In diffusion models, proper distributional learning entails reversing a diffusion process over underlying density functions, whereas a key failure is memorization, in which the reverse process instead learns diffusion from individual training samples. We provide an analytical characterization of such memorized reverse processes, along with a criterion that detects the associated thermodynamic singularity. The proposed criterion reliably distinguishes between properly learned and memorizing networks across a range of neural network models designed to promote or mitigate memorization.

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