Iteratively reweighted least squares minimization for sparse recovery
Bibliographic record
Abstract
Abstract Under certain conditions (known as therestricted isometry property, or RIP) on them×Nmatrix Φ (wherem<N), vectorsx∈ ℝNthat are sparse (i.e., have most of their entries equal to 0) can be recovered exactly fromy:= Φxeven though Φ−1(y) is typically an (N−m)—dimensional hyperplane; in addition,xis then equal to the element in Φ−1(y) of minimal 𝓁1‐norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determiningx, as the limit of aniteratively reweighted least squares(IRLS) algorithm. The main step of this IRLS finds, for a given weight vectorw, the element in Φ−1(y) with smallest 𝓁2(w)‐norm. Ifx(n)is the solution at iteration stepn, then the new weightw(n)is defined byw := [|x |2+ ε ]−1/2,i= 1, …,N, for a decreasing sequence of adaptively defined εn; this updated weight is then used to obtainx(n+ 1)and the process is repeated. We prove that when Φ satisfies the RIP conditions, the sequencex(n)converges for ally, regardless of whether Φ−1(y) contains a sparse vector. If there is a sparse vector in Φ−1(y), then the limit is this sparse vector, and whenx(n)is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast (linear convergencein the terminology of numerical optimization). The same algorithm with the “heavier” weightw = [|x |2+ ε ]−1+τ/2,i= 1, …,N, where 0 < τ < 1, can recover sparse solutions as well; more importantly, we show its local convergence is superlinear and approaches aquadraticrate for τ approaching 0. © 2009 Wiley Periodicals, Inc.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| 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".