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.
Data attribution seeks to quantify how individual training examples shape a model's predictions and underpins problems including data valuation, machine unlearning, and model interpretability. Despite having a long line of work, computationally scalable methods often struggle to predict the effect of removing training data in neural networks due to their non-convex nature. To overcome this challenge, metagradient-based methods such as MAGIC (Ilyas and Engstrom, 2025) differentiate each prediction through the entire training run and compute its exact influence with respect to the training data, but require a separate run for every prediction. To reduce this cost, we cast budgeted attribution as estimating a large influence matrix from a small number of measurements. We show that the measurements most appropriate for recovering this matrix differ from those best suited for attribution itself. We then present two algorithms, MAGE and SPELL, suited for reconstruction and attribution respectively, that run on existing metagradient machinery at no extra cost. Empirical studies demonstrate strong performance over existing baselines across training scales and measurement budgets.
Yuxi Chen, Hamza Golubovic, Han Tong +2
Carnegie Mellon University 1Department of Statistics & Data Science · Columbia University 2Department of Statistics · Columbia University 3Electrical Engineering Department +2
As large language models are increasingly trained and fine-tuned, practitioners need methods to identify which training data drive specific behaviors, particularly unintended ones. Training Data Attribution (TDA) methods address this by estimating datapoint influence. Existing approaches like influence functions are both computationally expensive and attribute based on single test examples, which can bias results toward syntactic rather than semantic similarity. To address these issues of scalability and influence to abstract behavior, we leverage interpretable structures within the model during the attribution. First, we introduce Concept Influence which attribute model behavior to semantic directions (such as linear probes or sparse autoencoder features) rather than individual test examples. Second, we show that simple probe-based attribution methods are first-order approximations of Concept Influence that achieve comparable performance while being over an order-of-magnitude faster. We empirically validate Concept Influence and approximations across emergent misalignment benchmarks and real post-training datasets, and demonstrate they achieve comparable performance to classical influence functions while being substantially more scalable. More broadly, we show that incorporating interpretable structure within traditional TDA pipelines can enable more scalable, explainable, and better control of model behavior through data.
Estimating the influence of training examples on model behavior is essential for data debugging, valuation, and attribution. Existing influence estimators often produce incompatible rankings, which are commonly ascribed to approximation error. We argue that a more fundamental source of disagreement is specification mismatch: influence depends on the behavior being attributed, the intervention applied to each training example, and the counterfactual training process that maps the intervention to a model response. These choices are especially important when the target behavior requires a tractable surrogate, such as query loss, a logit, or a margin. We formalize influence as a counterfactual estimand, distinguish specification mismatch across estimands from approximation error in estimating a fixed estimand, and organize representative estimators by their implied specifications. We further derive a local decomposition that exposes how behavior signals, training signals, and counterfactual parameter responses interact. Controlled experiments show that exact estimands under different specifications can induce different rankings, whereas approximation error grows as perturbations move farther from their linearization points. Experiments on noisy label detection and LLM attribution show that specification choices significantly affect attribution quality, especially for the choice of behavior surrogate. Behavior-aligned specifications can identify target-specific training examples obscured by default loss-based or similarity-based specifications. These results establish specification analysis as a necessary first step for interpreting and comparing data influence estimators.