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.
This paper provides mathematical foundations for regression methods used in forward initial margin approximation.
problem Developing robust methods for approximating forward initial margin.
method Introduces mathematical rigor to show that regression methods are variations of approximating the conditional expectation function.
result Each regression method is a numerical estimation of the conditional expectation with a different functional form.
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.
The paper develops new methods to approximate ruin probabilities in a perturbed risk model.
problem Calculating exact ruin probabilities in a perturbed risk model is complex.
method Adapted Cramér-Lundberg model with Wiener process, four approximation methods.
result Four approximation methods provide high accuracy for ruin probabilities.
Boosting Nyström improves accuracy of matrix approximations.
problem Generating low-rank approximations of large matrices efficiently.
method Iteratively generate multiple weak Nyström approximations, combine them to form a strong approximation.
result Boosting Nyström yields more efficient and accurate low-rank approximations.
We accelerate the power method for strong low-rank approximation using fast sketching.
problem Efficiency bottleneck in power method for large target ranks.
method Developed an algorithmic and theoretical framework for accelerating the power method using fast sketching.
result Simple and provably efficient methods for singular value decomposition, low-rank factorization, and Nyström approximation.
Nyström KPCA balances computational efficiency and statistical accuracy.
problem Computational burden in large sample situations for kernel methods.
method Theoretical analysis of Nyström approximate kernel principal component analysis (KPCA).
result Nyström approximate KPCA matches statistical performance of non-approximate KPCA while being computationally beneficial.
Efficient method for resampling problems using vector approximate message passing.
problem Computational demand in resampling techniques for statistical inference and ensemble learning.
method Combination of replica method from statistical physics and vector approximate message passing from information theory.
result Fast convergence and high approximation accuracy for variable selection problems.
Paper develops a new kernel approximation framework.
problem High time and space complexity of kernel methods for large datasets.
method Perturbation-based kernel approximation framework using classical perturbation theory.
result Framework generalizes and improves upon existing methods.
New RFs reduce kernel approximation variance and improve Transformer performance.
problem Efficient approximation of Gaussian and softmax kernels for kernel methods and Transformers.
method Parameterized, positive, non-trigonometric RFs optimized for variance reduction.
result Significant variance reduction in practice, outperforming previous methods.
New method tackles convergence issues in approximating FBSDEs.
problem Convergence issues in approximating coupled FBSDEs.
method Approximates initial condition for a family of FBSDEs, then uses it to approximate the original FBSDE.
result Method converges even when standard deep BSDE method fails.
New method improves credit risk estimation and pricing.
problem Estimating risk measures and CDO tranche prices in credit portfolios.
method Mod-Poisson approximation schemes based on mod-φ convergence.
result The method provides more accurate estimates with less computational time.
Method approximates Riemannian barycenter on manifolds.
problem Computing the exact Riemannian barycenter is computationally expensive.
method Uses under- and over-approximations of Riemannian distance to compute an approximate barycenter.
result Approximation method is more efficient than exact methods and steepest descent.
Deviation inequalities for stochastic approximation methods.
problem Establishing bounds on the deviation of stochastic approximation methods.
method Martingale approximation method for separately Lipschitz functions.
result Established various deviation inequalities for stochastic approximation by averaging and minimization.
Recently, variational approximations such as the mean field approximation have received much interest. We extend the standard mean field method by using an approximating distribution that factorises into cluster potentials. This includes undirected graphs, directed acyclic graphs and junction trees. We derive generaliz…
This paper introduces a spline-based method for nonparametric ADVI that handles complex posterior distributions.
problem Learning complex posterior distributions with skewness, multimodality, and bounded support.
method Develops a spline-based nonparametric approximation approach for ADVI.
result Establishes the asymptotic consistency of the derived lower bound for importance weighted autoencoder.
Paper develops fast low-rank approximation for smoothing splines.
problem Computational infeasibility of fitting cubic smoothing splines to large datasets.
method Low-rank approximation using eigensystem truncation.
result The method provides accurate, fast estimates with error bounds.
