On the nonexistence of dimension reduction for $\ell2_2$ metrics.
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Bibliographic record
Abstract
An ‘ 2 metric is a metric such that p can be embedded isometrically into R d endowed with Euclidean norm, and the minimal possible d is the dimension associated with . A dimension reduction of an ‘ 2 metric is an embedding of into another ‘ 2 metric µ so that distances in µ are similar to those in and moreover, the dimension associated with µ is small. Much of the motivation in investigating dimension reductions in ‘ 2 comes from a result of Goemans which shows that if such metrics have good dimension reductions, then they embed well into ‘1 spaces. This in turn yields a rounding procedure to a host of semidefinite programming with good approximation guarantees. In this work we show that there is no dimension reduction ‘ 2 metrics in the following strong sense: for every function D(n) and for every n there exists an n point ‘ 2 metric such that for all embeddings of into an ‘ 2 metric µ with distortion at most D(n), the associated dimension of µ is at least n 1. This stands in striking contrast to the Johnson Lindenstrauss lemma which provides a logarithmic dimension reduction for ‘2 metrics.
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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 | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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