Expansions for eigenfunction and eigenvalues of large-$n$ Toeplitz matrices
Bibliographic record
Abstract
This paper constructs methods for finding convergent expansions for eigenvectors and eigenvalues of large-$n$ Toeplitz matrices based on a situation in which the analogous infinite-$n$ matrix would be singular. It builds upon work done by Dai, Geary, and Kadanoff [H Dai et al., J. Stat. Mech. P05012 (2009)] on exact eigenfunctions for Toeplitz operators which are infinite-dimension Toeplitz matrices. One expansion for the finite-$n$ case is derived from the operator eigenvalue equations obtained by continuing the finite-$n$ Toeplitz matrix to plus infinity. A second expansion is obtained by continuing the finite-$n$ matrix to minus infinity. The two expansions work together to give an apparently convergent expansion for the finite-$n$ eigenvalues and eigenvectors, based upon a solvability condition for determining eigenvalues. The expansions involve an expansion parameter expressed as an inverse power of $n$. A variational principle is developed, which gives an approximate expression for determining eigenvalues. The lowest order asymptotics for eigenvalues and eigenvectors agree with the earlier work [H Dai et al., J. Stat. Mech. P05012 (2009)]. The eigenvalues have a $(\ln n)/n$ term as their leading finite-$n$ correction in the central region of the spectrum. The $1/n$ correction in this region is obtained here for the first time.Received: 19 October 2009; Accepted: 29 September 2010; Edited by: A. G. Green; Reviewed by: T. Ehrhardt, Math. Dept., Univ. California, Santa Cruz, USA; DOI: 10.4279/PIP.020003
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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 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".