Uncertainty quantification in high dimension: Sampling and noisy debiasing
Abstract
Reliably drawing inferences from noisy data requires to quantify the degree of certainty (or uncertainty) associated to a given clai m. Classical statistical theory provides a well established methodology to quantify uncertainty when the unknown object to be estima ted is low-dimensional, i.e. consists of a vector with a few entries. Within a frequentist framework, uncertainty quantification is achieved via the computation of p-values and confidence regions. Within a Bayesian approach, the posterior distribution encodes our uncertainty about the parameters of interest.Quantifying uncertainty in modern applications (in particular, in image and signal proc essing) requires to be able to carry out the same tasks in high dimension, i.e. when the dimension of the object to be estimated is comparable, or even larger than the number of measurements. This project aims at making fundamental progress on uncertainty quantifi cation, in both the directions mentioned above:1. We propose a new class ofalgorithms for sampling Bayesian posteriors. We describe the derivation of these algorithms using high-dimensional linear regression as a case study. To the best of our knowledge, these are the first sampling algorithms for which we can prove efficient sampling in a constant signal-to-noise ratio (SNR) regime.2. We prop ose a new method to compute confidence intervals in high-dimensional linear regression. Previous approaches assumed either that the correlation structure of the covariates is known exactly, or that it can be estimated accurately from data, possibly exploiting addi tional sparsity properties. We propose instead the first method that accounts for uncertainty in the correlation structure of the co variates.In both thrusts, our objective is to perform tasks that cannot be achieved altogether by current methods.Approved for Publi c Release
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Aug 20, 2021
- Source ID
- N000142112837
Entities
People
- Andrea Montanari
Organizations
- Office of Naval Research
- Stanford University
- United States Navy