On the Equivalence of Linear Discriminant Analysis and Least Squares
Bibliographic record
Abstract
Linear discriminant analysis (LDA) is a popular dimensionality reduction and classification method that simultaneously maximizes between-class scatter and minimizes within-class scatter. In this paper, we verify the equivalence of LDA and least squares (LS) with a set of dependent variable matrices. The equivalence is in the sense that the LDA solution matrix and the LS solution matrix have the same range. The resulting LS provides an intuitive interpretation in which its solution performs data clustering according to class labels. Further, the fact that LDA and LS have the same range allows us to design a two-stage algorithm that computes the LDA solution given by generalized eigenvalue decomposition (GEVD), much faster than computing the original GEVD. Experimental results demonstrate the equivalence of the LDA solution and the proposed LS solution.
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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 | 0.014 | 0.061 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.007 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.003 | 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".