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Gaussian Mean Testing under Truncation

Clément L. Canonne, Themis Gouleakis, Yuhao Wang, Joy Qiping Yang

TL;DR

The paper investigates Gaussian mean testing under truncation, formalizing the problem with a truncated $d$-dimensional Gaussian and a truncation set $S$ of mass $\varepsilon$. It reveals a phase transition in sample complexity for unknown truncation: when $\varepsilon\sqrt{\log(1/\varepsilon)}\lesssim\alpha$ standard testers achieve $n=\Theta(\sqrt{d})$, while in the regime $\varepsilon\lesssim\alpha\lesssim\varepsilon\sqrt{\log(1/\varepsilon)}$ the task requires $n=\Theta(d)$, i.e., as hard as learning; if $\alpha\lesssim\varepsilon$, testing is impossible. In contrast, when truncation is known, a gradient-based maximum likelihood tester attains the optimal $n=O(\sqrt{d})$ across all regimes. The results illuminate how truncation geometry and prior information fundamentally alter testing complexity and bridge connections to learning and robust statistics. The work thus provides precise, regime-dependent guidelines for efficient Gaussian mean testing in the presence of truncation, with implications for truncated data in economics and social sciences.

Abstract

We consider the task of Gaussian mean testing, that is, of testing whether a high-dimensional vector perturbed by white noise has large magnitude, or is the zero vector. This question, originating from the signal processing community, has recently seen a surge of interest from the machine learning and theoretical computer science community, and is by now fairly well understood. What is much less understood, and the focus of our work, is how to perform this task under truncation: that is, when the observations (i.i.d.\ samples from the underlying high-dimensional Gaussian) are only observed when they fall in an given subset of the domain $\R^d$. This truncation model, previously studied in the context of learning (instead of testing) the mean vector, has a range of applications, in particular in Economics and Social Sciences. As our work shows, sample truncations affect the complexity of the testing task in a rather subtle and surprising way.

Gaussian Mean Testing under Truncation

TL;DR

The paper investigates Gaussian mean testing under truncation, formalizing the problem with a truncated -dimensional Gaussian and a truncation set of mass . It reveals a phase transition in sample complexity for unknown truncation: when standard testers achieve , while in the regime the task requires , i.e., as hard as learning; if , testing is impossible. In contrast, when truncation is known, a gradient-based maximum likelihood tester attains the optimal across all regimes. The results illuminate how truncation geometry and prior information fundamentally alter testing complexity and bridge connections to learning and robust statistics. The work thus provides precise, regime-dependent guidelines for efficient Gaussian mean testing in the presence of truncation, with implications for truncated data in economics and social sciences.

Abstract

We consider the task of Gaussian mean testing, that is, of testing whether a high-dimensional vector perturbed by white noise has large magnitude, or is the zero vector. This question, originating from the signal processing community, has recently seen a surge of interest from the machine learning and theoretical computer science community, and is by now fairly well understood. What is much less understood, and the focus of our work, is how to perform this task under truncation: that is, when the observations (i.i.d.\ samples from the underlying high-dimensional Gaussian) are only observed when they fall in an given subset of the domain . This truncation model, previously studied in the context of learning (instead of testing) the mean vector, has a range of applications, in particular in Economics and Social Sciences. As our work shows, sample truncations affect the complexity of the testing task in a rather subtle and surprising way.

Paper Structure

This paper contains 26 sections, 18 theorems, 66 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation.
  3. Our contributions
  4. Related Works
  5. Learning from Truncated or Censored Samples
  6. Testing if samples are truncated
  7. Robust mean estimation
  8. Robust mean testing
  9. Notation and Preliminaries
  10. Notation.
  11. Truncated Gaussian Distribution.
  12. Testing under Unknown Truncation
  13. When Truncation Size is Much Smaller Than Accuracy $\varepsilon\xspace \sqrt{\log 1/ \varepsilon\xspace} \lesssim \alpha$
  14. When Truncation Size is Near Accuracy $\varepsilon\xspace \lesssim \alpha \lesssim \varepsilon\xspace \sqrt{\log 1/ \varepsilon\xspace}$
  15. Mean Testing Lower Bound
  16. ...and 11 more sections

Key Result

Lemma 1

Let $\boldsymbol{H}_{\ell}$ be the Hessian of the negative log likelihood function $\bar{\ell} (\boldsymbol{v})$, with the presence of arbitrary truncation $S$ such that $\mathcal{N} (\boldsymbol{\mu}, \boldsymbol{I}_d , S) \geqslant \beta$ for some $\beta \in (0, 1]$. Then it holds that where $C$ is a universal constant.

Theorems & Definitions (28)

  • Lemma 1: Strong convexity with truncation adapted daskalakis2018efficient
  • Proposition 1: Sample complexity lower bound for high-dimensional testing DBLP:journals/corr/DiakonikolasKS16c
  • Theorem 3.1
  • Lemma 1
  • Lemma 1: Truncated vs non-truncated parameters
  • Theorem 3.2
  • Lemma 1: Lower Bound for Mean Testing with Unknown Truncation When ( ≲ α ≲ √ (1/) )
  • Lemma 1
  • Lemma 1: Gap of Mean under Truncation
  • proof : Proof sketch
  • ...and 18 more