The matrix Dyson equation for machine learning: Correlated linearizations and the test error in random features regression

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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