Related papers: Goldilocks and the bootstrap
We develop new tools in the theory of nonlinear random matrices and apply them to study the performance of the Sum of Squares (SoS) hierarchy on average-case problems. The SoS hierarchy is a powerful optimization technique that has achieved…
We present a distributed asynchronous algorithm for approximating a single component of the solution to a system of linear equations $Ax = b$, where $A$ is a positive definite real matrix, and $b \in \mathbb{R}^n$. This is equivalent to…
We propose an interior point method (IPM) for solving semidefinite programming problems (SDPs). The standard interior point algorithms used to solve SDPs work in the space of positive semidefinite matrices. Contrary to that the proposed…
We propose a distributed bootstrap method for simultaneous inference on high-dimensional massive data that are stored and processed with many machines. The method produces an $\ell_\infty$-norm confidence region based on a…
We study the onset of the bootstrap percolation transition as a model of generalized dynamical arrest. We develop a new importance-sampling procedure in simulation, based on rare events around "holes", that enables us to access bootstrap…
Maximum A posteriori Probability (MAP) inference in graphical models amounts to solving a graph-structured combinatorial optimization problem. Popular inference algorithms such as belief propagation (BP) and generalized belief propagation…
We introduce a method based on semidefinite programming that produces rigorous two-sided bounds on ground state energy densities and correlation functions of translation-invariant classical spin models on infinite lattices. In this method,…
We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…
We study the singularity probability of n*n random matrices with i.i.d. entries from highly biased discrete distributions. We obtain sharp non-asymptotic bounds for this probability and derive estimates on the least singular values. Our…
We explore a new type of sparsity for the generalized moment problem (GMP) that we call ideal-sparsity. This sparsity exploits the presence of equality constraints requiring the measure to be supported on the variety of an ideal generated…
Closed-loop optimal control design for high-dimensional nonlinear systems has been a long-standing challenge. Traditional methods, such as solving the associated Hamilton-Jacobi-Bellman equation, suffer from the curse of dimensionality.…
The recent development of bootstrap methods based on semidefinite relaxations of positivity constraints has enabled rigorous two-sided bounds on local observables directly in the thermodynamic limit. However, these bounds inevitably become…
We describe an adaptive importance sampling algorithm for rare events that is based on a dual stochastic control formulation of a path sampling problem. Specifically, we focus on path functionals that have the form of cumulate generating…
In this paper we study the adaptivity of submodular maximization. Adaptivity quantifies the number of sequential rounds that an algorithm makes when function evaluations can be executed in parallel. Adaptivity is a fundamental concept that…
Matroids are a fundamental object of study in combinatorial optimization. Three closely related and important problems involving matroids are maximizing the size of the union of $k$ independent sets (that is, $k$-fold matroid union),…
The soft anomalous dimension governs the infrared singularities of scattering amplitudes to all orders in perturbative quantum field theory, and is a crucial ingredient in both formal and phenomenological applications of non-abelian gauge…
The paper studies a problem of constructing simultaneous likelihood-based confidence sets. We consider a simultaneous multiplier bootstrap procedure for estimating the quantiles of the joint distribution of the likelihood ratio statistics,…
We generalize and improve results of Andrews, Gravner, Holroyd, Liggett, and Romik on metastability thresholds for generalized two-dimensional bootstrap percolation models, and answer several of their open problems and conjectures.…
This paper studies the problem of selecting a submatrix of a positive definite matrix in order to achieve a desired bound on the smallest eigenvalue of the submatrix. Maximizing this smallest eigenvalue has applications to selecting input…
We first describe a general class of optimization problems that describe many natural, economic, and statistical phenomena. After noting the existence of a conserved quantity in a transformed coordinate system, we outline several instances…