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Found 25 papers, 3 papers with code
Fitting an ellipsoid to a quadratic number of random points
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 $n, d \to \infty$.
Injectivity of ReLU networks: perspectives from statistical physics
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.
On free energy barriers in Gaussian priors and failure of cold start MCMC for high-dimensional unimodal distributions
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.
The Franz-Parisi Criterion and Computational Trade-offs in High Dimensional Statistics
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.
Community Detection with a Subsampled Semidefinite Program
Semidefinite programming is an important tool to tackle several problems in data science and signal processing, including clustering and community detection.
The Average-Case Time Complexity of Certifying the Restricted Isometry Property
A matrix has the $(s,\delta)$-$\mathsf{RIP}$ property if behaves as a $\delta$-approximate isometry on $s$-sparse vectors.
Computationally efficient sparse clustering
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.
Experimental performance of graph neural networks on random instances of max-cut
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.
Notes on Computational Hardness of Hypothesis Testing: Predictions using the Low-Degree Likelihood Ratio
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.
Subexponential-Time Algorithms for Sparse PCA
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 $\rho \lesssim 1/\sqrt{n}$.
Stochastic Block Model for Hypergraphs: Statistical limits and a semidefinite programming approach
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).
Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models
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.
Notes on computational-to-statistical gaps: predictions using statistical physics
In these notes we describe heuristics to predict computational-to-statistical gaps in certain statistical problems.
Spurious Valleys in Two-layer Neural Network Optimization Landscapes
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.
Revised Note on Learning Algorithms for Quadratic Assignment with Graph Neural Networks
Inverse problems correspond to a certain type of optimization problems formulated over appropriate input distributions.
Statistical limits of spiked tensor models
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 $d$.
A polynomial-time relaxation of the Gromov-Hausdorff distance
The Gromov-Hausdorff distance provides a metric on the set of isometry classes of compact metric spaces.
Message-passing algorithms for synchronization problems over compact groups
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).
Optimality and Sub-optimality of PCA for Spiked Random Matrices and Synchronization
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.
The non-convex Burer-Monteiro approach works on smooth semidefinite programs
Semidefinite programs (SDPs) can be solved in polynomial time by interior point methods, but scalability can be an issue.
Optimization and Control Numerical Analysis
Multisection in the Stochastic Block Model using Semidefinite Programming
We consider the problem of identifying underlying community-like structures in graphs.
Non-unique games over compact groups and orientation estimation in cryo-EM
Let $\mathcal{G}$ be a compact group and let $f_{ij} \in L^2(\mathcal{G})$.
Relax, no need to round: integrality of clustering formulations
Under the same distributional model, the $k$-means LP relaxation fails to recover such clusters at separation as large as $\Delta = 4$.
Compressive classification and the rare eclipse problem
This paper addresses the fundamental question of when convex sets remain disjoint after random projection.
Open problem: Tightness of maximum likelihood semidefinite relaxations
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.
