Iteratively reweighted least squares minimization for sparse recovery
Bibliographic record
Abstract
Abstract Under certain conditions (known as the restricted isometry property , or RIP) on the m × N matrix Φ (where m < N ), vectors x ∈ ℝ N that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y := Φ x even though Φ −1 ( y ) is typically an ( N − m )—dimensional hyperplane; in addition, x is then equal to the element in Φ −1 ( y ) of minimal 𝓁 1 ‐norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x , as the limit of an iteratively reweighted least squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w , the element in Φ −1 ( y ) with smallest 𝓁 2 ( w )‐norm. If x ( n ) is the solution at iteration step n , then the new weight w ( n ) is defined by w := [| x | 2 + ε ] −1/2 , i = 1, …, N , for a decreasing sequence of adaptively defined ε n ; this updated weight is then used to obtain x ( n + 1) and the process is repeated. We prove that when Φ satisfies the RIP conditions, the sequence x ( n ) converges for all y , 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 when x ( n ) is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast ( linear convergence in the terminology of numerical optimization). The same algorithm with the “heavier” weight w = [| 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 a quadratic rate 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".