We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent low-dimensionality of the solution manifold to obtain approximation rates which are significantly…
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48 results for “parametric approximation”
This research uses DPPs to improve semi-parametric regression models.
problem Improving comprehensibility in semi-parametric regression models without sacrificing accuracy.
method Introduced a novel representation of finite DPPs and used it to derive a key identity illustrating implicit regularization.
result Demonstrated the implicit regularization effect of determinantal sampling for semi-parametric regression.
Paper proves CFlows can approximate any diffeomorphism and applies it in Bayesian optimization.
problem Proving the universality of CFlows in approximating diffeomorphisms.
method Deriving the universality of Para-CFlows through affine coupling layers and invertible linear transforms.
result Para-CFlows can approximate any diffeomorphism in C^k-norm.
Efficiently approximates neural network function space distance.
problem Estimating the average discrepancy between neural network outputs.
method Linearized Activation Function TRick (LAFTR) for ReLU networks.
result Parametric approximation outperforms nonparametric methods in memory and accuracy.
The paper provides approximation guarantees for neural networks trained with gradient flow.
problem Approximating neural networks trained with gradient flow in continuous -norm.
method NTK argument for non-convex second but last layer, under-parametrized regime.
result Gradient flow convergence guarantees for neural networks under Sobolev smoothness assumptions.
This work introduces the concept of parametric Gaussian processes (PGPs), which is built upon the seemingly self-contradictory idea of making Gaussian processes parametric. Parametric Gaussian processes, by construction, are designed to operate in "big data" regimes where one is interested in quantifying the uncertaint…
We use variational Gaussian approximations to analyze parametric models with unknown data-generating distributions.
problem Analyzing inference and learning in parametric models with unknown or intractable data-generating distributions.
method Replica method with variational Gaussian approximation in grand canonical formalism.
result Stationarity conditions adaptively determine parameters of the trial Hamiltonian for each dataset.
Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…
New ADANNs improve PDE approximations.
problem Approximating operators for parametric PDEs.
method Custom ANN architectures and initialization schemes.
result ADANNs significantly outperform existing methods.
The parametric complexity is the key quantity in the minimum description length (MDL) approach to statistical model selection. Rissanen and others have shown that the parametric complexity of a statistical model approaches a simple function of the Fisher information volume of the model as the sample size goes to in…
Study uses neural networks to solve complex equations efficiently.
problem Solving parametric partial differential equations.
method Machine learning and deep neural networks.
result Performance of the model is independent of parameter space dimension.
Kernel ridge regression (KRR) is a standard method for performing non-parametric regression over reproducing kernel Hilbert spaces. Given samples, the time and space complexity of computing the KRR estimate scale as and respectively, and so is prohibitive in many cases. We prop…
Proposes a non-parametric method for deep discrete latent variable models.
problem Learning sparse discrete latent representations in deep models.
method Iterative algorithm with Beta-Bernoulli process prior and local data scaling.
result Improves sparsity and scalability of deep discrete latent variable models.
Symbolic regression finds simple formulas for implied volatility.
problem Discovering accurate parametric representations for implied volatility.
method Symbolic regression to find analytic formulas from market data.
result Symbolic regression identifies compact parametrizations with competitive fitting performance.
Develops flexible non-parametric ACFs using B-spline kernels.
problem Flexible modelling of the autocovariance function (ACF) in time-series, spatial, and spatio-temporal analysis.
method Derives the inverse Fourier transform of B-spline spectral bases to create a general class of non-parametric ACFs.
result Provides a provably dense, flexible, and general class of non-parametric ACFs for various types of processes.
A parametric approach to the estimation of convex risk functionals based on Wasserstein distanceq-fin.RM
Estimates risk in finance using Wasserstein distance and parametric models.
problem Assessing risk in financial models with model uncertainty.
method Parametric approach based on Wasserstein distance for convex risk functionals.
result Developed a numerical method using neural networks to estimate risk and optimal perturbations.
