Average Kernel Sizes -- Computable Sharp Accuracy Bounds for Inverse Problems

TL;DR

This work introduces computable, sharp accuracy bounds for ill-posed inverse problems that are independent of the chosen reconstruction method. Central to the framework is the average kernel size, which quantifies the non-uniqueness of forward maps and yields a lower bound (half the average kernel size) on the reconstruction error for any approximate inverse map, with a complementary upper bound for optimal, empirical-limit maps. To compute these bounds, the authors develop two algorithms—one to approximate feasible error sets and one to aggregate them into an average kernel size—and provide a software library for practical use. The framework is validated on real-world-like inverse problems in fluorescence localization microscopy and multispectral satellite super-resolution, illustrating that the bounds hold before model development and can guide dataset and forward-model design. Overall, this approach enables principled, data-driven assessment of achievable accuracy, informing experimental design, early stopping, and evaluation in complex inverse problems.

Abstract

The reconstruction of an unknown quantity from noisy measurements is a mathematical problem relevant in most applied sciences, for example, in medical imaging, radar inverse scattering, or astronomy. This underlying mathematical problem is often an ill-posed (non-linear) reconstruction problem, referred to as an ill-posed inverse problem. To tackle such problems, there exist a myriad of methods to design approximate inverse maps, ranging from optimization-based approaches, such as compressed sensing, over Bayesian approaches, to data-driven techniques such as deep learning. For all stable approximate inverse maps, there are accuracy limits that are strictly larger than zero for ill-posed inverse problems, due to the accuracy-stability tradeoff [Gottschling et al., SIAM Review, 67.1 (2025)] and [Colbrook et al., Proceedings of the National Academy of Sciences, 119.12 (2022)]. The variety of methods that aim to solve such problems begs for a unifying approach to help scientists choose the approximate inverse map that obtains this theoretical optimum. Up to now there do not exist computable accuracy bounds to this optimum that are applicable to all inverse problems. We provide computable sharp accuracy bounds to the reconstruction error of solution methods to inverse problems. The bounds are method-independent and purely depend on the dataset of signals, the forward model of the inverse problem, and the noise model. To facilitate the use in scientific applications, we provide an algorithmic framework and an accompanying software library to compute these accuracy bounds. We demonstrate the validity of the algorithms on two inverse problems from different domains: fluorescence localization microscopy and super-resolution of multi-spectral satellite data. Computing the accuracy bounds for a problem before solving it, enables a fundamental shift towards optimizing datasets and forward models.

Figures (5)

• Figure 1: Inverse problem framework.
• Figure 2: Allocation of $x$-components of samples into feasible sets with Algorithm \ref{['alg:feasset']}. Note that the noise components can enlarge the feasible sets. For example, this occurs for additive and multiplicative noise models.
• Figure 2: RMSE for different methods on the two datasets and the corresponding lower accuracy bound - half of the average symmetric kernel size.
• Figure 3: RMSE of the $(x,y)-$location from different estimators against half of the average kernel size computed with feasible sets of size $N(1)= 15001$ with $K=1$ for $25$ different noisy measurements per microscope set-up $\{A_1,A_2,A_3,A_4\}$. The lower and upper gray lines ($y=x$ and $y=2x$) visualize the lower and upper accuracy bounds. All RMSE values are above the computed lower accuracy bounds and validate Theorem \ref{['thm:lowerboundapprx']}, $(i)$. The mean and median estimator's RMSE are below the upper accuracy bound. The different microscope set-ups $\{A_1,A_2,A_3,A_4\}$ highlight the increase of the kernel size with worsening imaging settings - a higher background photon flux and decreasing particle photon emission rate.
• Figure 4: RGB example visualizations of different versions of a multi-spectral satellite Croplands image from the NAIP data set. The images correspond to (a) the original high-resolution image, (b) a high-resolution version reflected around the orthogonal component of the downsampling maps kernel (according to \ref{['eq:reflected_null']}), and (c) the low-resolution image computed with the down sampling operator.

Theorems & Definitions (7)

• Remark 3.1: Does the Distribution of Noise Matter?
• Definition 3.2: Average Kernel Size
• Theorem 3.3: Accuracy Bounds for Computing Solutions to Inverse Problems
• Remark 3.4: Application to Finite Datasets $\mathcal{M}_1$
• Theorem 3.5: Fast Lower Bound for Functions on the Measurement Data
• proof : Proof of Theorem \ref{['thm:lowerboundapprx']}
• proof