Algorithms for approximate sparse regression and nearest induced hulls
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Bibliographic record
Abstract
The sparse regression problem, also known as best subset selection problem, can be cast as follows: Given a set $S$ of $n$ points in $\mathbb{R}^d$, a point $y\in\mathbb{R}^d$, and an integer $2 \leq k \leq d$, find an affine combination of at most $k$ points of $S$ that is nearest to $y$. We describe a $O(n^{k-1} \log^{d-k+1} n)$-time randomized $(1+\varepsilon)$-approximation algorithm for this problem with $d$ and $\varepsilon$ constant. This is the first algorithm for this problem running in time $o(n^k)$. Its running time is similar to the query time of a data structure recently proposed by Har-Peled, Indyk, and Mahabadi (ICALP'18), while not requiring any preprocessing. Up to polylogarithmic factors, it matches a conditional lower bound relying on a conjecture about affine degeneracy testing. In the special case where $k = d = O(1)$, we also provide a simple $O(n^{d-1+\delta})$-time deterministic exact algorithm, for any $\delta > 0$. Finally, we show how to adapt the approximation algorithm for the sparse convex regression problems with the same running time, up to polylogarithmic factors.
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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 | 0.001 |
| Open science | 0.000 | 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