Estimating Euclidean Distance to Linearity
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Given oracle access to a real-valued function on the n-dimensional Boolean cube, how many queries does it take to estimate the squared Euclidean distance to its closest linear function within ε? Our main result is that O(log³(1/ε) ⋅ 1/ε²) queries suffice. Not only is the query complexity independent of n but it is optimal up to the polylogarithmic factor. Our estimator evaluates f on pairs correlated by noise rates chosen to cancel out the low-degree contributions to f while leaving the linear part intact. The query complexity is optimized when the noise rates are multiples of Chebyshev nodes. In contrast, we show that the dependence on n is unavoidable in two closely related settings. For estimation from random samples, Θ(√n/ε + 1/ε²) samples are necessary and sufficient. For agnostically learning a linear approximation with ε mean-square regret under the uniform distribution, Ω(n/√ε) nonadaptively chosen queries are necessary, while O(n/ε) random samples are known to be sufficient (Linial, Mansour, and Nisan). Our upper bounds apply to functions with bounded 4-norm. Our lower bounds apply even to ± 1-valued functions.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.003 |
| 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 | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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