Speaker: Boumediene Hamzi

Abstract: Statistical learning theory measures capacity by smoothness. This talk develops an alternative in which compressibility plays that role, using kernel methods as the bridge between machine learning, dynamical systems, and algorithmic information theory (AIT).
A reproducing kernel is a measure of similarity, and therefore of compressibility. This observation yields Kolmogorov-complexity and Solomonoff kernels: bona fide Mercer kernels whose geometry is algorithmic. Because these are genuine reproducing kernels, the classical apparatus applies and the three universal spectral regimes of Cucker–Smale–Zhou (CSZ) are inherited. Complexity is then tied to that spectrum, and this is what converts the inherited CSZ regimes into complexity conditions rather than smoothness conditions.
Kernel selection then becomes Occam's razor made computable. Learning kernels from data using Sparse Kernel Flows is recovered as consistent with the MDL principle.
KC and Solomonoff kernels also allow us to construct Solomonoff Gaussian processes: Gaussian processes whose covariance is itself a Kolmogorov-complexity or Solomonoff kernel, so that the associated RKHS, which we call the Solomonoff Gaussian Hilbert Space, is one in which regularity means compressibility and algorithmic simplicity is encoded directly in the prior. These approximate Solomonoff induction: the right balance between Epicurus' principle of multiple explanations - retain every hypothesis consistent with the data - and Occam's razor, obtained by weighting each hypothesis by 2^(−K), so that all candidates survive but the simplest dominate exponentially.
We then argue that simplification in dynamical systems is already an algorithmic-information statement. We show that standard simplifying transformations are consistent with the MDL principle: the Cole-Hopf transformation, which takes Burgers' equation to the heat equation, and Poincaré normal-form reduction for the Brusselator ODE and for the Moore-Greitzer PDE.
Koopman is the part of the programme still under consideration, and we conjecture more broadly that spectral results in Koopman theory are statements about compression, in the same way that, for function approximation from random samples, the three Cucker–Smale–Zhou spectral regimes were reformulated as statements about compression from an AIT perspective.

Bio: Boumediene Hamzi is a Senior Scientist at Caltech's Department of Computing and Mathematical Sciences, a Visiting Reader in the Department of Mathematics at Imperial College London, and a Turing Fellow and External Researcher at the Alan Turing Institute (London, UK). Throughout his research career, he has pursued a central question: How can we analyze complex systems rigorously and at scale? His work spans three complementary mathematical frameworks: (1) Dynamical Systems Theory (DST) provides powerful analytical tools for systems with known models, offering deep insights into stability, long-term behavior, and bifurcations. Yet it reaches its limits in high dimensions and when models are unavailable. (2) Machine Learning (ML) excels precisely where DST struggles: it learns from data in high-dimensional spaces and requires no explicit model. But this empirical power comes at a cost—we lack theoretical understanding of why algorithms work, when they fail, and what guarantees they provide. (3) Algorithmic Information Theory (AIT) bridges this gap by providing rigorous foundations for understanding complexity, randomness, induction, and information. It offers conceptual clarity but grapples with computational feasibility.
His core insight is that these three areas are mutually complementary. By working at their intersection, one can equip Machine Learning with DST's analytical rigor and AIT's theoretical foundations, while enabling classical Dynamical Systems methods to scale to high-dimensional, data-driven settings. This convergence transforms each domain: theory gains practical reach, algorithms gain justification, and complex systems analysis gains both power and understanding.
His personal webpage is at https://sites.google.com/site/boumedienehamzi/

https://us06web.zoom.us/j/84636366540
MEETING ID: 84636366540
PASSCODE: 25862