Gradient descent trains shallow neural networks to approximate functions in 1D.
problem Approximating functions in 1D with shallow neural networks trained by gradient descent.
method Gradient descent optimization of non-convex weight space for finite width networks in 1D.
result Gradient descent can approximate functions in 1D with a minimal number of weights, balancing practical performance and theoretical capabilities.
We introduce a new framework for comparing parametric network families.
problem Comparing and analyzing data modeled as parameterized families of networks.
method A Gromov-Wasserstein variant of optimal transport for defining distances.
result Established foundational properties and theoretical approximation guarantees for the new distances.
Deep neural network approximates multivariate option pricing.
problem High-dimensional partial differential equations in option pricing.
method Deep parametric PDE method using neural networks.
result Option prices computed in milliseconds for up to 25 dimensions.
Method identifies shifts leading to large model performance differences.
problem Detecting shifts in distribution that affect model performance.
method Parametric changes in causal mechanisms define robustness sets; worst-case optimization problem approximated as non-convex quadratic.
result Second-order approximation of worst-case loss for small shifts, leading to efficient algorithms.
New algorithms for approximating stochastic processes efficiently.
problem Finding accurate finite approximations for stochastic processes.
method Develops new algorithms and fast implementations for approximating stochastic processes.
result Efficient approximations for stochastic processes can be found.
X-TFC solves parametric DEs with neural networks and physics constraints.
problem Solving parametric differential equations with physics constraints.
method Combines Theory of Functional Connections and Physics-Informed Neural Networks with a single-layer Extreme Learning Machine.
result Achieves high accuracy with low computational time.
Deep adaptive sampling improves surrogate modeling for complex systems.
problem Statistical errors in random sampling for high-dimensional problems.
method DAS^2 method, using deep generative models to refine training sets.
result Reduces statistical errors in approximating solutions for low-regularity problems.
Develops a new method for learning non-parametric DAGs using RKHS.
problem Challenges of learning non-parametric causal models with large combinatorial search space.
method Uses reproducing kernel Hilbert spaces (RKHS) and sparsity-inducing regularization terms based on partial derivatives to enforce acyclicity.
result Shows improved performance through simulations and data analyses.
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.
This note is concerned with accurate and computationally efficient approximations of moments of Gaussian random variables passed through sigmoid or softmax mappings. These approximations are semi-analytical (i.e. they involve the numerical adjustment of parametric forms) and highly accurate (they yield 5% error at most…
A neural network learns efficient parametrizations of product shape spaces.
problem Efficiently parametrize complex shape spaces with high computational costs.
method Developed a neural network architecture that separately learns approximations for low-dimensional factors and combines them.
result Demonstrated the effectiveness of the approach on synthetic and real data.
DebiNet uses over-parameterized neural networks to improve linear model performance and debiasing.
problem Improving linear model performance and debiasing in high-dimensional settings.
method Incorporates over-parameterized neural networks into semi-parametric models to estimate parameters consistently.
result DebiNet offers valid inference and accurate prediction by leveraging neural networks' universal approximation and linear model's interpretability.
Non-parametric approaches for analyzing network data based on exchangeable graph models (ExGM) have recently gained interest. The key object that defines an ExGM is often referred to as a graphon. This non-parametric perspective on network modeling poses challenging questions on how to make inference on the graphon und…
In this paper, we address the inverse problem, or the statistical machine learning problem, in Markov random fields with a non-parametric pair-wise energy function with continuous variables. The inverse problem is formulated by maximum likelihood estimation. The exact treatment of maximum likelihood estimation is intra…
Deep learning methods continue to have a decided impact on machine learning, both in theory and in practice. Statistical theoretical developments have been mostly concerned with approximability or rates of estimation when recovering infinite dimensional objects (curves or densities). Despite the impressive array of ava…
We present an approach of computing the intersection curve of two rational parametric surface and , one being projectable and hence can easily be implicitized. Plugging the parametric surface to the implicit surface yields a plane algebraic curve . By analyzing the topology …
A tutorial on variational inference for high-dimensional models.
problem Approximating marginal likelihood and posterior in Bayesian models.
method Parametric approach to variational inference.
result Variational inference is now preferred for high-dimensional models and large datasets.
