Recursive Karcher Expectation Estimators And Geometric Law of Large Numbers
Bibliographic record
Abstract
This paper studies a form of law of large numbers on Pn, the space of n × n symmet-ric positive-definite matrices equipped with Fisher-Rao metric. Specifically, we pro-pose a recursive algorithm for estimating the Karcher expectation of an arbitrary distribu-tion defined on Pn, and we show that the es-timates computed by the recursive algorithm asymptotically converge in probability to the correct Karcher expectation. The steps in the recursive algorithm mainly consist of mak-ing appropriate moves on geodesics in Pn, and the algorithm is simple to implement and it offers a tremendous gain in compu-tation time of several orders in magnitude over existing non-recursive algorithms. We elucidate the connection between the more familiar law of large numbers for real-valued random variables and the asymptotic conver-gence of the proposed recursive algorithm, and our result provides an example of a new form of law of large numbers for random vari-ables taking values in a Riemannian mani-fold. From the practical side, the computa-tion of the mean of a collection of symmetric positive-definite (SPD) matrices is a funda-mental ingredient in many algorithms in ma-chine learning, computer vision and medical imaging applications. We report an experi-ment using the proposed recursive algorithm for K-means clustering, demonstrating the al-gorithm’s efficiency, accuracy and stability.
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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 | 0.005 | 0.043 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 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".