Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models
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Bibliographic record
Abstract
Abstract We study the problem of recovering an unknown signal $${\varvec{x}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator $$\hat{\varvec{x}}^\mathrm{L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>L</mml:mi> </mml:msup> </mml:math> and a spectral estimator $$\hat{\varvec{x}}^\mathrm{s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:msup> </mml:math> . 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 $$\hat{\varvec{x}}^\mathrm{L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>L</mml:mi> </mml:msup> </mml:math> and $$\hat{\varvec{x}}^\mathrm{s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:msup> </mml:math> . At the heart of our analysis is the exact characterization of the empirical joint distribution of $$({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>L</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of $$\hat{\varvec{x}}^\mathrm{L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>L</mml:mi> </mml:msup> </mml:math> and $$\hat{\varvec{x}}^\mathrm{s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:msup> </mml:math> , given the limiting distribution of the signal $${\varvec{x}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> . When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form $$\theta \hat{\varvec{x}}^\mathrm{L}+\hat{\varvec{x}}^\mathrm{s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>L</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> </mml:math> and we derive the optimal combination coefficient. In order to establish the limiting distribution of $$({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>L</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , we design and analyze an approximate message passing algorithm whose iterates give $$\hat{\varvec{x}}^\mathrm{L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>L</mml:mi> </mml:msup> </mml:math> and approach $$\hat{\varvec{x}}^\mathrm{s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:msup> </mml:math> . Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.
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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 | 0.000 | 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