Recursive Karcher Expectation Estimators And Geometric Law of Large Numbers
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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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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 | 0.000 | 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 | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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