Geometrically reformulates the Laplace method for optimal transport.
problem Approximating integrals using the Laplace method without geometric interpretation.
method Introduces the Kim-McCann Riemannian metric to give a geometric formulation of the Laplace method.
result Expresses the first-order term of the Laplace method using geometric objects.
Corrects errors in ILA for Bayesian inference in LGMs.
problem Error in ILA for non-Gaussian likelihoods in LGMs.
method Importance sampling scheme to correct ILA errors.
result Corrected posterior converges to the true posterior with increased samples.
Bayesian neural networks speed up numerical integration.
problem Scalability of Bayesian quadrature methods.
method Bayesian Stein networks using neural networks and Laplace approximation.
result Orders of magnitude speed-up on benchmark functions and real-world problems.
Logistic Gaussian process (LGP) priors provide a flexible alternative for modelling unknown densities. The smoothness properties of the density estimates can be controlled through the prior covariance structure of the LGP, but the challenge is the analytically intractable inference. In this paper, we present approximat…
The future predictive performance of a Bayesian model can be estimated using Bayesian cross-validation. In this article, we consider Gaussian latent variable models where the integration over the latent values is approximated using the Laplace method or expectation propagation (EP). We study the properties of several B…
We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's pri…
We establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and…
Asymptotic Laplace transform for geometric Brownian motion applied to bond pricing.
problem Laplace transform of geometric Brownian motion time integral.
method Asymptotic analysis for σ2To0 limit. result Approximation for zero coupon bond prices in Dothan model.
Let M be a compact Riemannian manifold without boundary and let H be a self-adjoint generalized Laplace operator acting on sections in a bundle over M. We give a path integral formula for the solution to the corresponding heat equation. This is based on approximating path space by finite dimensional spaces of geodesic …
QLA improves Bayesian uncertainty estimation for DNNs without increasing computational cost.
problem Overconfident out-of-distribution predictions from DNNs.
method Proposes Quadratic Laplace Approximation (QLA) to improve Bayesian uncertainty quantification.
result QLA yields modest yet consistent uncertainty estimation improvements over Linearized Laplace Approximation (LLA) on five regression datasets.
Following Feynman's prescription for constructing a path integral representation of the propagator of a quantum theory, a short-time approximation to the propagator for imaginary time, N=1 supersymmetric quantum mechanics on a compact, even-dimensional Riemannian manifold is constructed. The path integral is interprete…
We give time-slicing path integral formulas for solutions to the heat equation corresponding to a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold with boundary. More specifically, we show that such a solution can be approximated by integrals over finite-dimens…
The paper studies Fourier-Laplace transforms in polynomial OU volatility models for option pricing.
problem Calibrating and pricing options in polynomial Ornstein-Uhlenbeck volatility models.
method Analyzes Fourier-Laplace transforms, connects to Riccati equations, and develops numerical schemes.
result Establishes existence and solution for Riccati equations and provides efficient numerical methods.
New methods improve Laplace approximations for deep neural networks by selecting key parameters.
problem Improving uncertainty quantification in deep neural networks using computationally feasible approximations.
method Gradient-Laplace and Greedy-Laplace methods for selecting parameters in sub-network Laplace approximations.
result Gradient-Laplace method outperforms existing heuristic approaches and provides formal optimality guarantees.
A new method combines Laplace and Variational Bayes for scalable inference.
problem Complex models and large datasets make exact inference infeasible.
method Low-Rank Variational Bayes Correction (VBC) using Laplace method and Variational Bayes correction in a lower dimension.
result The method ensures scalability in both model complexity and data size.
Bayesian PINNs optimize loss weights for PDEs and data.
problem Optimizing loss weights in physics-informed neural networks.
method Laplace approximation for efficient model evidence computation.
result Unified Bayesian setting for PDEs and noisy measurements.
Integration of the form ∫a∞f(x)w(x)dx, where w(x) is either sin(ωx) or cos(ωx), is widely encountered in many engineering and scientific applications, such as those involving Fourier or Laplace transforms. Often such integrals are approximated by a numerical integration…
Improves Laplace approximation for Bayesian inference on Riemannian manifolds.
problem Inaccurate Gaussian approximations for complex targets and finite-data posteriors.
method Develops alternative variants of the Laplace approximation using a Riemannian metric.
result Exact approximations at the limit of infinite data, improving practical performance.
