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

We develop an approximation formula for the cross-validation error (CVE) of a sparse linear regression penalized by ${\mathrm{\ell }}_1$-norm and total variation terms, which is based on a perturbative expansion utilizing the largeness of both the data dimensionality and the model. The developed formula allows us to reduce the necessary computational cost of the CVE evaluation significantly. The practicality of the formula is tested through application to simulated black-hole image reconstruction on the event-horizon scale with super resolution. The results demonstrate that our approximation reproduces the CVE values obtained via literally conducted cross-validation with reasonably good precision.