Many set-valued inputs are not arbitrary unordered collections, but finite discretizations of an underlying continuum object such as a field, surface, trajectory, or measure. We argue that set encoders in these regimes should be analyzed not only through permutation invariance, but through discretization invariance: the requirement that predictions and representations remain stable across different samplings of the same object. This perspective reveals a basic limitation of standard pooling rules. Mean-style aggregation may be permutation invariant while still behaving as a low-order numerical estimator whose output drifts under resampling, refinement, or density shift.
We propose a continuum view of set encoding in which additive aggregation is interpreted as numerical estimation of continuum functionals, and discretization error becomes a first-class object of analysis. From this viewpoint, geometry-aware and measure-aware aggregation are not implementation details, but principled corrections for sampling bias. We also advocate evaluation protocols based on same-object resampling rather than only conventional i.i.d. test accuracy, making it possible to measure robustness to changes in resolution and sensor layout directly.
Across synthetic point-cloud and signal-reconstruction benchmarks, this view explains large failures of uniform pooling under discretization shift and predicts substantial gains from geometry-aware aggregation in both task error and representation stability. More broadly, our framework connects set encoding to numerical analysis and positions discretization invariance as a foundational criterion for learned representations of sampled continuum data.