Semi-supervised learning (SSL) is the problem of finding missing labels from a partially labelled data set. The heuristic one uses is that “similar feature vectors should have similar labels”. The notion of similarity between feature vectors explored in this talk comes from a graph-based geometry where an edge is placed between feature vectors that are closer than some connectivity radius. A...
This talk presents results on scaling laws in hypergraph and multi-operator learning. Scaling considerations simultaneously guide both the theoretical foundations and the practical design of modern learning systems. The first part shows how a large-data asymptotic analysis identifies connectivity-scaling regimes in which semi-supervised learning on hypergraphs is effective and stable. This...
The fractional graph Laplacian, defined as a fractional power of the standard graph Laplacian, is one of the most popular tools for modeling non‑local diffusion on graphs. However, it is known to induce dynamics that, in some cases, are incompatible with the topology of the original network. To address this limitation, a regularized fractional Laplacian obtained through a combination of the...
Subdiffusion on networks can occur where overcrowding is present. It is studied through time-fractional diffusion equations, and admits an explicit solution through the Mittag-Leffler function. We give a representation of subdiffusion as a superposition of classical diffusion processes with subordination to a different timescale. Memory arises in subdiffusion, while the classical diffusion...
In many applied research fields—including Geophysics, Medicine, Engineering, Economics, and Finance— fundamental challenges involve extracting hidden information and meaningful features from complex signals, such as quasi-periodicities, time-varying frequency patterns, and underlying components like trends.
Classical signal processing techniques based on Fourier and Wavelet transforms,...
In signal processing, the time-frequency analysis of nonlinear and non-stationary processes, as well as the determination of the unknown number of active sub-signals in a blind-source composite signal, are generally challenging inverse problem tasks. If we consider data sampled on a sphere, things get even more complicated. This is the reason why just a few techniques have been developed so...
Accuracy–cost trade-offs and data and parameter efficiency are two fundamental aspects of machine learning for scientific computing. In the first part of this talk, we address test-time control of model performance. We introduce the Recurrent-Depth Simulator (RecurrSim), an architecture-agnostic framework that enables explicit control over accuracy–cost trade-offs in neural simulators without...
Recent empirical and theoretical results suggest that deep networks possess an implicit low-rank bias: their weight matrices naturally evolve toward approximately low-rank structure, and a structured pruning of small singular values can often reduce model size with little or no loss in accuracy. While this phenomenon is already understood in simplified settings, a complete theory accounting...
In this talk I will consider a method for learning the Lagrangian and forces for mechanical systems using the discrete Lagrange d'Alembert principle. The case of manifold valued data and data on Lie groups will also be discussed if time permits.
I will also describe a number of current projects with diverse applications where the main theme is learning vector fields from data.
Graph Neural Networks (GNNs) have emerged as powerful tools for nonlinear Model Order Reduction (MOR) of time-dependent parameterized Partial Differential Equations (PDEs) [1].
However, existing methodologies struggle to combine geometric inductive biases with interpretable latent dynamics, overlooking dynamics-driven features or disregarding geometric information, respectively.
In this...
Data-driven model discovery has become a powerful approach for identifying governing equations of dynamical systems using temporal data. The Sparse Identification of Nonlinear Dynamics (SINDy) algorithm, initially developed for ordinary differential equations (ODEs), has been extended to more general classes of problems, including partial differential equations (PDEs), stochastic differential...
The Hadamard decomposition is a powerful technique for data analysis and compression. Given an $m\times n$ matrix $X$ and two ranks $r_1,r_2\ll \min(m,n)$, we look for two low-rank matrices $X_1$ and $X_2$ of the same size of $X$ and with $\text{rank}(X_i)=r_i$ such that $X\approx X_1\circ X_2$, where $\circ$ is the element-wise product. In contrast with the well-known SVD, this decomposition...
This talk is about efficient solution methods for inverse problems, i.e., the task of recovering an object of interest from known but corrupted data available through a known but possibly corrupted model, formulated in a discrete and linear setting. Since these problems are ill posed, variational regularization methods are often applied to recover a meaningful solution. However, variational...
Non-linearity often leads to slow or unstable convergence in iterative solvers for nonlinear least-squares problems. In this work, we introduce a family of accelerated algorithms that leverage a periodically restarted variant of the Generalized Minimum Residual (GMRES) method to address these challenges. The restarting strategy keeps the computational cost under control and makes the method...
The log-determinant of a symmetric positive semi-definite matrix is a quantity that arises in different contexts, for instance in the evaluation of the log-marginal likelihood for Gaussian processes and in the normalization of the determinantal point processes for supervised learning.
We focus on randomized algorithms for estimating this quantity. The algorithms access the matrix only...
The nearest correlation matrix problem consists in finding the closest valid correlation matrix to a given symmetric matrix that may fail to be positive semi-definite. In other words, given a symmetric unit-diagonal matrix that is not a proper correlation matrix, one seeks the nearest positive semi-definite matrix with unit diagonal entries.
We address the problem of finding the nearest...
We propose two methods for the unsupervised detection of communities in undirected multiplex networks. These networks consist of multiple layers that record different relationships between the same entities or incorporate data from different sources. Both methods are formulated as gradient flows of suitable energy functionals: the first (MPBTV) builds on the minimization of a balanced total...
We introduce past-aware game-theoretic centrality, a class of centrality measures that captures the collaborative contribution of nodes in a network, accounting for both uncertain and certain collaborators. A general framework for computing standard game-theoretic centrality is extended to the past-aware case. As an application, we develop a new heuristic for different versions of the...
I will present two recent works, coauthored with Mei-Heng Yueh, in which we propose Riemannian optimization algorithms for computing spherical and toroidal area-preserving mappings of genus-zero and genus-one closed surfaces, respectively. The proposed framework is based on retraction-based Riemannian optimization, which provides an effective way to handle the geometric constraints of the...
We investigate the propagation of initial value perturbations along the solution of a linear ordinary differential equation $y'(t) = Ay(t)$. This propagation is analyzed using the relative error rather than the absolute error. Our focus is on the long-term behavior of this relative error, which differs significantly from that of the absolute error. Understanding the long-term behavior provides...
