Phase Diagram of Extensive-Rank Symmetric Matrix Denoising beyond Rotational Invariance
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Bibliographic record
Abstract
Matrix denoising is central to signal processing and machine learning. Its statistical analysis when the matrix to infer has a factorized structure with a rank growing proportionally to its dimension remains a challenge, except when it is rotationally invariant. In this case, the information-theoretic limits and an efficient Bayes-optimal denoising algorithm, called the rotational invariant estimator, are known. Beyond this setting, few results can be found. The reason is that the model is not a usual spin system because of the growing rank dimension, nor a matrix model (as appearing in high-energy physics) due to the lack of rotation symmetry, but rather a hybrid between the two. In this paper, we make progress toward the understanding of Bayesian matrix denoising when the hidden signal is a factored matrix <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mrow> <a:mi mathvariant="bold">X</a:mi> </a:mrow> <a:msup> <a:mrow> <a:mi mathvariant="bold">X</a:mi> </a:mrow> <a:mrow> <a:mo>⊺</a:mo> </a:mrow> </a:msup> </a:mrow> </a:math> that is not rotationally invariant. Monte Carlo simulations suggest the existence of a denoising-factorization transition separating a phase where denoising using the rotational-invariant estimator remains Bayes-optimal due to universality properties of the same nature as in random matrix theory, from one where universality breaks down and better denoising is possible, though algorithmically hard. We also argue that it is only beyond the transition that factorization, i.e., estimating <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mrow> <e:mi mathvariant="bold">X</e:mi> </e:mrow> </e:math> itself, becomes possible up to irresolvable ambiguities. On the theoretical side, we combine mean-field techniques in an interpretable multiscale fashion in order to access the minimum mean-square error and mutual information. Interestingly, our alternative method yields equations reproducible by the replica approach of Sakata and Kabashima. Using numerical insights, we delimit the portion of phase diagram where we conjecture the mean-field theory to be exact and correct it using universality when it is not. Our complete matches well the numerics in the whole phase diagram when considering finite-size effects.
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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 | 0.001 |
| 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 it