Interaction Budgets for Gated Wavelet Denoising: An EMPR Theory of Operator- Versus Correlation-Induced Coupling

Modern non-stationary signal denoisers increasingly replace a global wavelet threshold by a detect-then-act (gated) rule that first localises an artefact in the time–scale plane and then suppresses it selectively. Such gates couple sub-bands, and it has been unclear how to quantify, in a basis-intrinsic and interpretable way, how much of the denoising performance is produced by that coupling. We introduce the interaction budget I=1−∑jSj and prove that it equals the normalised L2-distance of the performance functional to the space of band-additive functions; hence, I=0 for any band-diagonal operator and I>0 exactly when the gate couples bands. We then (i) give an exact pairwise identity for a binary gate, a two-point lower bound, and a sharp if-and-only-if condition on the gate; (ii) identify a coupling-strength functional with the leading-order law I(κ)=(C/V0)κ2+o(κ2); and (iii) bound the second-order HDMR/EMPR truncation error, which vanishes for single-trigger gates. Finally, for coloured noise—where the sub-band factors are correlated, and I alone conflates the two effects—an EMPR product-support decomposition separates operator-induced from correlation-induced interaction, with a correlation-invariant operator signature and a permutation-based estimator. Controlled experiments confirm all results.