples.
result Infinite-dimensional divergences can be estimated from finite-dimensional versions with dimension-independent sample complexities.
Minimum energy paths for transitions such as atomic and/or spin rearrangements in thermalized systems are the transition paths of largest statistical weight. Such paths are frequently calculated using the nudged elastic band method, where an initial path is iteratively shifted to the nearest minimum energy path. The co…
This paper improves Bayesian optimization methods with tighter regret bounds and practical solutions.
problem Improving Bayesian optimization methods with tighter regret bounds and practical solutions.
method The paper analyzes and compares different acquisition functions (GP-UCB, TS, PIMS) to achieve tighter Bayesian cumulative regret bounds and address practical issues.
result PIMS achieves the tighter BCR bound and avoids hyperparameter tuning, unlike GP-UCB and TS.
For incomplete sub-Riemannian manifolds, and for an associated second-order hypoelliptic operator, which need not be symmetric, we identify two alternative conditions for the validity of Gaussian-type upper bounds on heat kernels and transition probabilities, with optimal constant in the exponent. Under similar conditi…