35 citations · 65 across the 9 of their papers we have counts for
10 papers
A Theory of Speciation in Generative Diffusion Models on Compact Riemannian Manifolds
Alessio Marta, Paola Causin
Speciation in generative diffusion models denotes the emergence of distinct stable branches during denoising, through which initially undifferentiated trajectories progressively co…
Estimating Dataset Dimension via Singular Metrics under the Manifold Hypothesis: Application to Inverse Problems
Paola Causin, Alessio Marta
High-dimensional datasets often exhibit low-dimensional geometric structures, as suggested by the manifold hypothesis, which implies that data lie on a smooth manifold embedded in…
Unveiling Transformer Perception by Exploring Input Manifolds
Alessandro Benfenati, Alfio Ferrara, Alessio Marta +2
This paper introduces a general method for the exploration of equivalence classes in the input space of Transformer models. The proposed approach is based on sound mathematical the…
A singular Riemannian Geometry Approach to Deep Neural Networks III. Piecewise Differentiable Layers and Random Walks on -dimensional Classes
Alessandro Benfenati, Alessio Marta
Neural networks are playing a crucial role in everyday life, with the most modern generative models able to achieve impressive results. Nonetheless, their functioning is still not…
Ground State for the Klein-Gordon field in anti-de Sitter spacetime with dynamical Wentzell boundary conditions
Claudio Dappiaggi, Benito A. Juárez-Aubry, Alessio Marta
We consider a real Klein-Gordon field in the Poincaré patch of -dimensional anti-de Sitter spacetime, PAdS, and impose dynamical boundary condition on the asymptotic…
A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations
Alessandro Benfenati, Alessio Marta
Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies emplo…
A singular Riemannian geometry approach to Deep Neural Networks II. Reconstruction of 1-D equivalence classes
Alessandro Benfenati, Alessio Marta
In a previous work, we proposed a geometric framework to study a deep neural network, seen as sequence of maps between manifolds, employing singular Riemannian geometry. In this pa…
Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes
Claudio Dappiaggi, Alessio Marta
We consider the Klein-Gordon operator on an -dimensional asymptotically anti-de Sitter spacetime together with arbitrary boundary conditions encoded by a self-adjoint ps…
A generalization of the propagation of singularities theorem on asymptotically anti-de Sitter spacetimes
Claudio Dappiaggi, Alessio Marta
In a recent paper O. Gannot and M. Wrochna considered the Klein-Gordon equation on an asymptotically anti-de Sitter spacetime subject to Robin boundary conditions, proving in parti…
Ground states of a Klein-Gordon field with Robin boundary conditions in global anti-de Sitter spacetime
Claudio Dappiaggi, Hugo Ferreira, Alessio Marta
We consider a real, massive scalar field both on the -dimensional anti--de Sitter (AdS) spacetime and on its universal cover CAdS. In the second scenario, we extend the…