Colóquio de Matemática

02/10/2026, 14h - Auditório EFI (!)

Evento conjunto com o III EncPos - Encontro de Pós-Graduandos da Matemática UFSC 2026

Prof. Dr. Tom Hanika

Information Systems and Machine Learning Lab, University of Hildesheim

Título: Regularization Methods for Deep Learning from Metric Measure Geometry

Resumo: This talk presents several regularization methods for deep learning that share a common origin: a notion of intrinsic dimension rooted in Gromov's metric measure geometry.

The approach rests on the observable diameter, which measures how much of the geometry of a metric measure space remains visible under real-valued observation; the intrinsic dimension of a set of points, or of the representation a network forms of them, is read off from its concentration behaviour. It requires neither an underlying manifold nor a metric of a particular form, and it can be computed for data sets and network representations alike.

From this one quantity we derive regularizers at three stages of the learning process. At the level of the data, it selects features by their resilience to the curse of dimensionality. At the level of the model, it prunes the components of an overparameterized network that contribute nothing to the intrinsic dimension of its representations. At the level of training, applied layer by layer, it regularizes the representations themselves. We present these methods as instances of a single geometric principle, discuss what distinguishes them in practice, and argue that where the complexity of a problem resides determines which of them to use.

This is joint work with Friedrich Martin Schneider and Vladimir Pestov.

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