A Perception vs. Distortion Perspective on Score-Based Generative Channel Estimation

Score matching is an alternative to maximum likelihood estimation when the normalizing constant is unknown or too costly to evaluate. However, vanilla score matching has shown to be inefficient relative to maximum likelihood estimation for multimodal distributions with well-separated modes, which are commonly encountered in practical applications. We compare a novel diffusion-based denoising score matching estimator (DDSME) to the vanilla score matching estimator (SME) in this scenario. In particular, we prove statistical guarantees for both estimators, showing that the error bound for the vanilla SME worsens when the separation between the modes increases, which can be avoided in case of the DDSME with suitable hyperparameter tuning. This provides a novel theoretical explanation for the superior behavior of diffusion-based score matching over the vanilla version.

Maximum likelihood (ML) estimation is a principled and statistically efficient approach for learning probabilistic models. However, for unnormalized models, ML estimation requires evaluating the partition function and differentiating through it, which may not always be tractable. Score matching provides a practically viable alternative that circumvents this obstacle by fitting the score in a way that eliminates dependence on the normalizing constant. We derive the generalized score matching objective on a convex subset of $\mathbb{R}^{d}$ constructively starting from Minimum Probability Flow (MPF) learning, and show how classical score matching as well as domain-adapted variants for non-negative data arise naturally within the proposed framework. We show that the resulting objective is a {\it proper local scoring rule} of second-order, which provides the theoretical guarantee that the true density is recovered when the objective is minimized. Furthermore, for a model belonging to the exponential family, we establish convexity of the objective together with consistency of the finite-sample estimator under standard regularity conditions. Our derivation sheds new light on the scope and applicability of generalized score matching in various problem settings. We compare generalized score matching-based estimators on constrained domains, where the partition function is analytically intractable. We provide experimental results on parameter estimation for model densities belonging to the exponential family defined over convex subsets of $\mathbb{R}^{d}$, and a generative modeling use-case to demonstrate broader applicability of the proposed generalized score matching framework.

Score-based diffusion models are typically trained by minimizing the $L^2$ score matching error, and standard theoretical analyses rely on this quantity to bound the sampling discrepancy between the learned and target distributions. We show the $L^2$ score error is not the right intrinsic measure of marginal distributional quality: a learned diffusion model can incur arbitrarily large $L^2$ score error while perfectly matching the target distribution. By decomposing score errors into a gradient and a solenoidal component (a Helmholtz-Hodge decomposition), we identify the geometric reason behind this: only the gradient component enters the marginal Fokker-Planck dynamics, while the solenoidal component is structurally invisible. We make this precise in three results. First, building on the corrected geometry, we prove an impossibility result: no monotone function of the $L^2$ score error can uniformly lower bound any divergence between the learned and target distributions. Second, we derive an upper bound on the Kullback-Leibler divergence that depends only on the observable gradient component of the error, tightening the standard Girsanov bound for generic score networks, and identifying its looseness as the cost of operating on path-space rather than marginal-space dynamics. Third, we give a tractable estimator of the gradient component via a dual Sobolev identity, which is shown to empirically correlate substantially better with sample quality than the full $L^2$ error.