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Traceable Spectral Inference via Influence Functions: Efficient Data Attribution and Error Proxies for the Ariel Mission
Nikki Grens, Luís F. Simões, Kai Hou Yip, Theresa Lueftinger
TL;DR
Ariel spectroscopy models must remain interpretable when operational ground truth is unavailable. The paper reformulates influence functions around predictions, exploits the closed-form ridge structure of an Extreme Learning Machine, and derives an influence-based error proxy. The proxy correlates monotonically with realised RMSPE and SAM errors, while influence analysis identifies influential and high-risk training samples and supports efficient operational assessment.
Problem
Ariel’s operational spectroscopy models require traceability and error assessment despite unavailable ground-truth transmission spectra and limited coverage of training-data attribution methods.
Method
The paper uses prediction-based influence functions with an Extreme Learning Machine’s closed-form ridge solution and combines influence sensitivities with training residuals to estimate harmfulness.
Results
Influence-based errors show a clear monotonic relationship with realised RMSPE and SAM errors, and influence functions efficiently approximate retraining effects and identify high-risk training samples.
Takeaways & Limitations
The results support influence functions as a framework for traceable spectral inference and operational risk assessment when ground truth is unavailable.
Takeaways & Limitations
Exact closed-form infinitesimal prediction influence relies on the ELM output layer’s ridge structure, while deeper architectures require approximate strategies.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper addresses interpretability for Ariel spectroscopy models operating without ground-truth transmission spectra. It focuses on training data attribution through influence analysis to make model behaviour and prediction drivers more traceable.
- Machine-learning models are increasingly applied to space-mission datasets, but growing complexity constrains interpretability.This limits understanding of model behaviour in scientific applications where physical consistency matters.
- ESA’s Ariel mission will analyse transit spectroscopy to conduct a chemical census of approximately 1000 exoplanets.Transmission spectra reveal wavelength-specific molecular signatures from planetary atmospheres.
- Ariel’s inflight observations will lack ground truth, making interpretability essential for assessing whether models learn the correct physical structure.
- Most explainable-AI methods attribute predictions to input features, whereas training data attribution examines which training samples shaped a prediction.The paper positions influence analysis as an approach to training data attribution.
- This work formalises influence as the change in a test prediction induced by a training instance rather than the conventional test-loss focus.The reformulation targets traceable spectral inference under Ariel’s operational constraints.
2 Methodology
The methodology reformulates influence as a label-free change in model prediction and exploits the ELM’s closed-form ridge structure for efficient training-data attribution. It also defines prediction influence, representer values, and harmfulness measures for tracing individual training samples.
- Influence Functions: Classical influence estimates how upweighting a training point changes test loss through the inverse Hessian, but its linear approximation can deviate when influence magnitudes are large.The approximation omits higher-order loss effects.
- Prediction Influence: Because runtime ground truth is unavailable, influence is reformulated from test loss to the model prediction, enabling label-free evaluation without new modelling assumptions.The target changes from L(z_test, θ_hat) to f(x_test, θ_hat).
- Extreme Learning Machines: The ELM fixes hidden-layer weights and learns output weights with ridge regression, yielding a constant Hessian that can be computed once and reused for influence evaluations.Its quadratic output-layer objective enables rapid closed-form computation.
- Prediction Influence Computation: Closed-form prediction influence is computed by calculating each training residual, projecting its gradient through H^-1, and taking the test activation dot product with the resulting parameter shift.The three stages use the training residual r_i, H^-1(ϕ_i · r_i), and test activations ϕ_test.
- Prediction Influence Computation: For prediction influence, values near zero indicate negligible output impact, while large absolute values identify samples producing the largest local prediction shifts.The sign indicates whether removing a training point raises or lowers the prediction.
- Theoretical Extensions: Representer values quantify each training sample’s contribution to a test prediction, while harmfulness additionally combines influence magnitude with training residual magnitude.Harmfulness is defined as |I(z_i,λ,x_test) · r_i,λ|, following the intuition of Cook’s Distance.
3 Results
The Ariel L2-L3 pipeline uses influence analysis to trace predictions to training examples and derive a conservative, relative error proxy from influence sensitivities. Evaluations on simulated spectra compare this proxy with realised spectral errors and visualise influential and harmful samples.
- Influence scores trace Ariel L2-L3 transmission-spectrum predictions back to their most influential training examples.
- The t-SNE analysis maps training points into two dimensions and highlights the most influential and harmful samples relative to each test prediction.
- 0.69 Spearman correlation links the proxy with RMSPE, while 0.81 links it with SAM across the 55 wavelength channels.The simulated dataset provides ground truth only for computing realised test-observation errors.
- The error proxy propagates training residuals through influence sensitivities and aggregates residual-weighted contributions into a non-cancelling perturbation estimate.
- The relative proxy is normalised for predicted amplitude, while RMS aggregation across 55 channels produces an observation-level risk score.This formulation reflects the larger impact of a fixed absolute deviation on shallow rather than deep transit depths.
4 Discussion
The discussion shows that influence-based attribution identifies training samples by their effect on predictions rather than simple spectral proximity, while providing efficient error assessment and agreement with leave-one-out retraining. Its broader applicability is conceptual, although exact closed-form efficiency depends on the ELM output-layer structure.
- Training-sample attribution: Influential training samples are distributed across the dataset rather than being limited to nearest neighbours.This indicates that attribution captures how samples drive predictions beyond visual similarity.
- Training-sample attribution: Top influential samples match nearest neighbours in predicted spectral shape but show a 6-fold increase in magnitude dissimilarity.The contrast supports influence being jointly determined by cross-leverage and training-residual magnitude.
- Error proxy: The influence-based error proxy has a clear monotonic relationship with realised RMSPE and SAM errors, with stronger Spearman correlation for SAM.This suggests the proxy aligns more closely with spectral shape errors and can support assessment without ground truth.
- Validation against retraining: Influence scores strongly correlate with actual prediction changes from leave-one-out retraining, while full removal amplifies effects beyond the infinitesimal approximation.The divergence is attributed to the sample’s leverage factor rather than unmodelled nonlinear output dynamics.
- Operational efficiency: Computing influence matrices across five folds required 1 minute and 15 seconds, whereas full leave-one-out retraining was estimated to require approximately 24 hours.The comparison supports influence functions as an efficient mechanism for operational risk assessment and training-data attribution.
- Generalisation and scope: The label-free influence formulation, residual-based error proxy, and harmfulness criterion remain conceptually applicable beyond ELMs, although exact closed-form computation relies on the ELM ridge output layer.Deeper architectures require alternative approximations such as last-layer, Hessian-vector, or random-projection methods.