The paper approximates supply curves using a one-step basis method.
problem Computing supply curves accurately and efficiently.
method Derives L2 approximation expression and proposes node selection procedure.
result Illustrates the approach with European electricity market bid curves.
Policy gradient methods with aggregated states can achieve better performance than approximate policy iteration.
problem Approximation errors in policy and value function approximations.
method State-aggregated representations and policy gradient methods.
result Policy gradient methods can achieve a per-period regret bounded by ε, while approximate policy iteration and value iteration have a higher regret.
Develops consistent approximations for composite optimization problems.
problem Significant errors in solutions due to approximations in optimization problems.
method Specifies conditions for well-behaved approximations in minimizers, stationary points, and level-sets for a broad class of composite problems.
result Framework of consistent approximations for composite problems, including stochastic, neural-network, and multi-objective optimization.
Proposes a new method for optimizing large-scale models using Nyström approximation of the Hessian.
problem Optimizing non-convex functions like deep learning models using second-order methods.
method Nyström-approximated curvature for stochastic optimization of large-scale empirical risk minimization.
result The proposed method achieves performance competitive with state-of-the-art first-order and stochastic quasi-Newton methods.
A new method for optimizing deep neural networks using TKFAC.
problem Optimizing deep neural networks with second-order methods.
method Proposes Trace-restricted Kronecker-factored Approximate Curvature (TKFAC) for Fisher information matrix approximation.
result TKFAC improves performance on deep network architectures compared to state-of-the-art algorithms.
Variational methods are widely used for approximate posterior inference. However, their use is typically limited to families of distributions that enjoy particular conjugacy properties. To circumvent this limitation, we propose a family of variational approximations inspired by nonparametric kernel density estimation. …
Paper approximates fractional harmonic maps with numerical methods.
problem Approximating fractional harmonic maps with constraints and nonlocality.
method Weak compactness results and numerical methods for various PDEs.
result Convergence of numerical approximations for fractional harmonic maps.
Proposes efficient Gaussian approximations for non-Gaussian likelihoods.
problem Computational challenges in learning and inference with non-Gaussian likelihoods.
method Variational inference and moment matching in transformed bases.
result Good approximation quality for binary and multiclass classification.
Paper improves kernel approximations for better statistical learning.
problem Improving kernel approximations for better statistical learning.
method Taylor series approximations of radial kernel functions.
result Establishes upper bounds for eigenfunctions, leading to better approximations.
We investigate how to train kernel approximation methods that generalize well under a memory budget. Building on recent theoretical work, we define a measure of kernel approximation error which we find to be more predictive of the empirical generalization performance of kernel approximation methods than conventional me…
We extend natural-gradient methods to mixtures of exponential-family distributions, improving inference speed.
problem Complex, multimodal posterior distributions are difficult to approximate with simple exponential-family distributions.
method We use minimal conditional-EF representations and derive simple natural-gradient updates.
result Our natural-gradient method converges faster than black-box methods with reparameterization gradients.
Collaborative filtering (CF) is a popular technique in today's recommender systems, and matrix approximation-based CF methods have achieved great success in both rating prediction and top-N recommendation tasks. However, real-world user-item rating matrices are typically sparse, incomplete and noisy, which introduce ch…
In this paper we derive an easily computed approximation to European basket call prices for a local volatility jump-diffusion model. We apply the asymptotic expansion method to find the approximate value of the lower bound of European basket call prices. If the local volatility function is time independent then there i…
Clarifies connections between Nyström and SVGP methods for scalable GPs.
problem Lack of understanding between GP and kernel methods communities.
method Investigates Nyström and SVGP methods for scalable Gaussian processes.
result Establishes connections and equivalences between Nyström and SVGP methods.