Revisits online Laplace methods for neural networks, showing they are sound under certain conditions.
problem Online Laplace methods violate the Laplace approximation's critical assumption.
method Re-derives online Laplace methods, showing they target a variational bound on a mode-corrected variant of the Laplace evidence.
result Online Laplace and its mode-corrected counterpart share stationary points that satisfy the Laplace method's assumption.
Bayesian neural networks approximate Gaussian, this method adapts to non-Gaussian posteriors.
problem Bayesian neural networks struggle with non-Gaussian posteriors, leading to poor performance.
method Proposes a Riemannian Laplace approximation to adapt to the shape of the true posterior.
result Consistently improves over conventional Laplace approximation across tasks.
New iterative methods improve Vecchia-Laplace approximations for large data sets.
problem Inaccurate and slow Vecchia-Laplace approximations for large data sets.
method Iterative methods to improve Vecchia-Laplace approximations, including preconditioners and novel methods for predictive variances.
result Order of magnitude speed-up and threefold increase in prediction accuracy compared to state-of-the-art methods.
Enhances predictive performance in Bayesian deep learning via generalized Laplace approximation.
problem Inconsistency in Bayesian deep learning.
method Interprets posterior tempering as a correction for model misspecification and recalibration of priors. Introduces generalized Laplace approximation.
result Generalized Laplace approximation enhances predictive performance.
In Bayesian Deep Learning, distributions over the output of classification neural networks are often approximated by first constructing a Gaussian distribution over the weights, then sampling from it to receive a distribution over the softmax outputs. This is costly. We reconsider old work (Laplace Bridge) to construct…
Bayesian deep learning method using subnetwork inference.
problem Improving deep neural networks' calibration and efficiency.
method Perform inference over a subset of model weights, keeping others as point estimates.
result Subnetwork inference enables accurate predictive posteriors without full network approximations.
New metrics using Laplace approximation improve Gaussian process model selection.
problem Finding a balance between model accuracy, interpretability, and simplicity.
method Introducing multiple metrics based on the Laplace approximation to evaluate Gaussian process models.
result Our metrics provide comparable performance to dynamic nested sampling but are significantly faster.
In this paper we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on 1 function of 1 variable. We describe linearization of such systems and their integration via Laplace transformation, relating this to Lie's integration theorem and formal th…
Fast approximate inference for non-Gaussian data.
problem Efficient inference for non-Gaussian data.
method Laplace Matching for fast approximate inference in latent Gaussian models.
result Achieves high approximation quality with low computational cost.
We give a short proof of a strong version of the short time asymptotic expansion of heat kernels associated to Laplace type operators acting on sections of vector bundles over compact Riemannian manifolds, including exponential decay of the difference of the approximate heat kernel and the true heat kernel. We use this…
The paper proves conditions for Darboux integrability in diagonal hydrodynamic systems.
problem Conditions for Darboux integrability in diagonal hydrodynamic systems.
method Proof of conditions using Laplace transformation sequences and geometric interpretations.
result Diagonal systems of hydrodynamic type are Darboux integrable if and only if the corresponding systems for commuting flows are Darboux integrable.
This thesis disentangles Gauss-Newton and variational approximations in Bayesian deep learning.
problem Understanding the interplay between the Gauss-Newton method and variational approximations in Bayesian deep learning.
method Analysis of the Gauss-Newton method and Laplace/Gaussian variational approximations for neural networks.
result The combination of the Gauss-Newton method with approximate inference can be cast as inference in a linear or Gaussian process model.
Proposes an INLA-based method for state and parameter estimation in nonlinear systems.
problem Difficulty in learning parameters accurately in nonlinear dynamical systems.
method Iterated INLA for state and parameter estimation in nonlinear dynamical systems.
result Outperforms existing methods on data assimilation tasks.
New algorithms improve likelihood of finding global optima in Bayesian inference.
problem Finding global optima in Bayesian inference is difficult due to nonconvexity.
method Developed two algorithms: consistent Laplace approximation (CLA) and consistent stochastic variational inference (CSVI).
result Both CSVI and CLA improve likelihood of obtaining global optima compared to standard methods.
