Lanczos, Householder transformations, and implicit deflation for fast and reliable dominant singular subspace computation
Bibliographic record
Abstract
Abstract Many applications, such as subspace‐based models in information retrieval and signal processing, require the computation of singular subspaces associated with the k dominant, or largest, singular values of an m×n data matrix A, where k≪min(m,n). Frequently, A is sparse or structured, which usually means matrix–vector multiplications involving A and its transpose can be done with much less than 𝒪(mn) flops, and A and its transpose can be stored with much less than 𝒪(mn) storage locations. Many Lanczos‐based algorithms have been proposed through the years because the underlying Lanczos method only accesses A and its transpose through matrix–vector multiplications. We implement a new algorithm, called KSVD, in the Matlab environment for computing approximations to the singular subspaces associated with the k dominant singular values of a real or complex matrix A. KSVD is based upon the Lanczos tridiagonalization method, the WY representation for storing products of Householder transformations, implicit deflation, and the QR factorization. Our Matlab simulations suggest it is a fast and reliable strategy for handling troublesome singular‐value spectra. Copyright © 2001 John Wiley & Sons, Ltd.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 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 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.002 |
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".