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When Compressibility Replaces Smoothness: Bridging Machine Learning, Dynamical Systems and Algorithmic Information Theory via Kolmogorov and Solomonoff Kernels

2026-10-20 02:30-03:30 理科群楼5号楼305

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.

References

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