Bayesian inference typically requires the computation of an approximation to the posterior distribution. An important requirement for an approximate Bayesian inference algorithm is to output high-accuracy posterior mean and uncertainty estimates. Classical Monte Carlo methods, particularly Markov Chain Monte Carlo, rem…
Paper introduces efficient top-k selection with differential privacy.
problem Efficiently selecting top-k elements with differential privacy.
method Oneshot Laplace mechanism, generalizing Report Noisy Max.
result Noise level of O(sqrt(k)/eps) for approximate differential privacy.
We introduce the Kronecker factored online Laplace approximation for overcoming catastrophic forgetting in neural networks. The method is grounded in a Bayesian online learning framework, where we recursively approximate the posterior after every task with a Gaussian, leading to a quadratic penalty on changes to the we…
The paper addresses the invariance issue in Bayesian neural networks using linearized Laplace approximation.
problem Bayesian neural networks fail to maintain invariance under reparameterization, leading to different posterior densities for identical functions.
method Developed a geometric view of reparameterizations and a Riemannian diffusion process to extend reparameterization invariance to neural network predictive.
result Empirically improved posterior fit through approximate posterior sampling.
Proposes sampling from reverse diffusion posteriors for contextual bandits.
problem Complex distributions in contextual bandits.
method Approximate posterior sampling with a diffusion model prior using Laplace approximation.
result Empirically consistent and efficient approximations for contextual bandits.
We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.
Efficiently simulates SABR model with novel sampling methods.
problem Sampling integrated variance and terminal forward price in SABR model.
method Moment-matched shifted lognormal approximation for integrated variance, CEV approximation for terminal forward price.
result Enhanced simulation scheme is highly efficient, accurate, and reliable.
Proves summability of state integrals for specific hyperbolic knots.
problem Summability of perturbative series for hyperbolic knots.
method Algorithm to compute Borel-Laplace resummation as state integrals.
result Complete description of resurgent structure and explicit computations of Stokes constants.
The Laplace-Beltrami operator (LBO) is a fundamental object associated to Riemannian manifolds, which encodes all intrinsic geometry of the manifolds and has many desirable properties. Recently, we proposed a novel numerical method, Point Integral method (PIM), to discretize the Laplace-Beltrami operator on point cloud…
Variational Laplace improves Bayesian neural networks performance.
problem Improving Bayesian neural networks performance.
method Develops variational Laplace for BNNs, exploiting curvature of likelihood.
result Variational Laplace outperforms other inference methods.
Combines Laplace approximations of deep networks for better uncertainty quantification.
problem Overconfident predictions on outliers in deep learning models.
method Gaussian mixture model posterior using weighted sum of Laplace approximations of pre-trained deep networks.
result Mitigates overconfidence 'far away' from training data.
Bayesian tensor train kernel machine uses Laplace approximation for scalable GP regression.
problem Scalability limitations of Gaussian process regression.
method Bayesian tensor train kernel machine with Laplace approximation and variational inference.
result VI replaces cross-validation and offers up to 65x faster training.
Variational Laplace improves Bayesian neural network performance without sampling.
problem Improving Bayesian neural network performance and calibration.
method Develops a new variational Laplace method for BNNs, exploiting curvature of likelihood.
result Variational Laplace outperforms standard VI methods in test performance and calibration.
In this paper we provide an integral representation of the fractional Laplace-Beltrami operator for general riemannian manifolds which has several interesting applications. We give two different proofs, in two different scenarios, of essentially the same result. One of them deals with compact manifolds with or without …
New method calculates geometric Brownian motion with affine drift and its integral.
problem Calculating the distribution of geometric Brownian motion with affine drift and its integral.
method Laplace transform approach and Heun differential equation.
result Joint distribution of geometric Brownian motion with affine drift and its integral can be determined.
LaLoRA prevents forgetting in LoRA fine-tuning.
problem Catastrophic forgetting in fine-tuned models.
method LaLoRA applies Laplace approximation to LoRA weights for regularization.
result Improved learning-forgetting trade-off with controllable regularization strength.