Study approximates nonlinear functionals using deep ReLU networks.
problem Approximating nonlinear continuous functionals with neural networks.
method Constructs continuous piecewise linear interpolation under simple triangulation, analyzes rates of approximation.
result Established rates of approximation for functional deep ReLU networks.
The study extends convergence guarantees for nonlinear TD learning, focusing on ReLU networks and reversibility.
problem Understanding convergence of nonlinear TD learning with function approximators.
method Analyzing TD(0) dynamics through nonlinear ODEs, considering function approximator geometry and reversibility.
result Global convergence to the true value function for well-conditioned function approximators in reversible environments.
Leveled Homomorphic Encryption (LHE) offers a potential solution that could allow sectors with sensitive data to utilize the cloud and securely deploy their models for remote inference with Deep Neural Networks (DNN). However, this application faces several obstacles due to the limitations of LHE. One of the main probl…
Method solves complex optimization problems with high probability bounds.
problem Nonlinear equality constrained stochastic optimization problems.
method Step-search sequential quadratic programming method.
result High-probability bound on iteration complexity for first-order stationarity.
Study on statistical inference for nonlinear stochastic approximation with Markovian data.
problem Statistical inference for nonlinear stochastic approximation algorithms with Markovian data.
method Established a functional central limit theorem for the partial-sum process of the target parameter estimate, providing asymptotic pivotal statistics for constructing confidence intervals.
result Valid and efficient asymptotic inference method for nonlinear stochastic approximation algorithms with Markovian data.
Lazy training and mean field regimes studied for TD learning with nonlinear function approximation.
problem Approximating value function for MRP with TD learning and nonlinear functions.
method Lazy training and mean field scaling of parameters analyzed for convergence.
result Lazy training leads to exponential convergence to local/global minimizers, while mean field scaling results in all fixed points being minimizers.
Tensor completion method identifies nonlinear systems from input-output data.
problem Identifying nonlinear functions from input-output data pairs.
method Formulated as tensor completion problem with smoothness regularization and solved using block coordinate descent.
result Provable correct nonlinear system identification under certain conditions.
Richberg technique adapted for nonlinear subequations.
problem Approximating strictly subharmonic functions in F-potential theory. method Adapting Richberg technique to F-potential theory. result Local approximation to global approximation for subharmonic functions.
A new model approximates complex functions in parameter space.
problem Complex and nonlinear functional regression problems.
method Mapping-to-Parameter function model with B-spline free knot placement.
result Robust knot placement algorithms improve model performance.
Kernel-based Bayesian filter for nonlinear systems using infinite-dimensional operators.
problem Modeling and predicting nonlinear dynamical systems.
method Functional Bayesian perspective, reproducing kernel Hilbert space, Gaussian kernel.
result Effective approximation and accurate results for nonlinear systems.
We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems - with a reasonable expectation of suc…
New Zap Q-learning accelerates reinforcement learning with neural networks.
problem Accelerate convergence of reinforcement learning algorithms.
method Introduces a new framework for analysis of stochastic approximation algorithms, proving consistency under non-degeneracy assumption.
result Zap Q-learning with neural network function approximation converges quickly and is robust to function approximation architecture choice.
The paper proves neural networks with ReLU and softmax can approximate any function.
problem Approximating functions and class labels in neural networks.
method Extended universal approximator theory to neural networks with ReLU and softmax.
result Neural networks with ReLU and softmax can approximate any function and class labels.
A new deep neural network tackles nonlinear functional regression with improved dimensionality reduction.
problem Nonlinear functional regression in infinite-dimensional functional data analysis.
method Functional deep neural network with adaptive kernel embedding and projection steps.
result Explicit rates of approximating nonlinear smooth functionals are derived, and the network is shown to be effective in both simulated and real datasets.
Improved bounds for function approximation in nonlinear sets.
problem Achieving high probability error with limited samples in nonlinear function approximation.
method Restricting model class to a neighbourhood of the best approximation and estimating sample complexity using tangent and normal spaces' complexities and curvature.
result Improved worst-case bounds for sample complexity in more general sets like tensor networks and neural networks.
DeepONet learns nonlinear operators from data to identify differential equations.
problem Learning nonlinear operators from data to identify differential equations.
method DeepONet architecture with branch and trunk nets.
result DeepONet significantly reduces generalization error compared to fully-connected networks.
