Accelerating cross-validation with total variation and its application to super-resolution imaging.

Area of Science:

• Computational mathematics
• Statistical learning theory
• Astrophysical imaging

Background:

• Sparse linear regression is widely used in high-dimensional data analysis.
• Estimating model performance using cross-validation error (CVE) can be computationally intensive.
• ℓ1-norm and total variation penalties are common in sparse regression for feature selection and regularization.

Purpose of the Study:

• To develop an approximation formula for the cross-validation error (CVE) of sparse linear regression.
• To reduce the computational cost associated with CVE evaluation.
• To validate the approximation's accuracy in a complex application like black-hole image reconstruction.

Main Methods:

• Developed a novel approximation formula for CVE using perturbative expansion.
• Leveraged the large dimensionality of data and model size in the expansion.
• Applied the formula to simulated super-resolution black-hole image reconstruction data.

Main Results:

• The approximation formula significantly reduces computational cost for CVE evaluation.
• The formula accurately reproduces CVE values obtained from traditional cross-validation.
• Achieved reasonably good precision in the context of event-horizon scale black-hole imaging.

Conclusions:

• The developed approximation offers an efficient alternative for CVE estimation in sparse linear regression.
• This method is practical and accurate, even for computationally demanding applications.
• Enables faster model selection and hyperparameter tuning in high-dimensional settings.