A Variational-Flow Analysis of Diffusion-Based Speech Enhancement under Noise-Power Mismatch

Diffusion-based speech enhancement architectures that pair a deterministic predictor with a learned score network, exhibit a sharp non-smooth transition (``kink’') in the SI-SDR degradation curve at the training-time noise amplitude. We give a pathwise variational-flow analysis that localizes this non-smoothness to the predictor stage. The central identity is an exact factorization of the parametric sensitivity, $\mathrm{\partial }{\text{sig}}^{(M)}\mathrm{/}\mathrm{\partial }M=K(M)\cdot \mathrm{\partial }{C}_M\mathrm{/}\mathrm{\partial }M$, where $K(M)$ is a continuous matrix-valued functional of the score Jacobian along the reverse trajectory and $C_M=\mathrm{\Pi }(y^{(M)})$ is the predictor output. Under three hypotheses on the reverse-process flow (score-Jacobian continuity, conditioning-Jacobian continuity, non-degeneracy of $K$), failure of $M\mapsto {\text{sig}}^{(M)}$ to be $C^1$ at $M^{\ast }$ holds if and only if $M\mapsto \mathrm{\Pi }(y^{(M)})$ fails to be $C^1$ at $M^{\ast }$. We extend the localization to the finite-step Euler–Maruyama sampler actually run at inference. The hypotheses translate into a concrete experimental program; this paper specifies the program and presents the variational structure. The empirical validation is deferred to a companion experimental report.