U-Turn Diffusion

We investigate diffusion models generating synthetic samples from the probability distribution represented by the ground truth (GT) samples. We focus on how GT sample information is encoded in the score function (SF), computed (not simulated) from the Wiener–Ito linear forward process in the artificial time (Formula presented.), and then used as a nonlinear drift in the simulated Wiener–Ito reverse process with (Formula presented.). We propose U-Turn diffusion, an augmentation of a pre-trained diffusion model, which shortens the forward and reverse processes to (Formula presented.) and (Formula presented.). The U-Turn reverse process is initialized at (Formula presented.) with a sample from the probability distribution of the forward process (initialized at (Formula presented.) with a GT sample) ensuring a detailed balance relation between the shortened forward and reverse processes. Our experiments on the class-conditioned SF of the ImageNet dataset and the multi-class, single SF of the CIFAR-10 dataset reveal a critical Memorization Time (Formula presented.), beyond which generated samples diverge from the GT sample used to initialize the U-Turn scheme, and a Speciation Time (Formula presented.), where for (Formula presented.), samples begin representing different classes. We further examine the role of SF nonlinearity through a Gaussian Test, comparing empirical and Gaussian-approximated U-Turn auto-correlation functions and showing that the SF becomes effectively affine for (Formula presented.) and approximately affine for (Formula presented.).