no code implementations • 3 Jul 2023 • Afonso S. Bandeira, Antoine Maillard, Shahar Mendelson, Elliot Paquette
We consider the problem $(\mathrm{P})$ of fitting $n$ standard Gaussian random vectors in $\mathbb{R}^d$ to the boundary of a centered ellipsoid, as $n, d \to \infty$.
1 code implementation • 27 Feb 2023 • Antoine Maillard, Afonso S. Bandeira, David Belius, Ivan Dokmanić, Shuta Nakajima
Recent work connects this problem to spherical integral geometry giving rise to a conjectured sharp injectivity threshold for $\alpha = \frac{m}{n}$ by studying the expected Euler characteristic of a certain random set.
no code implementations • 5 Sep 2022 • Afonso S. Bandeira, Antoine Maillard, Richard Nickl, Sven Wang
We exhibit examples of high-dimensional unimodal posterior distributions arising in non-linear regression models with Gaussian process priors for which MCMC methods can take an exponential run-time to enter the regions where the bulk of the posterior measure concentrates.
1 code implementation • 8 Dec 2020 • Antoine Maillard, Florent Krzakala, Yue M. Lu, Lenka Zdeborová
We consider the phase retrieval problem, in which the observer wishes to recover a $n$-dimensional real or complex signal $\mathbf{X}^\star$ from the (possibly noisy) observation of $|\mathbf{\Phi} \mathbf{X}^\star|$, in which $\mathbf{\Phi}$ is a matrix of size $m \times n$.
Information Theory Disordered Systems and Neural Networks Information Theory
1 code implementation • NeurIPS 2020 • Antoine Maillard, Bruno Loureiro, Florent Krzakala, Lenka Zdeborová
We consider the phase retrieval problem of reconstructing a $n$-dimensional real or complex signal $\mathbf{X}^{\star}$ from $m$ (possibly noisy) observations $Y_\mu = | \sum_{i=1}^n \Phi_{\mu i} X^{\star}_i/\sqrt{n}|$, for a large class of correlated real and complex random sensing matrices $\mathbf{\Phi}$, in a high-dimensional setting where $m, n\to\infty$ while $\alpha = m/n=\Theta(1)$.
no code implementations • 4 Dec 2019 • Antoine Maillard, Gérard Ben Arous, Giulio Biroli
Under a technical hypothesis, we obtain a rigorous explicit variational formula for the annealed complexity, which is the logarithm of the average number of critical points at fixed value of the empirical risk.
2 code implementations • NeurIPS 2019 • Benjamin Aubin, Bruno Loureiro, Antoine Maillard, Florent Krzakala, Lenka Zdeborová
Here, we replace the sparsity assumption by generative modelling, and investigate the consequences on statistical and algorithmic properties.
1 code implementation • NeurIPS 2018 • Benjamin Aubin, Antoine Maillard, Jean Barbier, Florent Krzakala, Nicolas Macris, Lenka Zdeborová
Heuristic tools from statistical physics have been used in the past to locate the phase transitions and compute the optimal learning and generalization errors in the teacher-student scenario in multi-layer neural networks.