More is different.

"The constructionist hypothesis breaks down when confronted with the twin difficulties of scale and complexity. The behavior of large and complex aggregates of elementary particles, it turns out, is not to be understood in terms of a simple extrapolation of the properties of a few particles. Instead, at each level of complexity entirely new properties appear, and the understanding of the new behaviors requires research which I think is as fundamental in its nature as any other."

P. W. Anderson

Research interests

My research currently focuses on applications of Mathematical and Statistical Physics techniques to Statistics and Information Theory, with a particular emphasis on spin-glasses, whose theory earned Giorgio Parisi the Nobel Prize for Physics in 2021. Some of these techniques can also be used to establish the fundamental limits of performance of some simple neural networks of Artificial Intelligence.

Publications and preprints

On the phase diagram of the multiscale mean-field spin-glass

Francesco Camilli, Pierluigi Contucci, Emanuele Mingione, Daniele Tantari, arXive e-prints (2024), https://doi.org/10.48550/arXiv.2412.00992

On the phase diagram of extensive-rank symmetric matrix denoising beyond rotational invariance

Jean Barbier, Francesco Camilli, Justin Ko, Koki Okajima, arXive e-prints (2024),
https://doi.org/10.48550/arXiv.2411.01974

Information limits and Thouless-Anderson-Palmer equations for spiked matrix models with structured noise

Jean Barbier, Francesco Camilli, Yizhou Xu, Marco Mondelli, Physical Review Research (2024) https://doi.org/10.1103/PhysRevResearch.7.013081

Beyond rotational invariance

The statistical limits of matrix denoising when the target matrix is not rotationally invariant have been haunting the Statistical Physics and Information Theory communities for more than a decade now.

In "On the phase diagram of extensive-rank symmetric matrix denoising beyond rotational invariance" we propose an innovative strategy to tackle this tough problem, with very encouraging results (the match on the right is not bad, right?).

Check out our small introductory article if you are curious!

Fundamental limits of overparametrized shallow neural networks for supervised learning

Francesco Camilli, Daria Tieplova, Jean Barbier, arXive e-prints (2023) https://doi.org/10.48550/arXiv.2307.05635

Central limit theorem for the overlaps on the Nishimori line

Francesco Camilli, Pierluigi Contucci, Emanuele Mingione, arXive e-prints (2023), https://doi.org/10.48550/arXiv.2305.19943

The Decimation Scheme for Symmetric Matrix Factorization

Francesco Camilli and Marc Mézard, arXive e-prints (2023), https://doi.org/10.48550/arXiv.2307.16564

Matrix factorization with neural networks

Francesco Camilli, Marc Mézard, Phys. Rev E 107, 064308 (2023), https://doi.org/10.1103/PhysRevE.107.064308

Fundamental limits in structured principal component analysis and how to reach them

Jean Barbier, Francesco Camilli, Marco Mondelli, Manuel Sáenz, PNAS 120 (30) e2302028120 (2023) https://doi.org/10.1073/pnas.230202812

An inference problem in a mismatched setting: a spin-glass model with Mattis interaction

Francesco Camilli, Pierluigi Contucci, Emanuele Mingione, SciPost Phys. 12, 125 (2022), doi: 10.21468/SciPostPhys.12.4.125

Enough with i.i.d.!

Check out our interviews on "Fundamental limits in structured principal component analysis and how to reach them."

"Exploiting the Structure of Noise in Big Data", by Charlotte Philips

The Multi-species Mean-Field Spin-Glass on the Nishimori Line

Alberici, D., Camilli, F., Contucci, P. et al., J Stat Phys 182, 2 (2021). https://doi.org/10.1007/s10955-020-02684-z

The Solution of the Deep Boltzmann Machine on the Nishimori Line

Alberici, D., Camilli, F., Contucci, P. et al. , Commun. Math. Phys. 387, 1191–1214 (2021). https://doi.org/10.1007/s00220-021-04165-0

A Statistical Physics approach to a multi-channel Wigner spiked model

Alberici, D., Camilli, F., Contucci, P. et al., Europhysics Letters 134 4 , 2022, DOI:10.1209/0295-5075/ac4794