A new machine learning method solves high-dimensional Kolmogorov PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs and SDEs.
method Stochastic weighted minimization and stochastic gradient descent with Malliavin weights.
result Accurate approximation of high-dimensional Kolmogorov PDEs and SDEs without curse of dimensionality.
A new method for high-dimensional RBDO using stochastic emulators.
problem Efficient RBDO in high-dimensional settings.
method Unified stochastic representation, stochastic emulators, deterministic mapping.
result Significant computational gains in high-dimensional settings.
Analyzes high-dimensional SGD dynamics using DMFT.
problem Understanding the high-dimensional behavior of multi-pass SGD with small batch sizes.
method Derives DMFT equations for high-dimensional SGD dynamics.
result Proves DMFT equations characterize the asymptotic distribution of SGF parameters.
Deep learning approximates high-dimensional stochastic control problems.
problem High-dimensional stochastic control problems with the curse of dimensionality.
method Approximates time-dependent controls as neural networks and trains them through model dynamics.
result Achieves satisfactory accuracy in high-dimensional problems.
Unified derivation of high-dimensional linear models using stochastic gradient descent.
problem Performance analysis of high-dimensional linear models trained with stochastic gradient descent.
method Derivation of a deterministic equivalence for the two-point function of a random matrix resolvent.
result Unified understanding of model performance including previously known and novel results.
New model estimates higher-order interactions in stochastic processes using lower-dimensional projections.
problem Estimating higher-order interaction effects in stochastic processes with limited data.
method Additive Poisson Process (APP) combines information geometry and generalized additive models to model intensity functions in lower dimensions.
result The model can estimate higher-order intensity functions with sparse data.
New model solves complex SDEs with high-dimensional spatial and stochastic spaces.
problem Solving SDEs with high-dimensional spatial and stochastic spaces.
method Physics-informed deep generative model (sPI-GeM) combining PI-BasisNet and PI-GeM.
result Scalable solution for high-dimensional SDE problems.
We expand volatility models for rough stochastic volatility.
problem Modeling rough stochastic volatility.
method Vol-of-vol expansion for potentially infinite dimensional models.
result Explicit representations of push-down Malliavin weights.
A new deep generative model uses BSDEs for high-dimensional data generation.
problem Generating high-dimensional complex data, especially images.
method Combines BSDEs with deep neural networks for training with MMD loss.
result BSDE-Gen effectively generates high-dimensional data with stochasticity.
This work bridges stochastic interpolants to infinite-dimensional Hilbert spaces.
problem Limited flexibility in generating arbitrary distributions for function-valued data.
method Establishes a rigorous framework for stochastic interpolants in infinite-dimensional Hilbert spaces.
result Achieves state-of-the-art results in conditional generation for complex PDE-based benchmarks.
A method constructs a stochastic surrogate from dimensionality reduction results for high-dimensional uncertainty quantification.
problem High-dimensional uncertainty quantification with physics-based models.
method Constructs a stochastic surrogate model from dimensionality reduction results.
result Preserves convenience of sequential dimensionality reduction and Gaussian process regression while overcoming limitations.
Geometric approach solves Euler equations with random forces.
problem Solving Euler equations with stochastic forcing.
method Infinite-dimensional geometric approach, combining stochastic analysis and Sobolev mappings.
result Local existence and uniqueness of strong solutions.
New method improves stochastic kriging for high-dimensional simulations.
problem High-dimensional simulation models require prohibitive sample sizes and computational costs.
method Tensor Markov kernels and sparse grid experimental designs.
result Sample complexity grows only slightly with dimensionality, improving accuracy and efficiency.
Study on stochastic flows on 7-dimensional spheres.
problem Stochastic processes on 7-dimensional spheres.
method Isometric stochastic flows of Stratonovich SDE on spheres.
result Properties of stochastic processes on Gromoll-Meyer exotic sphere.
Toolbox for stochastic Euler equations using Ebin-Marsden theory.
problem Applying geometric framework to stochastic PDEs.
method Combining infinite-dimensional geometry and stochastic analysis.
result Local well-posedness of maximal solutions for incompressible Euler equation with noise.
New algorithms improve GP inference without approximations, achieving better results.
problem Inexact stochastic optimization methods in Gaussian Processes leading to biased results.
method Exact stochastic inference for GPs with finite dimensional RKHS, extending to infinite dimensions.
result Achieves better experimental results than existing methods in constrained resource settings.
Deep learning for stochastic systems with multi-fidelity data.
problem Predicting stochastic, high-dimensional, and multi-fidelity systems with uncertainty.
method Probabilistic deep learning with variational inference for implicit distributions.
result Effective surrogate models for stochastic systems with quantified uncertainty.
