Traceable Spectral Inference via Influence Functions: Efficient Data Attribution and Error Proxies for the Ariel Mission
Authors: Nikki Grens, Luís F. Simões, Kai Hou Yip, Theresa Lueftinger
Organizations: ML Analytics, Lisbon, Portugal · Department of Space Engineering, Delft University of Technology, Delft, The Netherlands · Department of Physics, King’s College London, London, UK · European Space Agency, ESA-ESTEC, Noordwijk, The Netherlands
Interpretability is critical for machine learning models deployed in scientific space missions such as ESA's Ariel, where ground truth is unavailable during operations and physical plausibility must be assessed. While most explainable AI methods focus on feature attribution, this work investigates training data attribution through influence functions and introduces three key contributions for operational spectroscopy pipelines. First, influence is reformulated in terms of prediction rather than loss, enabling label-free deployment. Second, by leveraging the closed-form ridge solution of an Extreme Learning Machine, infinitesimal prediction influence is efficiently computed. Third, an influence-based conservative error proxy is derived by propagating training residuals through the influence sensitivities. Evaluated against simulated spectra, the proposed proxy correlates strongly with scale and shape-based spectral errors. Furthermore, influence functions enable the identification of the most influential samples and the approximation of the most harmful ones. Together, these results suggest that this approach can serve as an operational framework for scientific machine learning.
Figures & tables
Figure 1: t-SNE embedding of training samples, coloured by absolute influence on two specific output channels: 0.55 μ m (Visible, left) and 4.3 μ m (Infrared, right). These channels are selected to evaluate the model’s sensitivity across different physical regimes. The orange star denotes the test point under analysis. Influence values are clipped to enhance visualization, with magenta pluses and red crosses identifying the top 10 most influential and harmful samples, respectively. Notably, six samples overlap between the influential and harmful sets in both cases. Due to high proximity in the representation space, some markers appear superimposed in the figure.
Figure 2: Observation-level prediction error versus the influence-based proxy (log–log scale). Each point corresponds to a single test observation not seen during training. The dataset used consists of simulated data, so ground truth is available, which is used exclusively to compute the realised error. Left: RMSPE across the 55 wavelength channels shows a Spearman rank correlation of 0.69 with the influence-based error metric proxy. Right: SAM across the 55 wavelength channels shows a stronger correlation of 0.81.
Sample Subset ( k=50 )
RMSPE ↓
SAM ↓
Nearest Neighbors
0.1037
0.0309
Top Influential
0.6278
0.0380
Table 1: Spectral magnitude and shape dissimilarity metrics for nearest neighbours versus top influential samples.
Figure 3: Impact of removing a specific training sample on a given prediction output obtained through LOO retraining and influence functions. By leveraging the linear output optimisation of the ELM, influence functions allow for an additive decomposition of the test prediction into the respective contributions of the training instances, analogous to what SHAP achieves for input features.
Carnegie Mellon University 1Department of Statistics & Data Science · Columbia University 2Department of Statistics · Columbia University 3Electrical Engineering Department +2