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.
This Chapter, "Overview of Approximate Bayesian Computation", is to appear as the first chapter in the forthcoming Handbook of Approximate Bayesian Computation (2018). It details the main ideas and concepts behind ABC methods with many examples and illustrations.
In this paper, we apply the method of approximate transformation groups proposed by Baikov, Gaziziv and Ibragimov, to compute the first-order approximate symmetry for the Gardner equations with the small parameters. We compute the optimal system and analyze some invariant solutions of These types of equations. Particul…
The intrinsic error tolerance of neural network (NN) makes approximate computing a promising technique to improve the energy efficiency of NN inference. Conventional approximate computing focuses on balancing the efficiency-accuracy trade-off for existing pre-trained networks, which can lead to suboptimal solutions. In…
This work provides efficient algorithms for approximating ℓ_p sensitivities and related statistics.
problem Estimating the importance of datapoints in high-dimensional datasets.
method Efficient algorithms for computing α-approximation of ℓ_1 sensitivities and total sensitivity using importance sampling and sensitivity computations.
result Real-world datasets have significantly lower intrinsic effective dimensionality than theoretical predictions.
VISA improves inference efficiency for complex models.
problem Efficient approximate inference in computationally intensive models.
method Sequential sample-average approximations within a trust region.
result VISA achieves comparable accuracy with computational savings.
Accelerated RPCholesky speeds up kernel matrix approximations.
problem Efficiently approximating large kernel matrices.
method Accelerated randomly pivoted Cholesky (RPCholesky) with block matrix computations and rejection sampling.
result Approximates kernel matrices up to 40 times faster.
Neural network based approximate computing is a universal architecture promising to gain tremendous energy-efficiency for many error resilient applications. To guarantee the approximation quality, existing works deploy two neural networks (NNs), e.g., an approximator and a predictor. The approximator provides the appro…
Neural approximate computing gains enormous energy-efficiency at the cost of tolerable quality-loss. A neural approximator can map the input data to output while a classifier determines whether the input data are safe to approximate with quality guarantee. However, existing works cannot maximize the invocation of the a…
Develops a fast variational approximation for high-dimensional empirical Bayes posteriors.
problem Optimal posterior computation in high-dimensional settings with prior tails effect.
method Variational approximation of empirical Bayes posterior with data-driven centers and thin-tailed conjugate priors.
result Retains optimal concentration rate properties and superior performance compared to existing methods.
New GP methods account for both data and computational uncertainty.
problem Approximation error in Gaussian process models.
method Develops a new class of methods to estimate combined uncertainty.
result Proves convergence and decomposability of combined posterior covariance.
Hard to approximate critical points for simple nonconvex functions.
problem Approximating critical points of nonconvex functions.
method Proving hardness results for polynomial-time approximation of critical points.
result Proving that approximating critical points is intractable for simple nonconvex functions.
RCaGP improves robustness and computational efficiency in Gaussian processes.
problem Outliers in large datasets corrupt standard Gaussian process models.
method Combines robustness and approximation-awareness in a principled framework.
result Ensures more conservative and reliable uncertainty estimates.
We provide a fast approximation to eNTKs for neural networks.
problem Efficiently computing eNTKs for large networks.
method Developed and proved the 'sum of logits' approximation.
result The 'sum of logits' approximation converges to eNTKs at initialization.
Reduced modeling of a computationally demanding dynamical system aims at approximating its trajectories, while optimizing the trade-off between accuracy and computational complexity. In this work, we propose to achieve such an approximation by first embedding the trajectories in a reproducing kernel Hilbert space (RKHS…
The runtime for Kernel Partial Least Squares (KPLS) to compute the fit is quadratic in the number of examples. However, the necessity of obtaining sensitivity measures as degrees of freedom for model selection or confidence intervals for more detailed analysis requires cubic runtime, and thus constitutes a computationa…
The paper reduces xVA calculations by approximating sensitivities.
problem Nested expectation problem and computational expense in xVA calculations.
method Polynomial approximations of shocked and unshocked valuation functions, and their difference.
result High accuracy and remarkable computational cost reduction demonstrated.
Survey on learning Boolean functions in computational theory.
problem Learning Boolean function classes in computational theory.
method Overview of known results in PAC and related models.
result Discussion of various learning results for Boolean functions.
The paper proposes using path signatures for better inference in time series data.
problem Simulation models with time series data often lack tractable likelihood functions.
method Approximate Bayesian Computation with path signatures to handle sequential data.
result Theoretical guarantees on the resultant posteriors for Bayesian parameter inference.
Paper speeds up GP inference by reducing precision matrix computation.
problem High computational complexity in computing kernel precision matrices.
method Splitting precision matrix into Hankel-Toeplitz matrices and computing only unique entries.
result Precision matrix computation reduced from O(NM2) to O(NM). Computational method approximates homology groups of compact metric spaces.
problem Computing homology groups of compact metric spaces.
method Inversely sequence of finite topological spaces, inverse limit, homeomorphic copy, strong deformation retract.
result Approximates homology groups of compact metric spaces.
Matrix multiplication is a fundamental building block for large scale computations arising in various applications, including machine learning. There has been significant recent interest in using coding to speed up distributed matrix multiplication, that are robust to stragglers (i.e., machines that may perform slower …
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…
We construct algorithms via binomial approximations for computation of prices of game put options and obtain estimates of approximation errors.
