arXiv subjects

Federico Milanesio

Publications and source records attributed to Federico Milanesio.

4 recordsLinked to original sources

arXiv2026

The Advantage of Fine-Grained Training

In classification problems, models are trained to predict a class label based on the input data features. However, class labels are organized hierarchically in many datasets. While a classification task is often defined at a specific level of this hierarchy, training can utilize a finer granularity of labels. Empirical evidence suggests that such fine-grained training can enhance performance. In this work, we investigate the generality of this observation and explore its underlying causes using both real and synthetic datasets. We show that training on fine-grained labels does not universally improve classification accuracy. Instead, the effectiveness of this strategy depends on the geometric structure of the data and its relations with the label hierarchy. Specifically, we show that the advantage of fine-grained training crucially depends on the degree of alignment between the decision boundaries required for the fine- and coarse-grained tasks, a property that we term boundary redundancy. Additionally, factors such as dataset size and model capacity significantly influence whether fine-grained labels provide a performance benefit. Indeed, we identify a transition, whose location is largely controlled by the degree of overparameterization, separating regimes where fine-grained training improves performance from those where direct coarse-grained training is preferable.

cs.LG↗

arXiv2025

Tessellation Groups, Harmonic Analysis on Non-compact Symmetric Spaces and the Heat Kernel in view of Cartan Convolutional Neural Networks

In this paper, we continue the development of the Cartan neural networks programme, launched with three previous publications, by focusing on some mathematical foundational aspects that we deem necessary for our next steps forward. The mathematical and conceptual results are diverse and span various mathematical fields, but the inspiring motivation is unified. The aim is to introduce layers that are mathematically modeled as non-compact symmetric spaces, each mapped onto the next one by solvable group homomorphisms. In particular, in the spirit of Convolutional neural networks, we have introduced the notion of Tits Satake (TS) vector bundles where the TS submanifold is the base space. Within this framework, the tiling of the base manifold, the representation of bundle sections using harmonics, and the need for a general theory of separator walls motivated a series of mathematical investigations that produced both definite and partial results. Specifically, we present the group theoretical construction of the separators for all non-compact symmetric spaces $\mathrm{U/H}$, as well as of the $Δ_{8,3,2}$ tiling group and its normal Fuchsian subgroups, respectively yielding the uniformization of the genus $g=3$ Fermat Quartic and of the genus $g=2$ Bolza surface. The quotient automorphic groups are studied. Furthermore, we found a new representation of the Laplacian Green function and the Heat Kernel on Hyperbolic Spaces $\mathbb{H}^{n}$, and a setup for the construction of the harmonic functions in terms of the spinor representation of pseudo-orthogonal groups. Finally, to obtain an explicit construction of the Laplacian eigenfunctions on the Bolza Riemann surface, we propose and conjecture a new strategy relying on the Abel-Jacobi map of the Riemann surface to its Jacobian variety and the Siegel Theta function.

cs.LG↗

arXiv2025

Navigation through Non-Compact Symmetric Spaces: a mathematical perspective on Cartan Neural Networks

Recent work has identified non-compact symmetric spaces U/H as a promising class of homogeneous manifolds to develop a geometrically consistent theory of neural networks. An initial implementation of these concepts has been presented in a twin paper under the moniker of Cartan Neural Networks, showing both the feasibility and the performance of these geometric concepts in a machine learning context. The current paper expands on the mathematical structures underpinning Cartan Neural Networks, detailing the geometric properties of the layers and how the maps between layers interact with such structures to make Cartan Neural Networks covariant and geometrically interpretable. Together, these twin papers constitute a first step towards a fully geometrically interpretable theory of neural networks exploiting group-theoretic structures

cs.LG↗

arXiv2025

Cartan Networks: Group theoretical Hyperbolic Deep Learning

Hyperbolic deep learning leverages the metric properties of hyperbolic spaces to develop efficient and informative embeddings of hierarchical data. Here, we focus on the solvable group structure of hyperbolic spaces, which follows naturally from their construction as symmetric spaces. This dual nature of Lie group and Riemannian manifold allows us to propose a new class of hyperbolic deep learning algorithms where group homomorphisms are interleaved with metric-preserving diffeomorphisms. The resulting algorithms, which we call Cartan networks, show promising results on various benchmark data sets and open the way to a novel class of hyperbolic deep learning architectures.

cs.LG↗