Trading off 1-norm and sparsity against rank for linear models using mathematical optimization: 1-norm minimizing partially reflexive ah-symmetric generalized inverses
Bibliographic record
Abstract
The M-P (Moore–Penrose) pseudoinverse has as a key application the computation of least-squares solutions of inconsistent systems of linear equations. Irrespective of whether a given input matrix is sparse, its M-P pseudoinverse can be dense, potentially leading to high computational burden, especially when we are dealing with high-dimensional matrices. The M-P pseudoinverse is uniquely characterized by four properties, but only two of them need to be satisfied for the computation of least-squares solutions. Fampa and Lee (2018) and Xu, Fampa, Lee, and Ponte (2019) propose local-search procedures to construct sparse block-structured generalized inverses that satisfy the two key M-P properties, plus one more (the so-called reflexive property). That additional M-P property is equivalent to imposing a minimum-rank condition on the generalized inverse. (Vector) 1-norm minimization is used to induce sparsity and, importantly, to keep the magnitudes of entries under control for the generalized-inverses constructed. Here, we investigate the trade-off between low 1-norm and low rank for generalized inverses that can be used in the computation of least-squares solutions. We propose several algorithmic approaches that start from a 1 -norm minimizing generalized inverse that satisfies the two key M-P properties, and gradually decrease its rank, by iteratively imposing the reflexive property. The algorithms iterate until the generalized inverse has the least possible rank. During the iterations, we produce intermediate solutions, trading off low 1-norm (and typically high sparsity) against low rank.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 | 0.006 |
| 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 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 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".