When Can Diffusion Denoisers Share a Representation Across Noise Levels?

Diffusion models solve denoising problems across noise levels, yet typically reuse a single time-conditioned network. Motivated in part by latent diffusion, we ask when the data distribution itself justifies sharing a common representation across noise levels. For Gaussian diffusion, we establish an exact characterization: if the Bayes-denoiser Jacobian is diagonalizable in the same fixed frame at every input at just one positive noise level, then the clean distribution admits an affine orthogonal independent-component representation. Conversely, this structure guarantees a common coordinatewise Bayes-denoising frame across all noise levels. Beyond exact independence, we show that total correlation equals both the noiseintegrated excess Bayes risk of coordinatewise denoising and the generative KL discrepancy of an ideal product model. We distinguish this factorization cost from the additional cost of sharing a frame across noise levels. We derive sufficient finite-sample learning bounds for shared and untied frames, and show why approximate positive-noise Bayes-denoiser separability need not imply approximate clean independence in a prescribed frame. Experiments with independently trained denoisers and learned VAE-family and CIFAR-10 representations support these predictions, including the finite-sample crossover between shared and noisespecific representations.