➗Approximation Theory Unit 8 – Nonlinear approximation

Key Concepts and Definitions

• Nonlinear approximation involves approximating functions using nonlinear spaces or methods
• Differs from linear approximation which uses linear combinations of basis functions
• Adaptive approximation adjusts the approximation space based on the function being approximated
• Sparse approximation represents functions using a small number of basis elements
• Best $n$-term approximation finds the optimal approximation using $n$ basis functions
• Approximation spaces include Besov spaces and Triebel-Lizorkin spaces
  • Characterized by smoothness and integrability properties
• Nonlinear widths measure the error of best $n$-term approximations

Historical Context and Development

• Nonlinear approximation emerged as an alternative to linear approximation methods
• Developed to handle functions with singularities, discontinuities, or other irregular features
• Early work in the 1950s and 1960s laid the foundation for nonlinear approximation theory
• Significant contributions made by Ronald DeVore, Vladimir Temlyakov, and Yves Meyer
  • DeVore introduced adaptive approximation and characterized approximation spaces
  • Temlyakov studied best $n$-term approximations and greedy algorithms
• Wavelets and other basis systems played a crucial role in the development of nonlinear approximation
• Connections to harmonic analysis, functional analysis, and information theory were established

Types of Nonlinear Approximation Methods

• $n$-term approximation represents functions using a linear combination of $n$ basis functions
  • Basis functions are chosen adaptively based on the function being approximated
• Adaptive piecewise polynomial approximation uses piecewise polynomials with adaptive partition
• Free-knot spline approximation optimizes the placement of knots in spline functions
• Rational approximation uses rational functions (ratios of polynomials) for approximation
• Dictionary-based methods (matching pursuit, basis pursuit) select basis functions from a redundant dictionary
• Nonlinear wavelet approximation adapts the wavelet basis to the function being approximated
• Neural network approximation uses neural networks as the approximation model

Mathematical Foundations

• Function spaces provide the setting for studying approximation properties
  • $L^p$ spaces, Sobolev spaces, Besov spaces, and Triebel-Lizorkin spaces are commonly used
• Approximation spaces characterize the approximability of functions by nonlinear methods
  • Determined by the decay of best $n$-term approximation errors
• Jackson-type inequalities relate the smoothness of functions to their approximability
• Bernstein-type inequalities provide converse estimates relating approximability to smoothness
• Interpolation theory plays a role in understanding the relationships between approximation spaces
• Nonlinear widths quantify the optimal approximation errors achievable by nonlinear methods
• Basis selection algorithms (greedy algorithms, thresholding) are used to construct nonlinear approximations

Algorithms and Techniques

• Greedy algorithms (orthogonal matching pursuit, weak greedy algorithms) iteratively select basis functions
  • Aim to minimize the approximation error at each step
• Thresholding algorithms retain basis coefficients above a certain threshold
  • Wavelet thresholding is a prominent example
• Adaptive refinement strategies guide the selection of approximation spaces
  • Can be based on local error estimates or smoothness indicators
• Nonlinear optimization techniques are used to find the best approximation within a given space
• Regularization methods ($\ell^1$ regularization, total variation) promote sparsity in the approximation
• Compressed sensing techniques recover sparse approximations from limited measurements
• Fast algorithms have been developed for efficient computation of nonlinear approximations

Applications in Various Fields

• Signal and image processing use nonlinear approximation for compression, denoising, and feature extraction
  • Wavelet-based methods are widely used in these applications
• Numerical solutions of partial differential equations benefit from adaptive approximation
  • Finite element methods with adaptive mesh refinement are a prime example
• Machine learning and data analysis employ sparse representations for dimensionality reduction and feature selection
• Compressed sensing enables efficient acquisition and reconstruction of sparse signals
• Computer graphics and geometric modeling use nonlinear approximation for surface and curve fitting
• Uncertainty quantification relies on sparse approximations of high-dimensional functions
• Nonlinear approximation finds applications in control theory, system identification, and model reduction

Challenges and Limitations

• The choice of the appropriate approximation space depends on the function being approximated
  • Requires prior knowledge or adaptive strategies
• The computational complexity of nonlinear approximation algorithms can be high
  • Efficient implementations and fast algorithms are essential for practical use
• Stability and robustness of nonlinear approximations need to be considered
  • Sensitivity to perturbations and noise should be analyzed
• The theoretical analysis of nonlinear approximation can be challenging
  • Requires tools from functional analysis, approximation theory, and harmonic analysis
• The optimality of nonlinear approximations is not always guaranteed
  • Depends on the function class and the choice of approximation space
• The interpretation and visualization of nonlinear approximations may be more complex compared to linear methods

Recent Advances and Future Directions

• Deep learning and neural networks have emerged as powerful tools for nonlinear approximation
  • Provide flexible and expressive approximation models
• Combining nonlinear approximation with other paradigms (e.g., sparse grids, low-rank approximations) is an active area of research
• Adaptive and data-driven approaches are being developed to automate the selection of approximation spaces
• Theoretical foundations of deep learning from the perspective of approximation theory are being investigated
• Nonlinear approximation methods are being extended to handle high-dimensional and large-scale problems
• The interplay between nonlinear approximation and other fields (e.g., optimization, statistics) is being explored
• Applications in emerging areas such as data science, scientific machine learning, and uncertainty quantification are being pursued
• Efficient algorithms and implementations on modern computing architectures (e.g., GPUs) are being developed