We are interested in approximation of a multivariate function f(x1,…,xd) by linear combinations of products u1(x1)⋯ud(xd) of univariate functions ui(xi), i=1,…,d. In the case d=2 it is a classical problem of bilinear approximation. In the case of approximation in the L2 space the bili…
New matrix approximation method using RBF components for better memory efficiency.
problem Efficiently approximate any real matrix without being symmetric or positive definite.
method Formulate as an optimization problem with gradient descent methods.
result Significantly reduces memory usage for various matrix types.
Study efficient convergence of RL algorithm with function approximation.
problem Convergence of actor-critic algorithm with nonlinear function approximation.
method Stochastic gradient descent ascent with adaptive proximal term, Polyak-Łojasiewicz condition.
result First efficient convergence result with rate of O(sqrt{ln(N d G^2) / N}).
Paper improves image retrieval quality using nonlinear rank approximations.
problem Improving image retrieval quality in high-dimensional feature spaces.
method Computes normalized approximated ranks, converts to similarities, and uses them in a new loss function.
result Significant improvement in image retrieval quality on multiple datasets.
The purpose of this paper is to analyze and compute the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility which can be a function of the second derivative of the option price itself. A motivation for studying the nonlinear Black--Scholes equation with a nonlinear vola…
New method for separating mixed signals with nonlinear functions.
problem Recovering source signals from nonlinear mixtures.
method Optimisation-based function approximation to minimize mutual statistical dependence.
result The method can recover source signals from nonlinear mixtures under certain conditions.
Kernel discriminant analysis uses nonlinear embeddings to improve classification.
problem Limited effectiveness of linear discriminant analysis in capturing nonlinear features.
method Study of nonlinear embeddings in kernel discriminant analysis using polynomial and Gaussian kernels, solving generalized eigenvalue problems.
result Polynomial and Gaussian discriminants capture class differences through population moments and randomized projections.
We study the expressivity of deep neural networks. Measuring a network's complexity by its number of connections or by its number of neurons, we consider the class of functions for which the error of best approximation with networks of a given complexity decays at a certain rate when increasing the complexity budget. U…
Method solves Bayesian inverse problems in function space without assuming log-concavity.
problem Bayesian inverse problems in infinite-dimensional nonlinear settings.
method Score-based diffusion models as a prior, Langevin-type MCMC on function spaces.
result Provable convergence bound for posterior sampling, dependent on score approximation.
In this paper we propose a new robust algorithm to find the optimal static replicating portfolios for general nonlinear payoff functions and give the estimate of the rate of convergence that is absent in the literature. We choose the static replication by minimizing the error bound between the nonlinear payoff function…
This paper introduces a new method for semi-supervised learning on high dimensional nonlinear manifolds, which includes a phase of unsupervised basis learning and a phase of supervised function learning. The learned bases provide a set of anchor points to form a local coordinate system, such that each data point x on…
EnSF improves accuracy in tracking high-dimensional nonlinear systems.
problem Low accuracy in high-dimensional, nonlinear filtering problems.
method Score-based diffusion model, mini-batch Monte Carlo estimator.
result EnSF outperforms state-of-the-art methods in tracking high-dimensional systems.
Given a function dictionary D and an approximation budget N∈N+, nonlinear approximation seeks the linear combination of the best N terms {Tn}1≤n≤N⊆D to approximate a given function f with the minimum approximation error\[\varepsilon_{L,f}:=\min_{\{g_n\}\subseteq{\ma…
Enhances RSCNs with hybrid regularization for nonlinear dynamics.
problem Modeling nonlinear dynamic systems with uncertainties.
method Recurrent stochastic configuration networks with hybrid regularization.
result The method outperforms other models in nonlinear system identification and industrial tasks.
New methods for scalable causal discovery from complex data.
problem Learning causal structures from nonlinear, continuous or mixed data.
method BF-BIC score and BF-LRT test for scalable causal discovery.
result BF-BIC score and BF-LRT test enable scalable causal discovery with competitive accuracy and runtime.
Temporal Difference Learning analysis under non-i.i.d. data and nonlinear approximation.
problem Finite-sample behavior of TD(0) under non-i.i.d. data and nonlinear approximation.
method High-probability, finite-sample analysis of vanilla TD(0) on polynomially mixing Markov data, assuming Holder continuity and bounded generalized gradients.
result Bounds on the convergence rate of TD(0) with high probability, matching known i.i.d. rates and holding even with nonstationary initialization.