We propose a general algorithm for approximating nonstandard Bayesian posterior distributions. The algorithm minimizes the Kullback-Leibler divergence of an approximating distribution to the intractable posterior distribution. Our method can be used to approximate any posterior distribution, provided that it is given i…
We consider Bayesian inference problems with computationally intensive likelihood functions. We propose a Gaussian process (GP) based method to approximate the joint distribution of the unknown parameters and the data. In particular, we write the joint density approximately as a product of an approximate posterior dens…
Posterior refinement improves sample efficiency in Bayesian neural networks.
problem Bayesian neural networks suffer from poor predictive performance due to inaccurate posterior approximations.
method Propose refining Gaussian approximate posteriors with normalizing flows to improve predictive distributions.
result Posterior refinement yields competitive predictive performance with minimal computational overhead.
Bandit methods for black-box optimisation, such as Bayesian optimisation, are used in a variety of applications including hyper-parameter tuning and experiment design. Recently, \emph{multi-fidelity} methods have garnered considerable attention since function evaluations have become increasingly expensive in such appli…
We are concerned with an approximation problem for a symmetric positive semidefinite matrix due to motivation from a class of nonlinear machine learning methods. We discuss an approximation approach that we call {matrix ridge approximation}. In particular, we define the matrix ridge approximation as an incomplete matri…
In this paper, we propose a low-rank approximation method based on discrete least-squares for the approximation of a multivariate function from random, noisy-free observations. Sparsity inducing regularization techniques are used within classical algorithms for low-rank approximation in order to exploit the possible sp…
Approximate multipliers boost CNN training speed, power, and area at slight accuracy cost.
problem Improving CNN training performance in terms of speed, power, and area.
method Simulation of approximate multipliers' impact on CNN training, hybrid training method combining approximate and exact multipliers.
result Using approximate multipliers for most of training enhances speed, power, and area with minimal accuracy loss.
Compact bilinear pooling approximates covariance features for faster training.
problem Efficiently approximating covariance features for faster training.
method Compact bilinear pooling extended to polynomial approximations of covariance features.
result The proposed method achieves comparable accuracy with fewer dimensions.
A quasiclassical method approximates magnetic monopole eigenvalues.
problem Describing eigenvalues of magnetic Laplacian in non-exact magnetic fields.
method Applying multidimensional WKB method to nontrivial line bundles.
result Demonstrated for Dirac magnetic monopole on a sphere.
Good sparse approximations are essential for practical inference in Gaussian Processes as the computational cost of exact methods is prohibitive for large datasets. The Fully Independent Training Conditional (FITC) and the Variational Free Energy (VFE) approximations are two recent popular methods. Despite superficial …
A new method improves ICA performance by approximating MDI.
problem Improving F astICA's performance with nonlinear functions.
method Second-order approximation of MDI for joint maximization.
result Efficiency validated through experiments compared to other ICA algorithms.
Efficiently reduces tensor ranks using mean-field approximation.
problem Low-rank approximation of non-negative tensors.
method Mean-field approximation of tensor rank reduction.
result Our algorithm achieves faster and competitive tensor rank reduction.
Paper applies ANOVA decomposition for interpretable data approximation.
problem High-dimensional data interpretation and dimensionality reduction.
method ANOVA decomposition and Grouped Transformations for interpretability.
result Ability to rank variable interactions and unimportant variables.
Efficiently selects predictors in sparse regression without approximations.
problem High computational cost in subset selection for sparse regression.
method Conditional uncorrelation formula and efficient non-approximate method.
result Significant reduction in computational complexity for subset selection.
We consider variants of trust-region and cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated. Under mild conditions on the inexact Hessian, and using approximate solution of the corresponding sub-problems, we provide iteration complexity to achieve ε-approximate seco…
We propose a black-box variational inference method to approximate intractable distributions with an increasingly rich approximating class. Our method, termed variational boosting, iteratively refines an existing variational approximation by solving a sequence of optimization problems, allowing the practitioner to trad…
We propose a new analytical approximation to the χ2 kernel that converges geometrically. The analytical approximation is derived with elementary methods and adapts to the input distribution for optimal convergence rate. Experiments show the new approximation leads to improved performance in image classification and …