The paper analyzes SGD in high-dimensional networks, revealing new scaling limits.
problem Understanding SGD dynamics in high-dimensional networks.
method Analyzing the effective dynamics of SGD using recent work on the subject.
result A new correction term emerges at the critical scaling regime, changing the phase diagram.
Study BSΔE on lattices for asset price analysis.
problem Optimal investment and market equilibrium analysis in asset price models.
method Backward stochastic difference equations on lattices.
result Applications to optimal investment and market equilibrium analysis.
The paper tackles efficient dimensionality reduction for time series data using stochastic optimization.
problem Estimating the principle component of stationary time series data with nonconvex and dependent data points.
method Proposes a variant of Oja's algorithm combined with downsampling to control bias in stochastic gradient.
result Proves asymptotic rate of convergence and near optimal sample complexity for the proposed algorithm.
Paper tackles infinite-dimensional optimization and Bayesian learning for stochastic differential equations.
problem Learning the drift function of stochastic differential equations with uncertainty quantification.
method Combines infinite-dimensional optimization results with Bayesian hierarchical framework, incorporating shrinkage priors for sparse learning.
result Systematic approach for accurate learning of stochastic differential equations with uncertainty quantification.
The paper develops optimal strategies for high-dimensional statistical arbitrage using factor models and stochastic control.
problem Optimal strategies for high-dimensional statistical arbitrage in a factor model setting.
method Combines factor models with stochastic control to derive optimal strategies.
result Closed-form optimal strategies for market-neutral portfolios in a high-dimensional setting.
Improved NMF using variance-reduced MU rule.
problem Slow convergence of multiplicative update in NMF.
method Introduces variance-reduced stochastic multiplicative update.
result Robustly outperforms state-of-the-art algorithms.
Method uses neural networks for high-dimensional committor function calculations.
problem Computing committor functions for high-dimensional stochastic processes.
method Parameterizes committor function with neural networks and optimizes weights using stochastic algorithms.
result Achieves moderate accuracy for high-dimensional problems.
APAC-Net solves high-dimensional stochastic MFGs using neural networks.
problem High-dimensional stochastic mean-field games.
method Alternating population and control neural networks, parameterizing value and density functions.
result Solves up to 100-dimensional MFG problems.
Proposes flexible auto-encoders for varying data dimensions.
problem Fixed latent dimensions limit data flexibility.
method Stochastic bottleneck with weighted dropouts.
result Seamless variable dimensionality reduction with high performance.
Develops robust methods for infinite-dimensional stochastic processes.
problem Measuring covariations in stochastic evolution equations in infinite dimensions.
method Asymptotic theory for jump robust measurement of covariations.
result Identifies scaling limits for realized covariations.
Expands learning paradigm to stochastic orders using Choquet-Toland distance and Variational Dominance Criterion.
problem Learning high-dimensional distributions with stochastic orders.
method Introduces Choquet-Toland distance and Variational Dominance Criterion, uses input convex maxout networks (ICMNs).
result Proposes surrogates for Choquet-Toland distance and Variational Dominance Criterion with parametric rates.
Paper formalizes multi-dimensional FSD using geometric methods.
problem Complex measure theory and calculus barriers to formalization in proof assistants.
method Geometric framework for first-order stochastic dominance in N dimensions.
result Geometric approach bypasses complex integration theory for direct comparison of survival probabilities.
Study stochastic processes on surfaces in contact sub-Riemannian manifolds using Riemannian approximations.
problem Analyzing stochastic processes on surfaces in contact sub-Riemannian manifolds.
method Employing Riemannian approximations, a second order partial differential operator is derived on the surface. The stochastic process moves along the characteristic foliation induced by the contact distribution.
result Elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from separatrices.
New approach solves utility maximization problems using Delta family.
problem Utility maximization in stochastic control problems.
method Directly solving DP equation with Delta function representation.
result Explicit series representation of value function.
The paper solves optimal bounds for separating data points in high dimensions.
problem Correcting AI errors and analyzing vulnerabilities in high-dimensional data.
method General stochastic separation theorems with optimal probability estimates.
result Explicit and optimal estimates of separation probabilities for important classes of distributions.
Two algorithms optimize high-dimensional convex functions using sparse gradient or function value queries.
problem Optimizing high-dimensional convex functions with sparse gradient or function value queries.
method Two algorithms: successive component/feature selection and noisy mirror descent using Lasso gradient estimates.
result Both algorithms have logarithmically dependent convergence rates on the problem's dimensionality.
