A Data-Adaptive RKHS Prior for Bayesian Learning of Kernels in Operators
Bibliographic record
Abstract
Kernels effectively represent nonlocal dependencies and are extensively employed in formulating operators between function spaces. Thus, learning kernels in operators from data is an inverse problem of general interest. Due to the nonlocal dependence, the inverse problem is often severely ill-posed with a data-dependent normal operator. Traditional Bayesian methods address the ill-posedness by a non-degenerate prior, which may result in an unstable posterior mean in the small noise regime, especially when data induces a perturbation in the null space of the normal operator. We propose a new data-adaptive Reproducing Kernel Hilbert Space (RKHS) prior, which ensures the stability of the posterior mean in the small noise regime. We analyze this adaptive prior and showcase its efficacy through applications on Toeplitz matrices and integral operators. Numerical experiments reveal that fixed non-degenerate priors can produce divergent posterior means under errors from discretization, model inaccuracies, partial observations, or erroneous noise assumptions. In contrast, our data-adaptive RKHS prior consistently yields convergent posterior means. ©2024 Neil K. Chada, Quanjun Lang, Fei Lu and Xiong Wang.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.006 | 0.023 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".