Nicolás Alvarado
Especialidad:
Fundamentos teóricos de la IA, redes neuronales y geometría
Email:
nfalvarado@mat.uc.cl
Nicolás estudió licenciatura y magister en matemática en la UC, y luego realizó un doctorado en ciencias de la computación en la misma universidad. Al entrar a computación se dio cuenta de la importancia que tiene la inteligencia artificial en el mundo actual, y se dedicó a estudiar los fundamentos teóricos de ésta, en particular las relaciones entre redes neuronales y geometría. Estuvo alrededor de 5 años trabajando con los alumnos con necesidades educativas especiales de la UC, en donde trabajó principalmente con alumnos con discapacidad sensorial y motora.
PUBLICACIONES
Hyperbolic neural networks have attracted increasing attention within the community in recent years, with various empirical studies on the subject standing out. However, there is little theoretical research on this topic. In this work, we use results from Avelin and Karlsson to ensure convergence of hyperbolic neural networks defined in the Lorentz hyperboloid model. Also, we extend this result to any Riemannian manifold.
We analyze the long term behavior of hyperbolic neural networks through subhomogeneous layer maps, focusing on stability, growth control, and robustness under stochastic perturbations. This work unifies the standard hyperbolic models via explicit isometries and Möbius operations, allowing statements to be transported across representations without loss of geometric meaning. Within this model invariant view, we study iterated, noise perturbed transformations and develop an ergodic theoretic framework that characterizes their asymptotic behavior, including conditions that promote stability and convergence of averaged iterates. Beyond theory, these insights inform practical design choices for training procedures that remain well-behaved in the presence of noise and avoid unbounded parameter growth, thereby supporting more reliable use of hyperbolic representations in hierarchical and graph structured learning tasks.
We introduce Quasiconformal Neural Networks (QNNs), a novel framework that integrates quasiconformal maps into neural architectures, providing a rigorous mathematical basis for handling non-Euclidean data. QNNs control geometric distortions using bounded maximal dilatation across network layers, preserving essential data structures. We present theoretical results that guarantee the stability and geometric consistency of QNNs. This work opens new avenues in geometric deep learning, particularly for applications involving complex topologies, with significant implications for fields such as image registration and medical imaging.
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Vicuña Mackenna 4860
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Vicuña Mackenna 4860
Macul, Chile