Neural networks approximate likelihood ratios for complex models.
problem Difficulty in computing likelihood ratios for modern models.
method Applying the likelihood ratio trick with neural network classifiers.
result Different neural network setups can approximate likelihood ratios with varying performance.
Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the present paper is to provide a short overview of some results and open problems on s…
Gaussian processes with differential privacy protect both inputs and outputs.
problem Previous DP methods only protected model outputs, not inputs.
method Sparse GP with private variational approximation, adjusting covariance for DP noise.
result Accurate models can be produced under strong privacy protection with sufficient data.
Develops neural network approximations for infinite-dimensional input-output maps.
problem Approximating input-output maps between infinite-dimensional spaces.
method Combines neural networks and model reduction techniques.
result Proves convergence of the proposed approximation methodology.
Weyl law for 1-cyclesmath.DG
Proves the Weyl law for 1-cycles in manifolds.
problem Proving the Weyl law for the volume spectrum of 1-cycles in n-dimensional manifolds.
method Using parametric versions of the coarea inequality and isoperimetric inequality, along with a localized approximation method.
result Proves the Weyl law for 1-cycles in manifolds.
We develop several deep learning algorithms for approximating families of parametric PDE solutions. The proposed algorithms approximate solutions together with their gradients, which in the context of mathematical finance means that the derivative prices and hedging strategies are computed simulatenously. Having approx…
A scalable algorithm approximates Bayesian posteriors in RKHS with improved efficiency.
problem Scalable inference for Bayes posteriors in infinite-dimensional spaces.
method Approximate Langevin diffusion projection onto first M components, using law of total probability and sufficiency assumption.
result The method recovers SVGP as a special case and is provably close to optimal for convex and Lipschitz continuous likelihoods.
Neural networks estimate SDEs with jump noise using a Tamed-Milstein scheme.
problem Estimating drift and diffusion functions in SDEs with jump noise.
method Tamed-Milstein scheme with neural networks as non-parametric approximators.
result Flexible estimation of complex nonlinear dynamics in systems with state-dependent noise.
Optimized -posteriors reduce KL divergence from true posterior in parametric misspecification.
problem Reduction of KL divergence from true posterior in parametric model misspecification.
method Derivation of Bernstein-von Mises theorem and optimization of -posteriors.
result Optimized -posteriors minimize KL divergence from true posterior, especially in severe misspecification.
Recent policy optimization approaches have achieved substantial empirical success by constructing surrogate optimization objectives. The Approximate Policy Iteration objective (Schulman et al., 2015a; Kakade and Langford, 2002) has become a standard optimization target for reinforcement learning problems. Using this ob…
Physics-informed neural networks and neural operators speed up solving parametric PDEs by orders of magnitude.
problem Solving PDEs for varying parameters is computationally expensive.
method Physics-informed neural networks and neural operators learn solution mappings across parameter spaces.
result Neural operators achieve computational speedups of 10^3 to 10^5 times faster than traditional methods.
Neural Drift Estimation for Ergodic Diffusions: Non-parametric Analysis and Numerical Explorationmath.ST
Estimates neural drift for stochastic equations, improving inference on noisy data.
problem Estimating drift in stochastic differential equations with neural networks.
method Non-parametric estimation using ReLU neural networks, enforcing theoretical bounds.
result Practical method for inference on noisy and rough functional data.
We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with infinitesimal quads. It is shown that choosing the points of discrete nets on the smooth surface one can obtain second-order approximation g…
Adversarially robust machine learning has received much recent attention. However, prior attacks and defenses for non-parametric classifiers have been developed in an ad-hoc or classifier-specific basis. In this work, we take a holistic look at adversarial examples for non-parametric classifiers, including nearest neig…
New method reduces over-parametrization in neural networks, ensuring sparsity and finite network size.
problem Over-parametrization leads to too many active neurons in neural networks, especially with large data.
method Investigates a nonconvex regularization method for shallow ReLU networks.
result Locally optimal networks are finite even with infinite data, maintaining approximation guarantees and network size bounds.