Instance-Wise Adaptive Sampling for Dataset Construction in Approximating Inverse Problem Solutions

TL;DR

This work proposes an instance-wise adaptive sampling framework for constructing compact and informative training datasets for supervised learning of inverse problem solutions and demonstrates the effectiveness of the approach in the inverse scattering problem under two types of structured priors.

Abstract

We propose an instance-wise adaptive sampling framework for constructing compact and informative training datasets for supervised learning of inverse problem solutions. Typical learning-based approaches aim to learn a general-purpose inverse map from datasets drawn from a prior distribution, with the training process independent of the specific test instance. When the prior has a high intrinsic dimension or when high accuracy of the learned solution is required, a large number of training samples may be needed, resulting in substantial data collection costs. In contrast, our method dynamically allocates sampling effort based on the specific test instance, enabling significant gains in sample efficiency. By iteratively refining the training dataset conditioned on the latest prediction, the proposed strategy tailors the dataset to the geometry of the inverse map around each test instance. We demonstrate the effectiveness of our approach in the inverse scattering problem under two types of structured priors. Our results show that the advantage of the adaptive method becomes more pronounced in settings with more complex priors or higher accuracy requirements. While our experiments focus on a particular inverse problem, the adaptive sampling strategy is broadly applicable and readily extends to other inverse problems, offering a scalable and practical alternative to conventional fixed-dataset training regimes.

Figures (6)

• Figure 1: Schematic of the instance-wise adaptive sampling method. The upper-left portion of the diagram in the dashed box depicts the typical machine learning approach to inverse problems, resulting in a base model for the inverse operator and its prediction of the unknown parameter corresponding to a given measurement instance. In the adaptive sampling method, the base model and its prediction are iteratively refined, as depicted in the upper-right portion of the diagram. It is important to note that these iterative refinements are specifically tailored to the given measurement instance. The bottom of the figure shows the progression of the method in the parameter space.
• Figure 2: A schematic of the inverse scattering problem. Left: Illustration of the experimental setup, in which an incident wave scatters off the medium and is detected at $N_t$ receivers. A total of $N_d$ incident waves, sent from different directions, are used to obtain the full measurement. Right: The resulting measurement matrix $m\in \mathbb C^{N_d\times N_t}$. For visualization purposes, only the real part of $m$ is displayed.
• Figure 3: Visualization of field progression for a test case under the $N_\text{disk} \in [4,6]$ disk prior setting. Top: Ground truth field. Middle: Predicted fields from the base model and subsequent refinement rounds. Bottom: Projections of the predicted fields onto the disk prior manifold. The projections from the base model and round 2 include extra disks (highlighted by red dashed squares) that are not present in the true field. Nevertheless, these errors are progressively corrected in later rounds.
• Figure 4: Data efficiency comparison between adaptive and non-adaptive training in the disk prior setting. Left: Average relative error $\varepsilon_\text{rel}$ versus total training dataset size. Dashed lines show a log-linear fit for the non-adaptive method across varying dataset sizes. Right: Data efficiency factor of the adaptive method, defined as the ratio of non-adaptive to adaptive dataset sizes required to reach the same error level, plotted as a function of target accuracy $1 - \varepsilon_\text{rel}$.
• Figure 5: Visualization of field progression for a test case under the $N_F=3$ Fourier prior setting. Top: Ground truth field. Bottom: Predicted fields from the base model and subsequent refinement rounds.