Near-optimal sample complexity for convex tensor completion
Bibliographic record
Abstract
Abstract We study the problem of estimating a low-rank tensor when we have noisy observations of a subset of its entries. A rank-$r$, order-$d$, $N \times N \times \cdots \times N$ tensor, where $r=O(1)$, has $O(dN)$ free variables. On the other hand, prior to our work, the best sample complexity that was achieved in the literature is $O\left(N^{\frac{d}{2}}\right)$, obtained by solving a tensor nuclear-norm minimization problem. In this paper, we consider the ‘M-norm’, an atomic norm whose atoms are rank-1 sign tensors. We also consider a generalization of the matrix max-norm to tensors, which results in a quasi-norm that we call ‘max-qnorm’. We prove that solving an M-norm constrained least squares (LS) problem results in nearly optimal sample complexity for low-rank tensor completion (TC). A similar result holds for max-qnorm as well. Furthermore, we show that these bounds are nearly minimax rate-optimal. We also provide promising numerical results for max-qnorm constrained TC, showing improved recovery compared to matricization and alternating LS.
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How this classification was reachedexpand
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) | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".