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Debiased LASSO under Poisson-Gauss Model

Pedro Abdalla, Gil Kur

TL;DR

The paper addresses uncertainty quantification for high-dimensional sparse regression under a Poisson-Gauss noise model. It extends the debiased LASSO to a Poisson inverse problem, deriving asymptotically Gaussian debiased estimators and establishing near-optimal sample complexity with $n$ large relative to $s log^2 p$. It also develops practical procedures for unknown $|x^*|_1$ and $\sigma$, via robust estimation and a median-of-means design, and proves consistency and asymptotic normality results that support honest confidence intervals. Numerical experiments on synthetic data corroborate the theoretical guarantees, demonstrating accurate coverage and reliable estimation of nuisance parameters. The approach enables principled uncertainty quantification in Poisson-type inverse problems relevant to imaging and related applications.

Abstract

Quantifying uncertainty in high-dimensional sparse linear regression is a fundamental task in statistics that arises in various applications. One of the most successful methods for quantifying uncertainty is the debiased LASSO, which has a solid theoretical foundation but is restricted to settings where the noise is purely additive. Motivated by real-world applications, we study the so-called Poisson inverse problem with additive Gaussian noise and propose a debiased LASSO algorithm that only requires $n \gg s\log^2p$ samples, which is optimal up to a logarithmic factor.

Debiased LASSO under Poisson-Gauss Model

TL;DR

The paper addresses uncertainty quantification for high-dimensional sparse regression under a Poisson-Gauss noise model. It extends the debiased LASSO to a Poisson inverse problem, deriving asymptotically Gaussian debiased estimators and establishing near-optimal sample complexity with large relative to . It also develops practical procedures for unknown and , via robust estimation and a median-of-means design, and proves consistency and asymptotic normality results that support honest confidence intervals. Numerical experiments on synthetic data corroborate the theoretical guarantees, demonstrating accurate coverage and reliable estimation of nuisance parameters. The approach enables principled uncertainty quantification in Poisson-type inverse problems relevant to imaging and related applications.

Abstract

Quantifying uncertainty in high-dimensional sparse linear regression is a fundamental task in statistics that arises in various applications. One of the most successful methods for quantifying uncertainty is the debiased LASSO, which has a solid theoretical foundation but is restricted to settings where the noise is purely additive. Motivated by real-world applications, we study the so-called Poisson inverse problem with additive Gaussian noise and propose a debiased LASSO algorithm that only requires samples, which is optimal up to a logarithmic factor.
Paper Structure (9 sections, 12 theorems, 132 equations, 1 figure, 2 algorithms)

This paper contains 9 sections, 12 theorems, 132 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Uncertainty Quantification under Poisson Noise
  3. The case when $\|x^{\ast}\|_1$ is known
  4. The case when $\|x^{\ast}\|_1$ is unknown
  5. Consistency in the Poisson-Gauss Model
  6. Consistency of the LASSO
  7. Consistency of the Scaled LASSO
  8. The Debiased Algorithm under the Poisson-Gauss noise
  9. Numerical Experiments

Key Result

Theorem 1

Consider the model of eq:main_regression_model above with a design matrix $A \sim Bern(q)$ for some absolute constant $q\in (0,1/2)$, and assume that there is a sufficiently large constant $C > 0$ for which $\|x^{\ast}\|_{1}\le Cn/\log p$. Then, in the regime of $n,p\rightarrow \infty$ and $s\log^2 where $\sigma(q) = (1/\sqrt{q(1-q)})\sqrt{1+(\sigma^2\Tilde{A}^T\Tilde{A}/(n\|x^{\ast}\|_1))}$. Fur

Figures (1)

  • Figure 1: Confidence intervals for the $100$-largest coefficients arranged in the non-increasing order.

Theorems & Definitions (13)

  • Theorem 1: Main Result
  • Theorem 2
  • Proposition 3
  • Lemma 1
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Definition 1: Restricted Eigenvalue Condition
  • Proposition 7
  • Proposition 8
  • ...and 3 more