A closer look at some new lower bounds on the minimum singular value of a matrix
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Bibliographic record
Abstract
There is an extensive body of literature on estimating the eigenvalues of the sum of two symmetric matrices, P + Q , in relation to the eigenvalues of P and Q . Recently, the authors introduced two novel lower bounds on the minimum eigenvalue, λ min ( P + Q ) , under the conditions that matrices P and Q are symmetric positive semi-definite and their sum P + Q is non-singular. These bounds rely on the Friedrichs angle between the range spaces of matrices P and Q , which are denoted by R ( P ) and R ( Q ) , respectively. In addition, both results led to the derivation of several new lower bounds on the minimum singular value of full-rank matrices. One significant aspect of the two novel lower bounds on λ min ( P + Q ) is the distinction of the case where R ( P ) and R ( Q ) have no principal angles between 0 and π 2 . This work offers an explanation for the aforementioned scenario and presents a classification of all matrices that meet the specified criteria. Additionally, we offer insight into the rationale behind selecting the decomposition for the subspace R ( Q ) , which is employed to formulate the lower bounds for λ min ( P + Q ) . At last, an example that showcases the potential for improving these two lower bounds is presented.
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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 | 0.001 | 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 | 0.001 | 0.000 |
| Open science | 0.001 | 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