insight - High-dimensional function approximation - # Learning sparse polynomial and deep neural network approximations of holomorphic functions

Learning Efficient Approximations of High-Dimensional Smooth Functions from Limited Data

Q: What are some potential applications beyond parametric PDEs where the (b,ε)-holomorphic function class and the associated learning theory could be relevant

The (b,ε)-holomorphic function class and the associated learning theory developed in this work could find applications beyond parametric PDEs in various fields. One potential application could be in image processing, where functions representing image features or characteristics could be approximated using sparse polynomial or deep neural network methods. Another application could be in financial modeling, where functions representing complex financial relationships or market behaviors could be learned from limited data using these techniques. Additionally, in natural language processing, functions representing language patterns or semantics could be approximated using these methods to improve language understanding and generation models.

Q: How might the theory and methods developed in this work extend to other function classes beyond holomorphic functions, such as non-smooth or discontinuous functions

The theory and methods developed for (b,ε)-holomorphic functions could be extended to other function classes beyond holomorphic functions, such as non-smooth or discontinuous functions. For non-smooth functions, techniques like sparse polynomial approximation could be used to capture the discontinuities or irregularities in the function. Deep neural networks, on the other hand, could be employed to learn complex mappings for discontinuous functions by leveraging their ability to approximate arbitrary functions. By adapting the learning algorithms and architectures to suit the characteristics of non-smooth or discontinuous functions, the theory and methods could be applied effectively in these domains.

Q: What are the key practical challenges that remain in deploying sparse polynomial and deep neural network methods for high-dimensional function approximation in real-world applications

Deploying sparse polynomial and deep neural network methods for high-dimensional function approximation in real-world applications faces several key practical challenges. One challenge is the computational complexity associated with training deep neural networks on large datasets, requiring significant computational resources and time. Another challenge is the interpretability of the models generated by these methods, especially in scenarios where explainability is crucial. Additionally, ensuring the robustness and generalization of the learned models to unseen data is a critical challenge, as overfitting and underfitting can occur in high-dimensional spaces. Addressing these challenges will be essential for the successful deployment of these methods in real-world applications.

Conceitos Básicos

Functions that are holomorphic in high-dimensional parameter spaces can be efficiently approximated using sparse polynomial expansions or deep neural networks, even when only limited training data is available.

Resumo

The content discusses the problem of learning approximations to smooth, high-dimensional functions from finite data. Key points:

Motivation: This problem arises in parametric models and computational uncertainty quantification, where the target function represents a quantity of interest that depends on many parameters.
Function class: The target functions are assumed to be (b,ε)-holomorphic, meaning they admit holomorphic extensions to certain complex regions in the parameter space. This class captures the smoothness of many parametric PDE solutions.
Benchmark: The best s-term polynomial approximation provides a theoretical benchmark for the approximation error, showing algebraic convergence rates that are free from the curse of dimensionality.
Limits of learnability: It is shown that no learning method can achieve the best s-term approximation rates from finite data, highlighting a fundamental gap between approximation theory and practical learning.
Sparse polynomial learning: A weighted sparse polynomial approximation method is described that achieves near-optimal learning rates, bridging this gap.
Deep neural networks: Existence theorems for DNN approximations are reviewed, and a "practical existence theory" is developed to show that certain DNN architectures and training strategies can also achieve near-optimal learning rates.

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Estatísticas

The content does not provide any specific numerical data or statistics to support the key points. It focuses on theoretical results and conceptual insights.

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Principais Insights Extraídos De

Learning smooth functions in high dimensions

by Ben Adcock,S... às arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.03761.pdf

Learning smooth functions in high dimensions

Perguntas Mais Profundas

What are some potential applications beyond parametric PDEs where the (b,ε)-holomorphic function class and the associated learning theory could be relevant

The (b,ε)-holomorphic function class and the associated learning theory developed in this work could find applications beyond parametric PDEs in various fields. One potential application could be in image processing, where functions representing image features or characteristics could be approximated using sparse polynomial or deep neural network methods. Another application could be in financial modeling, where functions representing complex financial relationships or market behaviors could be learned from limited data using these techniques. Additionally, in natural language processing, functions representing language patterns or semantics could be approximated using these methods to improve language understanding and generation models.

How might the theory and methods developed in this work extend to other function classes beyond holomorphic functions, such as non-smooth or discontinuous functions

The theory and methods developed for (b,ε)-holomorphic functions could be extended to other function classes beyond holomorphic functions, such as non-smooth or discontinuous functions. For non-smooth functions, techniques like sparse polynomial approximation could be used to capture the discontinuities or irregularities in the function. Deep neural networks, on the other hand, could be employed to learn complex mappings for discontinuous functions by leveraging their ability to approximate arbitrary functions. By adapting the learning algorithms and architectures to suit the characteristics of non-smooth or discontinuous functions, the theory and methods could be applied effectively in these domains.

What are the key practical challenges that remain in deploying sparse polynomial and deep neural network methods for high-dimensional function approximation in real-world applications

Deploying sparse polynomial and deep neural network methods for high-dimensional function approximation in real-world applications faces several key practical challenges. One challenge is the computational complexity associated with training deep neural networks on large datasets, requiring significant computational resources and time. Another challenge is the interpretability of the models generated by these methods, especially in scenarios where explainability is crucial. Additionally, ensuring the robustness and generalization of the learned models to unseen data is a critical challenge, as overfitting and underfitting can occur in high-dimensional spaces. Addressing these challenges will be essential for the successful deployment of these methods in real-world applications.