➗Approximation Theory Unit 7 – Hilbert Space Approximation

What's Hilbert Space Again?

• Hilbert space is a complete inner product space, a generalization of Euclidean space to any finite or infinite number of dimensions
• Consists of vectors with well-defined notions of length, angle, and orthogonality determined by the inner product
• Completeness ensures that Cauchy sequences converge to a limit within the space, a crucial property for approximation theory
• Includes many function spaces used in functional analysis (L^2 spaces) and quantum mechanics (L^2(R^n))
• Provides a framework for studying linear operators, eigenvalues, and eigenvectors
  • Linear operators map elements of the Hilbert space to other elements within the same space
  • Eigenvalues and eigenvectors help characterize the behavior of linear operators
• Serves as a foundation for various approximation techniques, such as least squares approximation and orthogonal projection
• Hilbert spaces can be separable (possessing a countable dense subset) or non-separable, with separable spaces being more commonly used in approximation theory

Key Concepts in Hilbert Space Approximation

• Best approximation: finding an element in a subspace that minimizes the distance to a given element in the Hilbert space
• Orthogonal projection: projecting an element onto a closed subspace, resulting in the best approximation within that subspace
• Basis functions: a set of linearly independent elements that span the Hilbert space or a subspace, used to represent elements as linear combinations
  • Orthonormal basis functions are particularly useful due to their orthogonality and unit norm properties
• Parseval's identity: relates the norm of an element to the sum of the squares of its Fourier coefficients with respect to an orthonormal basis
• Least squares approximation: minimizing the sum of squared residuals between the approximation and the target function
• Convex optimization: many approximation problems in Hilbert spaces can be formulated as convex optimization problems, which have unique global minima
• Regularization: adding constraints or penalty terms to the approximation problem to prevent overfitting and ensure stability
• Convergence rates: characterizing how quickly the approximation error decreases as the approximation space grows or the number of basis functions increases

Important Theorems and Proofs

• Projection Theorem: states that for any element in a Hilbert space and any closed subspace, there exists a unique best approximation within the subspace
  • Proves the existence and uniqueness of the orthogonal projection onto a closed subspace
• Riesz Representation Theorem: establishes a one-to-one correspondence between a Hilbert space and its dual space (the space of continuous linear functionals)
  • Allows the representation of continuous linear functionals as inner products with elements of the Hilbert space
• Bessel's Inequality: provides an upper bound for the sum of squares of the Fourier coefficients of an element with respect to an orthonormal set
• Best Approximation Theorem: characterizes the best approximation in terms of orthogonality to the error vector
  • States that an element is the best approximation if and only if the error vector is orthogonal to the approximation subspace
• Pythagorean Theorem in Hilbert Spaces: relates the norms of an element, its best approximation, and the error vector
  • Proves that the squared norm of an element equals the sum of the squared norms of its best approximation and the error vector
• Convergence Theorems: various results that establish the convergence of approximations under specific conditions (density of the approximation space, smoothness of the target function)

Practical Applications

• Signal and image processing: approximating signals or images using basis functions (Fourier, wavelets) for compression, denoising, or feature extraction
• Numerical solutions of partial differential equations (PDEs): approximating the solution of a PDE using finite element methods or spectral methods
  • Galerkin methods project the PDE onto a finite-dimensional subspace to obtain a discrete approximation
• Machine learning and data analysis: approximating complex functions or decision boundaries using linear combinations of basis functions (polynomial regression, radial basis functions)
• Compressed sensing: recovering sparse signals from a limited number of linear measurements by solving an approximation problem with sparsity constraints
• Quantum mechanics: approximating wavefunctions or operators using truncated expansions in orthonormal basis functions (eigenfunctions of the Hamiltonian)
• Control theory: approximating the optimal control policy or the value function in reinforcement learning using linear combinations of basis functions
• Computational finance: approximating the solution of stochastic differential equations (SDEs) for option pricing or risk management using Monte Carlo methods or finite difference schemes

