8.4 Regularization methods for linear problems

Regularization methods are crucial for solving linear inverse problems that are ill-posed. They stabilize solutions by adding constraints or prior information, balancing data fit with solution complexity. This approach reduces sensitivity to noise and improves model performance.

Common techniques include Tikhonov regularization, truncated SVD, and iterative methods. Each method has unique strengths, like preserving edges or promoting sparsity. Choosing the right approach and parameter values is key to getting reliable solutions in various applications.

Regularization for Ill-Posed Problems

Understanding Ill-Posed Problems and Regularization

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• Ill-posed problems characterized by solutions that are not unique, not stable, or do not exist for all data
• Regularization stabilizes solutions of ill-posed inverse problems by incorporating additional information or constraints
• Regularization balances fitting data and satisfying additional constraints or prior information about the solution
• Regularization methods typically add penalty term to objective function, controlling solution complexity or smoothness
• Regularization parameter controls trade-off between data fidelity and regularization term (larger values result in smoother solutions)
• Regularization reduces sensitivity of solution to noise in data and improves model generalization performance

Common Regularization Techniques

• Tikhonov regularization (ridge regression) widely used for solving ill-posed linear inverse problems
• Truncated singular value decomposition (TSVD) filters out small singular values, reducing solution space dimension
• Iterative methods like conjugate gradient implicitly regularize through early termination of iterative process
• Total variation (TV) regularization preserves edges by penalizing L1 norm of gradient
• Sparse regularization techniques (L1 regularization, elastic net) promote sparse solutions by penalizing L1 norm
• Regularization by projection methods (Krylov subspace methods) reduce problem dimensionality through subspace projection

Tikhonov Regularization in Inverse Problems

Formulation and Implementation

• Tikhonov regularized solution minimizes combination of data misfit and regularization term (typically L2 norm of solution or derivatives)
• Standard form of Tikhonov regularization solves minimization problem: $\min ||Ax - b||^2 + \lambda^2 ||Lx||^2$
  • A: forward operator
  • b: data
  • x: solution
  • λ: regularization parameter
  • L: regularization matrix
• Regularization matrix L choice depends on prior solution information (identity matrix for zero-order, finite difference operators for first-order or second-order)
• Regularized solution computed explicitly using normal equations: $x = (A^T A + \lambda^2 L^T L)^{-1} A^T b$
  • ^T denotes transpose
  • ^(-1) denotes inverse

Computational Considerations

• Large-scale problems use iterative methods (conjugate gradient, LSQR) to efficiently compute Tikhonov regularized solution
• Effectiveness of Tikhonov regularization depends on choice of regularization parameter λ and regularization matrix L
• Tikhonov regularization promotes smooth solutions by penalizing L2 norm of solution or derivatives
• Computational efficiency varies based on problem size and structure (direct methods for small problems, iterative methods for large-scale problems)

Parameter Selection for Regularization

Graphical and Cross-Validation Methods

• L-curve method: graphical tool for selecting regularization parameter
  • Plots norm of regularized solution against norm of residual for different parameter values
  • Optimal parameter chosen at corner of L-shaped curve, balancing trade-off between solution norm and residual norm
• Cross-validation techniques select regularization parameter by minimizing prediction error on held-out data
  • k-fold cross-validation: divides data into k subsets, trains on k-1 subsets and validates on remaining subset
  • Leave-one-out cross-validation: uses single observation for validation and remaining observations for training
• Generalized cross-validation (GCV) method selects regularization parameter by minimizing function approximating leave-one-out cross-validation error

Other Parameter Selection Approaches

• Discrepancy principle chooses largest regularization parameter such that residual norm consistent with data noise level
• Bayesian methods provide framework for estimating regularization parameter along with solution
  • Hierarchical Bayes: models regularization parameter as random variable with prior distribution
  • Empirical Bayes: estimates regularization parameter from data using maximum likelihood or moment matching
• Choice of parameter selection method depends on factors like problem structure, computational resources, and prior knowledge of data noise level

Regularization Techniques: A Comparison

Smoothness and Sparsity Promoting Methods

• Tikhonov regularization promotes smooth solutions (penalizes L2 norm of solution or derivatives)
• Total variation (TV) regularization preserves edges (penalizes L1 norm of gradient)
• Sparse regularization techniques promote sparse solutions
  • L1 regularization (Lasso): penalizes L1 norm of solution
  • Elastic net: combines L1 and L2 penalties for both sparsity and smoothness
• Choice between smoothness and sparsity-promoting methods depends on expected solution characteristics (smooth functions, piecewise constant functions, sparse representations)

Dimensionality Reduction and Iterative Methods

• Truncated singular value decomposition (TSVD) filters out small singular values, reducing solution space dimension
• Tikhonov regularization dampens contribution of small singular values without explicit truncation
• Iterative regularization methods implicitly regularize through early termination
  • Conjugate gradient least squares (CGLS)
  • Landweber iteration
• Regularization by projection methods reduce problem dimensionality
  • Krylov subspace methods project onto lower-dimensional subspace
  • Tikhonov regularization modifies full-dimensional problem
• Choice between dimensionality reduction and iterative methods influenced by problem size, computational resources, and desired solution properties