Papers learn from data to make decisions without interacting, improving on previous methods.
problem Achieving optimal decision-making from offline data with non-linear function approximation.
method Pessimistic Nonlinear Least-Square Value Iteration (PNLSVI) with three innovative components.
result Achieves minimax optimal instance-dependent regret for non-linear function approximation.
A novel adaptive kernel improves RBF neural networks performance.
problem Improving performance of RBF neural networks.
method Adaptive fusion of Euclidean and cosine distance measures using gradient descent.
result The method outperforms manual fusion on three major problems.
Deep neural networks can approximate rough functions with high accuracy.
problem Approximating rough functions with neural networks.
method Proved that ENO interpolation can be cast as a deep ReLU neural network, transferring ENO's high-order accuracy.
result Deep neural networks can achieve high-order accuracy in approximating Lipschitz functions.
New method recovers latent sources from multiple noisy views using deep neural networks.
problem Recovering a common latent source from multiple nonlinearly mixed views.
method Novel identifiability proofs using deep neural networks.
result Independent latent sources can be recovered from multiple noisy views using deep neural networks.
We simplify complex regression coefficients using linearization and feature comparison.
problem Interpreting high-dimensional regression coefficients from nonlinear responses.
method Developed a linearization method to derive feature coefficients and compare them with regression coefficients.
result Shows how regression coefficients relate to linearized feature coefficients and how they change under regularization.
Develops a method for causal inference with noisy confounders.
problem Noisy measurements of confounders in treatment effects models.
method Local principal subspace approximation combining K-nearest neighbors matching and PCA.
result Estimators of treatment effects and counterfactual distributions are constructed.
A deep learning method solves nonlinear filtering problems efficiently.
problem Nonlinear filtering problem
method Deep splitting method combined with energy-based neural network approximation
result Computational efficiency and performance comparable to Kalman and bootstrap filters
Reduces function approximation dimensions from high to low with sparse data.
problem Function approximation from sparse data.
method Nonlinear Level Set Learning (NLL) with geometric information.
result Reduces input dimension to theoretical lower bound with minor accuracy loss.
Paper develops a new state estimation method for nonlinear systems.
problem State estimation for nonlinear state-space models is intractable.
method Developed a variational inference approach based on Gaussian approximations.
result The method outperforms alternative Gaussian approaches in various examples.
Neural networks with random hidden nodes have gained increasing interest from researchers and practical applications. This is due to their unique features such as very fast training and universal approximation property. In these networks the weights and biases of hidden nodes determining the nonlinear feature mapping a…
Rational neural networks approximate functions more efficiently with less depth.
problem Choosing optimal nonlinear activation functions in neural networks.
method Rational activation functions with optimal bounds and efficiency proofs.
result Rational neural networks approximate smooth functions more efficiently than ReLU networks with exponentially smaller depth.
Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.
problem Quantifying the efficiency of neural operators for solving nonlinear parabolic PDEs.
method Deriving approximation rates by transferring PDEs to integral equations and leveraging Picard's iteration.
result Neural operators can efficiently approximate solution operators of nonlinear PDEs without exponential complexity growth.
RVFL NNs perform well without direct links and output bias for regression.
problem Effect of direct links and output bias on RVFL performance.
method Classical and two new methods for generating hidden nodes' parameters tested.
result Direct links and output bias do not significantly improve RVFL accuracy for typical nonlinear regression problems.
Deep residual networks can approximate any continuous function using control theory.
problem Universal approximation capabilities of deep residual neural networks.
method Relating residual networks to control systems and using Lie algebraic techniques.
result Deep residual networks with adequately deep layers can approximate any continuous function on a compact set.
In reinforcement learning, temporal difference (TD) is the most direct algorithm to learn the value function of a policy. For large or infinite state spaces, exact representations of the value function are usually not available, and it must be approximated by a function in some parametric family. However, with \emph{no…
New machine learning methods solve complex PDEs with improved accuracy.
problem Solving fully nonlinear PDEs with convex Hamiltonian.
method Rewriting PDE in dual stochastic control form, estimating optimal feedback control with neural network, approximating value function with neural networks.
result Improved estimation of PDE solution and its derivatives, especially the second derivative.