Hypergraphs provide a natural mathematical framework for modeling systems with higher-order interactions. Motivated by the heterogeneity of many real-world networks, we study a non-uniform inhomogeneous Erdős-Rényi hypergraph model, in which hyperedges of different sizes are included with possibly different probabilities. In a suitable non-sparse regime, we characterize the limiting spectral distribution of an associated adjacency matrix and identify the relative contributions of the different hyperedge sizes.
Joint work with Luca Avena and Elia Bisi (University of Florence).

The Wiener (MMSE) estimator of a Brownian motion (a.k.a. Wiener process) from noisy samples is a linear spline. Similarly, a variational reconstruction problem with second-order total variation regularization yields an adaptive linear spline solution. Deep ReLU networks can also be interpreted as spline functions, albeit in higher-dimensional spaces. In this talk, I will show how these seemingly disparate results are unified through a common variational framework. I will further relate this perspective to stochastic modeling: kernel methods naturally correspond to Gaussian processes, while neural network approaches can be interpreted in terms of sparse stochastic processes.

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