Beyond MSE: Differentiable Complexity Priors for Structure-Preserving Neural Denoising

Neural denoisers trained with pointwise distortion losses such as MSE often suppress the temporal regularities that downstream tasks depend on---a failure that becomes acute under non-Gaussian, impulsive, or non-stationary disturbances. We propose a training framework that lifts a family of classical complexity descriptors---permutation entropy, sample entropy, Lempel-Ziv complexity, and Higuchi fractal dimension---into end-to-end differentiable objectives via calibrated soft surrogates with straight-through gradients. Three complementary mechanisms emerge from this lifting: (i) early fusion injects local structural priors as auxiliary inputs, (ii) a complexity-consistency regularizer matches the multi-scale complexity of denoiser outputs to clean references, and (iii) a curriculum sampler prioritizes examples whose noise-induced complexity inflation is largest. We prove that additive noise strictly inflates ordinal uncertainty for bounded piecewise-smooth signals, and that, under bounded surrogate calibration, complexity consistency yields a strictly larger expected decision margin than MSE-only training. On OFDM/QAM communication signals under AWGN, $1/f$ colored, and $\alpha$-stable impulsive noise, our framework reduces post-detection bit error rate by 29--47% over Transformer (PatchTST, iTransformer), GNN, and CNN baselines, with statistically significant gains across five seeds and 95% bootstrap confidence intervals. The improvements transfer to over-the-air conditions on the public RadioML 2018.01A benchmark (single-carrier QAM portion) and a new 50-session in-house SDR testbed, yielding 22--26% relative BER reduction without fine-tuning. The framework is architecture-agnostic and adds $\leq 3.5\%$ FLOPs over the underlying backbone.