An Operator-Theoretic Characterisation of Label Noise Detection

Label noise detectors inspect neighbours, model predictions, or training history, yet ask the same question: does a recorded label agree with evidence from other examples? Designing a detector means choosing what evidence to use, how examples interact, and how their agreement is judged. Exploring these choices through repeated experiments is expensive. We give a unified operator-theoretic characterisation that organises them into a common search space for detector design. The theory identifies the information needed to characterise a detector's response to a proposed change and bounds the error when its effect is approximated. This enables a constructive strategy: predict how alternatives will behave before fully evaluating them, and run exact experiments where uncertainty could change the choice. In experiments varying training and scoring while keeping features fixed, this strategy matches the selection accuracy of evaluating every alternative across all 18 cases, with up to $6.83\times$ speed-up over an optimised implementation. The result connects unification to algorithm design for label noise detection through a shared language for proposing changes and a principled way to make the search cheaper. Experimentation code: https://anonymous.4open.science/r/label-noise-operator-164B.