Common Approximation Techniques

• Fourier series approximation: representing a periodic function as an infinite sum of trigonometric basis functions (sines and cosines)
• Polynomial approximation: approximating a function using polynomials of increasing degree
  • Includes Taylor series approximation, which uses the derivatives of the function at a single point
  • Also includes Chebyshev approximation, which minimizes the maximum absolute error over an interval
• Wavelet approximation: representing a function using a linear combination of scaled and translated versions of a mother wavelet
  • Allows for multi-resolution analysis and efficient representation of local features
• Spline approximation: approximating a function using piecewise polynomials with specified smoothness conditions at the knots
  • Commonly used in interpolation problems and curve fitting
• Radial basis function (RBF) approximation: approximating a function using linear combinations of radially symmetric basis functions centered at different points
  • Useful for scattered data interpolation and machine learning applications
• Rational approximation: approximating a function using a ratio of two polynomials
  • Padé approximation is a specific type of rational approximation that matches the Taylor series coefficients up to a certain order
• Least squares approximation: finding the best approximation that minimizes the sum of squared errors between the approximation and the target function

Numerical Methods and Algorithms

• Gram-Schmidt orthogonalization: constructing an orthonormal basis from a linearly independent set of vectors
  • Used to obtain orthonormal basis functions for approximation problems
• QR factorization: decomposing a matrix into an orthogonal matrix Q and an upper triangular matrix R
  • Useful for solving least squares problems and computing orthogonal projections
• Singular Value Decomposition (SVD): factoring a matrix into the product of three matrices (U, Σ, V^T), where U and V are orthogonal, and Σ is diagonal with non-negative entries
  • Provides a way to compute the pseudoinverse of a matrix and solve ill-conditioned least squares problems
• Iterative methods: algorithms that generate a sequence of approximations converging to the solution of an approximation problem
  • Examples include the conjugate gradient method for solving linear systems and the LSQR algorithm for least squares problems
• Regularization methods: techniques for adding constraints or penalty terms to an approximation problem to prevent overfitting and ensure stability
  • Tikhonov regularization is a common approach that adds a quadratic penalty term to the least squares objective function
• Adaptive approximation: methods that automatically adjust the approximation space or the number of basis functions based on the local properties of the target function
  • Includes adaptive mesh refinement in finite element methods and adaptive wavelet approximation

Challenges and Limitations

• Curse of dimensionality: the exponential growth of the approximation space with the number of dimensions, leading to computational and storage challenges
• Ill-conditioning: approximation problems with poorly conditioned matrices or basis functions, resulting in numerical instability and sensitivity to perturbations
• Gibbs phenomenon: the overshooting of approximations near discontinuities or sharp transitions in the target function, particularly in Fourier series approximations
• Overfitting: the approximation becomes too complex and fits the noise in the data rather than the underlying function, leading to poor generalization performance
• Choosing the right basis functions: selecting an appropriate set of basis functions that can efficiently represent the target function and capture its essential features
  • Requires prior knowledge or assumptions about the properties of the function (smoothness, periodicity, sparsity)
• Balancing accuracy and complexity: finding the right trade-off between the approximation error and the number of basis functions or the complexity of the approximation space
• Handling noisy or incomplete data: developing robust approximation methods that can cope with measurement errors, missing data, or outliers in the target function
• Computational complexity: the time and space requirements of approximation algorithms can become prohibitive for large-scale problems or high-dimensional spaces

Real-world Examples

• Audio compression (MP3): using psychoacoustic models and Fourier-based approximations to compress audio signals while preserving perceptual quality
• Image compression (JPEG): applying discrete cosine transform (DCT) to image blocks and quantizing the coefficients to achieve lossy compression
• Finite element analysis in engineering: approximating the solution of PDEs governing the behavior of structures, fluids, or electromagnetic fields using a finite element mesh
• Machine learning models: approximating complex decision boundaries or regression functions using linear combinations of basis functions (support vector machines, neural networks)
• Quantum chemistry: approximating the electronic structure of molecules using linear combinations of atomic orbitals (LCAO) or plane wave basis functions
• Computerized tomography (CT) reconstruction: approximating the 3D density distribution of an object from a series of 2D projections using techniques like filtered back-projection or iterative reconstruction algorithms
• Robotics and motion planning: approximating the configuration space of a robot using probabilistic roadmaps or rapidly-exploring random trees (RRTs) to find feasible paths
• Climate modeling: approximating the solution of coupled PDEs governing the atmosphere, oceans, and land surface using finite difference methods or spectral methods on a global grid