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.
no code implementations • 19 May 2022 • Afonso S. Bandeira, Ahmed El Alaoui, Samuel B. Hopkins, Tselil Schramm, Alexander S. Wein, Ilias Zadik
We define a free-energy based criterion for hardness and formally connect it to the well-established notion of low-degree hardness for a broad class of statistical problems, namely all Gaussian additive models and certain models with a sparse planted signal.
no code implementations • 2 Feb 2021 • Pedro Abdalla, Afonso S. Bandeira
Semidefinite programming is an important tool to tackle several problems in data science and signal processing, including clustering and community detection.
no code implementations • 22 May 2020 • Yunzi Ding, Dmitriy Kunisky, Alexander S. Wein, Afonso S. Bandeira
A matrix has the $(s,\delta)$-$\mathsf{RIP}$ property if behaves as a $\delta$-approximate isometry on $s$-sparse vectors.
no code implementations • 21 May 2020 • Matthias Löffler, Alexander S. Wein, Afonso S. Bandeira
We study statistical and computational limits of clustering when the means of the centres are sparse and their dimension is possibly much larger than the sample size.
no code implementations • 15 Aug 2019 • Weichi Yao, Afonso S. Bandeira, Soledad Villar
In particular we consider Graph Neural Networks (GNNs) -- a class of neural networks designed to learn functions on graphs -- and we apply them to the max-cut problem on random regular graphs.
no code implementations • 26 Jul 2019 • Dmitriy Kunisky, Alexander S. Wein, Afonso S. Bandeira
These notes survey and explore an emerging method, which we call the low-degree method, for predicting and understanding statistical-versus-computational tradeoffs in high-dimensional inference problems.
no code implementations • 26 Jul 2019 • Yunzi Ding, Dmitriy Kunisky, Alexander S. Wein, Afonso S. Bandeira
Prior work has shown that when the signal-to-noise ratio ($\lambda$ or $\beta\sqrt{N/n}$, respectively) is a small constant and the fraction of nonzero entries in the planted vector is $\|x\|_0 / n = \rho$, it is possible to recover $x$ in polynomial time if $\rho \lesssim 1/\sqrt{n}$.
no code implementations • 8 Jul 2018 • Chiheon Kim, Afonso S. Bandeira, Michel X. Goemans
We study the problem of community detection in a random hypergraph model which we call the stochastic block model for $k$-uniform hypergraphs ($k$-SBM).
no code implementations • 2 Jul 2018 • Amelia Perry, Alexander S. Wein, Afonso S. Bandeira, Ankur Moitra
Our results leverage Le Cam's notion of contiguity, and include: i) For the Gaussian Wigner ensemble, we show that PCA achieves the optimal detection threshold for certain natural priors for the spike.
no code implementations • 29 Mar 2018 • Afonso S. Bandeira, Amelia Perry, Alexander S. Wein
In these notes we describe heuristics to predict computational-to-statistical gaps in certain statistical problems.
no code implementations • 18 Feb 2018 • Luca Venturi, Afonso S. Bandeira, Joan Bruna
Focusing on a class of two-layer neural networks defined by smooth (but generally non-linear) activation functions, we identify a notion of intrinsic dimension and show that it provides necessary and sufficient conditions for the absence of spurious valleys.
3 code implementations • 22 Jun 2017 • Alex Nowak, Soledad Villar, Afonso S. Bandeira, Joan Bruna
Inverse problems correspond to a certain type of optimization problems formulated over appropriate input distributions.
no code implementations • 22 Dec 2016 • Amelia Perry, Alexander S. Wein, Afonso S. Bandeira
Finally, for priors (i) and (ii) we carry out the replica prediction from statistical physics, which is conjectured to give the exact information-theoretic threshold for any fixed $d$.
no code implementations • 17 Oct 2016 • Soledad Villar, Afonso S. Bandeira, Andrew J. Blumberg, Rachel Ward
The Gromov-Hausdorff distance provides a metric on the set of isometry classes of compact metric spaces.
no code implementations • 14 Oct 2016 • Amelia Perry, Alexander S. Wein, Afonso S. Bandeira, Ankur Moitra
Various alignment problems arising in cryo-electron microscopy, community detection, time synchronization, computer vision, and other fields fall into a common framework of synchronization problems over compact groups such as Z/L, U(1), or SO(3).
no code implementations • 19 Sep 2016 • Amelia Perry, Alexander S. Wein, Afonso S. Bandeira, Ankur Moitra
Our results include: I) For the Gaussian Wigner ensemble, we show that PCA achieves the optimal detection threshold for a variety of benign priors for the spike.
1 code implementation • NeurIPS 2016 • Nicolas Boumal, Vladislav Voroninski, Afonso S. Bandeira
Semidefinite programs (SDPs) can be solved in polynomial time by interior point methods, but scalability can be an issue.
Optimization and Control Numerical Analysis
no code implementations • 8 Jul 2015 • Naman Agarwal, Afonso S. Bandeira, Konstantinos Koiliaris, Alexandra Kolla
We consider the problem of identifying underlying community-like structures in graphs.
no code implementations • 14 May 2015 • Afonso S. Bandeira, Yutong Chen, Amit Singer
Let $\mathcal{G}$ be a compact group and let $f_{ij} \in L^2(\mathcal{G})$.
no code implementations • 18 Aug 2014 • Pranjal Awasthi, Afonso S. Bandeira, Moses Charikar, Ravishankar Krishnaswamy, Soledad Villar, Rachel Ward
Under the same distributional model, the $k$-means LP relaxation fails to recover such clusters at separation as large as $\Delta = 4$.
no code implementations • 11 Apr 2014 • Afonso S. Bandeira, Dustin G. Mixon, Benjamin Recht
This paper addresses the fundamental question of when convex sets remain disjoint after random projection.
no code implementations • 10 Apr 2014 • Afonso S. Bandeira, Yuehaw Khoo, Amit Singer
We have observed an interesting, yet unexplained, phenomenon: Semidefinite programming (SDP) based relaxations of maximum likelihood estimators (MLE) tend to be tight in recovery problems with noisy data, even when MLE cannot exactly recover the ground truth.