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…
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48 results for “Gaussian paths”
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.
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.
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 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…
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.
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.
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.
Generalized Gaussian Bridgesmath.PR
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 …
Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughnessmath.PR
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…
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…
Differential privacy of Gaussian process posterior sampling
problem Privacy of posterior sample paths from Gaussian process
method Intrinsic randomness yields DP guarantees
result Intrinsic randomness yields DP guarantees
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).
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…
The calculation of minimum energy paths for transitions such as atomic and/or spin re-arrangements is an important task in many contexts and can often be used to determine the mechanism and rate of transitions. An important challenge is to reduce the computational effort in such calculations, especially when ab initio …
Using integration by parts on Gaussian space we construct a Stein Unbiased Risk Estimator (SURE) for the drift of Gaussian processes using their local and occupation times. By almost-sure minimization of the SURE risk of shrinkage estimators we derive an estimation and de-noising procedure for an input signal perturbed…
We introduce a model for the dynamics of stock prices based on a non quadratic path integral. The model is a generalization of Ilinski's path integral model, more precisely we choose a different action, which can be tuned to different time scales. The result is a model with a very small number of parameters that provid…
Transformers can solve complex filtering problems for non-Gaussian signals.
problem Non-linear and non-Markovian filtering problems for conditionally Gaussian signals.
method Continuous-time transformer models called filterformers.
result Filterformers can approximate the conditional law of non-Markovian and conditionally Gaussian signal processes.
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.
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.
New BdryMatérn GP model for reliable boundary integration on irregular domains.
problem Incorporating boundary information in Gaussian process models for complex phenomena.
method Proposes a novel BdryMatérn GP framework with a new covariance kernel derived via path integral and stochastic PDE.
result Sample paths from the BdryMatérn GP satisfy desired boundaries with smoothness control on derivatives.
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.
A Machine Learning Framework for Computing the Most Probable Paths of Stochastic Dynamical Systemsmath.DS
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.
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.
Considering the driving habits which are learned from the naturalistic driving data in the path-tracking system can significantly improve the acceptance of intelligent vehicles. Therefore, the goal of this paper is to generate the prediction results of lateral commands with confidence regions according to the reference…
We propose a representation of Gaussian processes (GPs) based on powers of the integral operator defined by a kernel function, we call these stochastic processes integral Gaussian processes (IGPs). Sample paths from IGPs are functions contained within the reproducing kernel Hilbert space (RKHS) defined by the kernel fu…
Minimalistic model captures head direction system properties.
problem Representing head direction system in a high-dimensional space.
method A minimalistic representation model of the rotation group U(1), including fully connected and convolutional versions.
result Emergence of Gaussian-like tuning profiles and 2D circle geometry in both model versions.
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.
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.
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.
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.
HTFM improves mode coverage and tail-statistic recovery for heavy-tailed data.
problem Tackles heavy-tailed data in various domains with rare events.
method Proposes a framework using clock-conditioned Gaussian sources and truncated logsignature features.
result Improves mode coverage, sample quality, and tail-statistic recovery over Gaussian flow matching and baselines.
Gaussian-Dirichlet Random Fields for Inference over High Dimensional Categorical Observationsstat.ML
Generative model for high-dimensional categorical data using Gaussian-Dirichlet fields.
problem Efficiently modeling and predicting high-dimensional categorical data.
method Combines Dirichlet and Gaussian processes for spatio-temporal modeling.
result Model accurately approximates categorical data in unobserved locations.
GIST adapts HMC by tuning parameters based on position and momentum.
problem Locally adaptive sampling in Hamiltonian Monte Carlo.
method GIST uses Gibbs sampling to adaptively tune HMC parameters.
result GIST improves sampling efficiency for high-dimensional models.
Paper explores rough path theory for frictionless markets, linking NCFL to unbiased rough integrators.
problem Tackles the limits of rough path theory in frictionless markets.
method Investigates the capacity of rough path theory to support No Free Lunch markets.
result Establishes a 'Rough Kreps-Yan' theorem linking NCFL to unbiased rough integrators.
Sharp Gaussian isoperimetry proven along Ricci flow.
problem Proving sharp Gaussian isoperimetric inequality for Ricci flow.
method Using monotonicity formula to prove inequality.
result Exact Gaussian enlargement theorem and concentration estimates.
Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosionsq-fin.MF
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…
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…
For any ReLU network there is a representation in which the sum of the absolute values of the weights into each node is exactly , and the input layer variables are multiplied by a value coinciding with the total variation of the path weights. Implications are given for Gaussian complexity, Rademacher complexity,…
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.
This paper introduces a method to approximate Gaussian process regression by representing the problem as a stochastic differential equation and using variational inference to approximate solutions. The approximations are compared with full GP regression and generated paths are demonstrated to be indistinguishable from …
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.
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…
New imputation strategies improve signature models for irregular time series.
problem Applying signature models to irregular time series requires continuous path construction.
method Characterized imputation as a problem, evaluated various strategies, proposed GP-PoM.
result Gaussian process adapters improve predictive performance and robustness.
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.
We study the problem of learning the support of transition matrix between random processes in a Vector Autoregressive (VAR) model from samples when a subset of the processes are latent. It is well known that ignoring the effect of the latent processes may lead to very different estimates of the influences among observe…
We present the first treatment of the arc length of the Gaussian Process (GP) with more than a single output dimension. GPs are commonly used for tasks such as trajectory modelling, where path length is a crucial quantity of interest. Previously, only paths in one dimension have been considered, with no theoretical con…