Reconstructing high-dimensional Hilbert-valued functions via compressed\n sensing
Bibliographic record
Abstract
We present and analyze a novel sparse polynomial technique for approximating\nhigh-dimensional Hilbert-valued functions, with application to parameterized\npartial differential equations (PDEs) with deterministic and stochastic inputs.\nOur theoretical framework treats the function approximation problem as a joint\nsparse recovery problem, where the set of jointly sparse vectors is possibly\ninfinite. To achieve the simultaneous reconstruction of Hilbert-valued\nfunctions in both parametric domain and Hilbert space, we propose a novel\nmixed-norm based $\\ell_1$ regularization method that exploits both energy and\nsparsity. Our approach requires extensions of concepts such as the restricted\nisometry and null space properties, allowing us to prove recovery guarantees\nfor sparse Hilbert-valued function reconstruction. We complement the enclosed\ntheory with an algorithm for Hilbert-valued recovery, based on standard\nforward-backward algorithm, meanwhile establishing its strong convergence in\nthe considered infinite-dimensional setting. Finally, we demonstrate the\nminimal sample complexity requirements of our approach, relative to other\npopular methods, with numerical experiments approximating the solutions of\nhigh-dimensional parameterized elliptic PDEs.\n
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 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".