The matrix Dyson equation for machine learning: Correlated linearizations and the test error in random features regression
Bibliographic record
Abstract
Contemporary machine learning models, particularly deep learning models, are frequently trained on large datasets within high-dimensional feature spaces, presenting challenges for traditional analytical approaches.Notably, the effective generalization of highly overparameterized models contradicts conventional statistical wisdom.Furthermore, the presence of non-linear activations in artificial neural networks adds complexity to their analysis.To simplify theoretical analysis, it is often assumed that training data is sampled from an unstructured distribution.While such analyses offer insights into certain aspects of machine learning, they fall short in elucidating how neural networks extract information from the structure of the data, crucial for their success in real-world applications.Fortunately, random matrix theory has emerged as a valuable tool for theoretically understanding certain machine learning procedures.Various techniques have been employed to explore large random matrices through asymptotic deterministic equivalents.One such approach involves substituting the random resolvent associated with a large random matrix with the solution of a deterministic fixed-point equation known as the matrix Dyson equation.Another effective technique, known as the linearization trick, involves embedding a matrix expression into a larger random matrix, termed a linear matrix pencil, with a simplified correlation structure.In this thesis, we extend the matrix Dyson equation framework to derive an anisotropic global law for a broad class of pseudo-resolvents with general correlation structures.This extension enables the analysis of spectral properties of a wide range of random matrices using a simpler and deterministic solution to the matrix Dyson equation.Through the development of this theory, we address critical aspects such as existence-uniqueness, spectral support bounds, and stability properties.These considerations are essential for constructing i I wish to extend my deepest gratitude to Professors Courtney and Elliot Paquette, my supervisors, for their guidance and support throughout my academic journey.I consider myself incredibly fortunate to have had the opportunity to learn from them, and I am grateful for their constant availability, advice, and encouragement.Their mentorship has been instrumental in shaping me into the researcher I am today.They exemplify excellence in both research and mentorship, serving as role models for aspiring researchers like myself.I would also like to express my heartfelt thanks to Courtney for generously funding my studies and research.Courtney and Elliot, thank you for everything.I am also grateful to Professor Tim Hoheisel for introducing me to the field of mathematical optimization and providing continuous guidance.Although optimization is underrepresented in this thesis, it has significantly shaped my academic journey and greatly influenced my research interests.While I cannot name everyone, I would like to express my gratitude to the numerous brilliant students and faculty members who have made my time at McGill University memorable and enjoyable.Last but not least, I extend my heartfelt thanks to my family, friends, and partner for their unwavering support, encouragement, and understanding throughout this journey.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.018 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.004 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".