New algorithm reduces dimensionality in stochastic optimization.
problem Stochastic optimization in high-dimensional problems.
method Proposes a sparsity-inducing stochastic gradient-free (SI-SGF) algorithm.
result Proves dimension-free query complexity in convex and strongly convex cases.
Optimizes CM for stochastic convex optimization with progressive precision.
problem Stochastic nature of objective function in convex optimization.
method Iterative coordinate minimization with optimal precision control.
result Order-optimal regret performance for strongly convex and nonsmooth functions.
Study pricing options on forward contracts using infinite-dimensional affine models.
problem Pricing European-style options on forward contracts in complex stochastic volatility models.
method Model forward price curves using stochastic partial differential equations modulated by stochastic volatility processes. Analyze two classes of affine stochastic volatility models: Gaussian and pure-jump. Derive conditions for existence of exponential moments and develop semi-closed pricing formulas.
result Developed semi-closed Fourier-based pricing formulas for vanilla call and put options in infinite-dimensional affine models.
New deep learning solver for high-dimensional derivative pricing.
problem High-dimensional derivatives pricing problems.
method Combines deep learning with least square regression for backward SDE solving.
result Accurate and efficient pricing of complex derivatives.
SPP improves partitioning of sparse regions in multi-dimensional arrays.
problem Existing partition models cause unnecessary dissections in sparse regions.
method SPP uses an 'enclosing' strategy to attach patches to dense regions, making it self-consistent for infinite arrays.
result SPP outperforms state-of-the-arts in relational modeling.
A novel method reduces dimensionality for filtering SRNs with observed variables.
problem Challenges in estimating hidden state variables in SRNs with limited observations.
method Filtered Markovian Projection (Filtered MP) for dimensionality reduction in filtering.
result Filtered MP guarantees consistency and superior computational efficiency in high dimensions.
The generalized 5D Black-Scholes differential equation with stochastic volatility is derived. The projections of the stochastic evolutions associated with the random variables from an enlarged space or superspace onto an ordinary space can be achieved via higher-dimensional operators. The stochastic nature of the secur…
Algorithm reduces high-dimensional SLB regret by exploiting hidden low-rank structure.
problem High-dimensional stochastic linear bandits with hidden low-rank structure.
method Projective Stochastic Linear Bandit (PSLB) using PCA projection.
result PSLB achieves tighter regret bound and faster convergence.
Pathwise uniqueness shown for specific stochastic equations.
problem Stochastic Volterra equations with singular kernels and Hölder coefficients.
method Established pathwise uniqueness through Hölder continuity of coefficients.
result Pathwise uniqueness and existence of unique strong solutions.
Paper uses Stochastic Mirror Descent for large-scale sparse recovery problems.
problem Statistical estimation of high-dimensional sparse parameters.
method Non-Euclidean Composite Stochastic Mirror Descent (CSMD) algorithm for solving penalized stochastic optimization problems.
result The proposed algorithm achieves optimal convergence in sparse Generalized Linear Regression problems.
We show that stochastic interpolation flow maps are Lipschitz with a sharp constant.
problem High dimensional sampling and transport problems.
method Investigating stochastic interpolation flow for generating data samples.
result Stochastic interpolation flow maps are Lipschitz with a sharp constant matching optimal transport maps.
A new flow-based Bayesian filter tackles high-dimensional nonlinear stochastic systems.
problem Bayesian filtering for high-dimensional nonlinear systems is challenging due to non-Gaussian distributions and computational limitations.
method Integrates normalizing flows to construct a latent linear state-space model with efficient density estimation and sampling.
result Demonstrates superior accuracy and efficiency in numerical experiments.
The paper develops stochastic models for mortality rates using infinite dimensional processes.
problem Uncertainty in demographic projections of future mortality rates.
method Forward mortality models driven by Wiener process and Poisson random measure.
result Consistency conditions for forward mortality improvements and mortality rates.
Efficiently simulates slow dynamics of high-dimensional stochastic systems.
problem Simulating high-dimensional stochastic systems with slow dynamics and fast modes.
method Designs an algorithm to estimate an invariant manifold and its dynamics, averaging out fast modes.
result Efficient simulator of effective dynamics on low-dimensional invariant manifold.
Paper analyzes adaptive Lasso for high-dimensional diffusion processes, improving support recovery and bias.
problem Support recovery for high-dimensional diffusion processes under sparsity constraints.
method Adaptive Lasso estimator for d-dimensional ergodic diffusion process, focusing on linear models.
result Adaptive Lasso achieves support recovery and asymptotic normality for drift parameter under certain conditions.