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…
Nystrom approximation speeds up kernel model training.
problem Slow convergence in kernel models due to poor conditioning.
method Spectral preconditioning with Nystrom approximation for scalability.
result Nystrom approximation accelerates gradient descent nearly as well as exact preconditioner.
The paper examines how kernel approximations affect Gaussian process regression in large data applications.
problem Effect of kernel approximations on Gaussian process regression in large data applications.
method Unified framework to analyze Gaussian process regression under computational and epistemic misspecification.
result Theoretical analysis of Gaussian process regression under various misspecifications.
Markov chains and diffusion processes are indispensable tools in machine learning and statistics that are used for inference, sampling, and modeling. With the growth of large-scale datasets, the computational cost associated with simulating these stochastic processes can be considerable, and many algorithms have been p…
Bayesian inference requires approximation methods to become computable, but for most of them it is impossible to quantify how close the approximation is to the true posterior. In this work, we present a theorem upper-bounding the KL divergence between a log-concave target density f(θ) and its La…
Using classical Taylor series techniques, we develop a unified approach to pricing and implied volatility for European-style options in a general local-stochastic volatility setting. Our price approximations require only a normal CDF and our implied volatility approximations are fully explicit (ie, they require no spec…
Efficient algorithm approximates discrete random variables with minimal Kolmogorov distance.
problem Estimating the probability of missing deadlines in series-parallel schedules.
method An efficient algorithm that computes a random variable with minimal Kolmogorov distance to a given discrete random variable.
result The algorithm efficiently approximates the probability of missing deadlines with minimal Kolmogorov distance.
New method approximates CV efficiently for large-scale problems.
problem High computational cost of standard CV in large-scale problems.
method Iterative first-order algorithm to approximate CV solution.
result Extends CV approximation guarantees to non-converged solutions.
We develop methods to efficiently approximate data in metric spaces without additional assumptions.
problem Efficiently approximating data in metric spaces without imposing structural assumptions.
method Identify discrete modulus of continuity, investigate consistency, propose algorithm, and develop approximation theory.
result Consistent approximation of data in metric spaces without structural assumptions.
Machine learning impacts computational math, offering new functions approximations.
problem Machine learning's black box nature hinders further progress in computational math.
method Analyzes machine learning's impact on computational math and vice versa.
result Integrating computational math with machine learning can enhance both fields.
Two-layer networks struggle with high frequencies due to numerical and computational limitations.
problem High frequency approximation and learning in shallow networks.
method Mathematical and computational analysis focusing on numerical error, computational cost, and stability.
result Explicit answers to fundamental computational issues in shallow networks' high frequency handling.
Stochastic Volatility in Mean models with heavy-tailed distributions using Hidden Markov Models
problem Accurate inference for Stochastic Volatility in Mean models with heavy-tailed distributions
method Numerically stable estimation procedure and parallel computing
result Significant reduction in computational times
We develop a Markovian approximation for SVV models to compute hedging strategies.
problem Computing optimal hedging strategies for SVV models with non-Markovian noise.
method Develop a Markovian approximation of the Volterra noise kernel to compute hedging strategies.
result Error estimates for the approximation of volatility, prices, and optimal hedge.
Paper tackles robust optimal transport with improved computational complexity and barycenter approximation.
problem Computing robust optimal transport and its barycenter efficiently.
method Sinkhorn-based algorithms for robust optimal transport and iterative Bregman projections for barycenter approximation.
result Improved computational complexity for robust optimal transport and barycenter approximation.
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.
We propose an efficient algorithm for approximate computation of the profile maximum likelihood (PML), a variant of maximum likelihood maximizing the probability of observing a sufficient statistic rather than the empirical sample. The PML has appealing theoretical properties, but is difficult to compute exactly. Inspi…
New method speeds up analysis of computer experiments.
problem Computational infeasibility of direct GP inference for large datasets.
method Adapted Vecchia's ordered conditional approximation to scaled input space.
result Significant performance improvement over existing methods.
Method computes harmonic and conformal maps from point clouds.
problem Computing maps from irregular point cloud data.
method Meshless method using cubic lattice approximations.
result Harmonic and conformal maps computed accurately.
New methods accelerate NCGP inference by trading computation for uncertainty.
problem Prohibitively expensive exact inference in NCGPs for large datasets.
method Iterative methods explicitly modeling approximation error, leveraging parallel computing.
result Significant acceleration of posterior inference compared to baselines.
Paper proposes diagnostics for error and variance estimation in randomized matrix computations.
problem Safe use of randomized matrix algorithms in applications.
method Leave-one-out error estimator and jackknife resampling method.
result Provides rapid diagnostics to assess quality of randomized matrix computations.
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.
Study compares methods for computing hypergradients in machine learning problems.
problem Computing exact hypergradients in machine learning is difficult.
method Investigates reverse mode iterative differentiation and approximate implicit differentiation methods.
result Unified analysis provides iteration complexity bounds and hierarchy of methods.
It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maxim…
A new method for uncertainty estimation in neural networks using existing optimization steps.
problem Uncertainty quantification in deep neural networks.
method L2M: Practical posterior Laplace approximation with optimization-driven second moment estimation.
result L2M method yields reasonable results without requiring changes in models or extra computational steps.