Abstract
We develop an approximation formula for the cross-validation error (CVE) of a sparse linear regression penalized by ℓ1-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.
Highlights
At present, in many practical situations of science and technology, large high-dimensional observational datasets are created and accumulated on a continuous basis
Reducing the cross-validation error (CVE) computational cost could have a significant impact on a broad range of sparse modeling applications in various disciplines. Considering these circumstances, in this paper, we provide a CVE approximation formula for a statistical model of linear regression penalized by the l1 and total variation (TV) terms, to efficiently reduce the computational cost
We have developed an approximation formula for the CVE of a sparse linear regression penalized by l1 and TV terms, and demonstrated its usefulness in the reconstruction of simulated black hole images
Summary
In many practical situations of science and technology, large high-dimensional observational datasets are created and accumulated on a continuous basis. Sparse modeling [1, 2] is a promising framework for overcoming this difficulty, which has recently been utilized in many disciplines [3, 4] In this framework, a statistical or machine-learning model with a large number of parameters (explanatory variables) is fitted to the data, in conjunction with a certain sparsity-inducing penalty. The TV yields “continuity” of the neighboring variables, which is suitable for the processing of certain datasets expected to have such continuity, such as natural images [4, 7,8,9] Another common difficulty associated with the use of statistical models is model selection. Note that our formula will be applied to actual EHT observations to be conducted after April 2017
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