Maximum Likelihood Estimation of Regularization Parameters in High-Dimensional Inverse Problems: An Empirical Bayesian Approach Part I: Methodology and Experiments
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Bibliographic record
Abstract
Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularizing the estimation problem to make it well- posed. This often requires setting the value of the so-called regularization parameters that control the amount of regularization enforced. These parameters are notoriously difficult to set a priori and can have a dramatic impact on the recovered estimates. In this work, we propose a general empirical Bayesian method for setting regularization parameters in imaging problems that are convex w.r.t. the unknown image. Our method calibrates regularization parameters directly from the observed data by maximum marginal likelihood estimation and can simultaneously estimate multiple regularization parameters. Furthermore, the proposed algorithm uses the same basic operators as proximal optimization algorithms, namely gradient and proximal operators, and it is therefore straightforward to apply to problems that are currently solved by using proximal optimization techniques. Our methodology is demonstrated with a range of experiments and comparisons with alternative approaches from the literature. The considered experiments include image denoising, nonblind image deconvolution, and hyperspectral unmixing, using synthesis and analysis priors involving the $\ell_1$, total-variation, total-variation and $\ell_1$, and total-generalized-variation pseudonorms. A detailed theoretical analysis of the proposed method is presented in our companion paper [V. De Bortoli et al., SIAM J. Imaging Sci., 13 (2020), pp. 1990--2028].
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it