Simultaneous Frequentist Calibration of Confidence Regions for Multiple Functionals in Constrained Inverse Problems

TL;DR

This work develops a unified test-inversion framework for constructing finite-sample, simultaneous confidence regions for multiple functionals in constrained linear inverse problems. It extends classical unconstrained and nonnegative inference to the multi-functional setting, introducing three test statistics, λ^2_c, λ^2_u, and λ^1, and deriving calibration constants via quantile optimization to guarantee 1−α coverage. Practical high-dimensional methods are provided, including generalized TFM reductions and row–null splitting, enabling tractable regions that are smaller than traditional SSB bounds while maintaining uniform coverage under Gaussian or general log-concave noise. Theoretical results on convexity, recession monotonicity, and chi-bar-squared-like behavior at the origin underpin the calibrations, with numerical experiments showing meaningful gains in region volume and interpretability for complex, constrained inverse problems.

Abstract

Many scientific analyses require simultaneous comparison of multiple functionals of an unknown signal at once, calling for multidimensional confidence regions with guaranteed simultaneous frequentist under structural constraints (e.g., non-negativity, shape, or physics-based). This paper unifies and extends many previous optimization-based approaches to constrained confidence region construction in linear inverse problems through the lens of statistical test inversion. We begin by reviewing the historical development of optimization-based confidence intervals for the single-functional setting, from "strict bounds" to the Burrus conjecture and its recent refutation via the aforementioned test inversion framework. We then extend this framework to the multiple-functional setting. This framework can be used to: (i) improve the calibration constants of previous methods, yielding smaller confidence regions that still preserve frequentist coverage, (ii) obtain tractable multidimensional confidence regions that need not be hyper-rectangles to better capture functional dependence structure, and (iii) generalize beyond Gaussian error distributions to generic log-concave error distributions. We provide theory establishing nominal simultaneous coverage of our methods and show quantitative volume improvements relative to prior approaches using numerical experiments.

Figures (4)

• Figure 1.1: Illustration of the problem setup. We seek to construct confidence region $\mathcal{R}_{\alpha}(y) \subseteq \mathbb{R}^k$ for $H x^* \in \mathbb{R}^k$ from an observation ${\bm{y}} \in \mathbb{R}^n$ sampled from $P_{{\bm{x}}^*}$ that satisfies a frequentist coverage guarantee in finite sample while being as small (in volume) as possible.
• Figure 5.1: Comparison between the fixed chi-square quantile $Q_{\chi^2_n,\,1-\alpha}$ (dashed line) and the distribution $Q_{\chi^2_r,\,1-\alpha}+\xi_{n,r}$ (boxplots), for $n=100$ and $r\in\{25,50,75,99\}$. The top panel corresponds to $1-\alpha=0.05$, and the bottom panel to $1-\alpha=0.25$.
• Figure 7.1: Confidence regions of at level $1-\alpha = 68\%$ for $Hx^*$ in the problem setup \ref{['eq:toy_problem_setup']}, comparing the different methods in \ref{['subsec:allthemethods']} when $y = (0,0)$ (left) and $y = (20,10)$ (right), which are the noiseless observations for $x^* = (0,0,0)$ and $x^* = (5,5,5)$, respectively.
• Figure 7.2: Coverage probabilities and area distributions of the constructed confidence sets for two scenarios: (top) $x^\star = (0,0,0)$, and (bottom) $x^\star = (5,5,5)$. Each panel shows the empirical coverage (left) and the distribution of the areas (right) for different methods, with $N = 10^5$ samples of $y \sim \mathcal{N}(Kx^*, I)$. In each plot, the diamond shows the empirical average.

Theorems & Definitions (23)

• Proposition 2.1: Equivalence between $x$-description and $\mu$-description
• proof
• Proposition 2.2: Exact null laws for both test statistics and orthogonal decomposition
• proof
• Remark 1
• Lemma 2.3: Noiseless versus noisy boundedness equivalence
• proof
• Theorem 2.4: Boundedness characterization
• proof
• Theorem 2.5: Coordinate-wise boundedness via the recession cone