Séminaire de Vahid Shaverdi (KTH, Sweden)

Title: Neuroalgebraic Geometry: An Algebraic Perspective on Neural Networks

In this talk, I introduce neuroalgebraic geometry: a unified algebraic-geometric framework for analyzing neural networks via polynomial activations. Our central object is the neuromanifold, the function space realized by a network, which forms a semi-algebraic variety when activations are polynomial.

I begin by explaining how algebraic invariants, particularly the dimension and degree of the neuromanifold, relate to fundamental learning properties such as expressivity and sample complexity. Then, by analyzing both fully connected (MLP) and convolutional neural network (CNN) architectures, we uncover a striking phenomenon: training dynamics tend to concentrate near singular loci of the neuromanifold, revealing an implicit bias toward simpler or sparse solutions.

Finally, I discuss how the global geometry of neuromanifolds, captured by algebraic invariants, influences the structure of the loss landscape. This perspective offers new insights into the optimization dynamics and generalization behavior of neural networks.

Place: Seminar room at 4th floor of the FST 1er cycle (Henri Poincaré) building

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