Behind the Noise: Conformal Quantile Regression Reveals Emergent Representations

TL;DR

The paper tackles the challenge of high-noise, time-intensive scientific imaging by introducing a lightweight ensemble of randomly wired SMSNets trained with conformal quantile regression to provide denoising with calibrated uncertainty. A shared latent space across ensemble members yields emergent, semantically meaningful representations without supervision, enabling unsupervised segmentation-like insights. The method is validated on real SEM-EDX and XCT soil imaging data, showing reliable denoising, uncertainty quantification, and actionable latent structure that guides experimental design under resource constraints. This approach offers a data-efficient path to faster imaging and automated prioritization of subsequent measurements while paving the way for self-supervised foundational representations in geobiochemical imaging.

Abstract

Scientific imaging often involves long acquisition times to obtain high-quality data, especially when probing complex, heterogeneous systems. However, reducing acquisition time to increase throughput inevitably introduces significant noise into the measurements. We present a machine learning approach that not only denoises low-quality measurements with calibrated uncertainty bounds, but also reveals emergent structure in the latent space. By using ensembles of lightweight, randomly structured neural networks trained via conformal quantile regression, our method performs reliable denoising while uncovering interpretable spatial and chemical features -- without requiring labels or segmentation. Unlike conventional approaches focused solely on image restoration, our framework leverages the denoising process itself to drive the emergence of meaningful representations. We validate the approach on real-world geobiochemical imaging data, showing how it supports confident interpretation and guides experimental design under resource constraints.

Figures (12)

• Figure 1: An instance of a random sparse mixed-scale network (SMSNet) with tethered quantile projection heads. Arrows represent convolutional kernels and activation functions, nodes are feature channel concatenators. The the bottom (black) skip connections connect the input and hidden layers directly to the output. The top (colored) connections between nodes are formed vi a stochastic process.
• Figure 2: Panels (A), (B), and (C) display the effect of ensemble size on model performance, focusing on three key metrics: correlation coefficient, quantile width reduction, and coverage. Panel (A) shows the median correlation coefficient as a function of ensemble size, with shaded regions representing the 25%-75% and 10%-90% quantiles. Panel (B) illustrates the reduction in quantile width, calibrated via conformal prediction, with similar quantile ranges shown. Panel (C) presents the coverage statistics for the 90% quantile, again as a function of ensemble size. In each panel, the results for the SMSNet ensemble (blue) are compared with those for the UNET ensemble (orange), demonstrating the improved performance of the SMSNet ensemble across all ensemble sizes. The data in all panels highlight the benefits of using larger ensembles for improved accuracy and robustness in model predictions.
• Figure 3: Denoising results for SEM-EDX elemental maps of a soil core. Top: Low-exposure (7 s) input images for Si, Al, and Fe. Bottom: Predicted median outputs. Denoising reveals fine structure and material boundaries otherwise lost in noise. The displayed patch is 2.5 x 2.5 mm, an area over 50 times larger then the training data.
• Figure 4: Emergent segmentation from latent space analysis. (A) XCT slice and (B) 3D rendering with clustered latent tokens.
• Figure S1: (A) Visual representation of network architectures under different hyperparameter settings $\alpha$ and $\gamma$. The networks consist of an input (0), output (11) and 10 intermediate nodes (1-10) with various dilated convolutions represented by colored arrows. Black dashed lines represent skip connections between the input and hidden nodes, and hidden and output nodes. As $\alpha$ increases from 0.0 to 1.5, connections become increasingly localized between neighboring nodes. Increasing $\gamma$ to 1.5 reduces the overall connection density. Each network feeds into a latent space that produce a median predictions and quantile offset values. B. Parallel coordinate plot showing relationships between hyperparameters $\alpha$, $\gamma$, depth, emergent network characteristics (parameter count, longest path, average degree), and performance (correlation). Colors represent performance quantiles, with warmer colors (orange/red) indicating superior performance. Networks with moderate complexity achieve optimal results. C. Histograms showing the distribution of network properties across performance bands. While there is no clear optimal performance setting, low complexity networks systematically under-perform as compared to larger networks.