Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models
Bibliographic record
Abstract
Abstract We study the problem of recovering an unknown signal $${\varvec{x}}$$ x given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator $$\hat{\varvec{x}}^\mathrm{L}$$ x ^ L and a spectral estimator $$\hat{\varvec{x}}^\mathrm{s}$$ x ^ s . The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine $$\hat{\varvec{x}}^\mathrm{L}$$ x ^ L and $$\hat{\varvec{x}}^\mathrm{s}$$ x ^ s . At the heart of our analysis is the exact characterization of the empirical joint distribution of $$({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})$$ ( x , x ^ L , x ^ s ) in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of $$\hat{\varvec{x}}^\mathrm{L}$$ x ^ L and $$\hat{\varvec{x}}^\mathrm{s}$$ x ^ s , given the limiting distribution of the signal $${\varvec{x}}$$ x . When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form $$\theta \hat{\varvec{x}}^\mathrm{L}+\hat{\varvec{x}}^\mathrm{s}$$ θ x ^ L + x ^ s and we derive the optimal combination coefficient. In order to establish the limiting distribution of $$({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})$$ ( x , x ^ L , x ^ s ) , we design and analyze an approximate message passing algorithm whose iterates give $$\hat{\varvec{x}}^\mathrm{L}$$ x ^ L and approach $$\hat{\varvec{x}}^\mathrm{s}$$ x ^ s . Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.
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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 | 0.018 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| 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".