The Landscape of the Spiked Tensor Model
Why this work is in the frame
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Bibliographic record
Abstract
We consider the problem of estimating a large rank‐one tensor u ⊗ k ∈ ( ℝ n ) ⊗ k , k ≥ 3 , in Gaussian noise. Earlier work characterized a critical signal‐to‐noise ratio λ Bayes = O (1) above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably, no polynomial‐time algorithm is known that achieved this goal unless λ ≥ Cn ( k − 2)/4 , and even powerful semidefinite programming relaxations appear to fail for 1 ≪ λ ≪ n ( k − 2)/4 . In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree‐ k homogeneous polynomial over the unit sphere in n dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions n , and give exact formulas for the exponential growth rate. We show that (for λ larger than a constant) critical points are either very close to the unknown vector u or are confined in a band of width Θ( λ −1/( k − 1) ) around the maximum circle that is orthogonal to u . For local maxima, this band shrinks to be of size Θ( λ −1/( k − 2) ) . These “uninformative” local maxima are likely to cause the failure of optimization algorithms. © 2019 Wiley Periodicals, Inc.
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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 | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.002 |
| 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