Closed-Form Diffusion Models
Christopher Scarvelis Haitz Sáez de Ocáriz Borde Justin Solomon
Score-based generative models (SGMs) sample from a target distribution by
iteratively transforming noise using the score function of the perturbed
target. For any finite training set, this score function can be evaluated in
closed form, but the resulting SGM memorizes its training data and does not
generate novel samples. In practice, one approximates the score by training a
neural network via score-matching. The error in this approximation promotes
generalization, but neural SGMs are costly to train and sample, and the
effective regularization this error provides is not well-understood
theoretically. In this work, we instead explicitly smooth the closed-form score
to obtain an SGM that generates novel samples without training. We analyze our
model and propose an efficient nearest-neighbor-based estimator of its score
function. Using this estimator, our method achieves sampling times competitive
with neural SGMs while running on consumer-grade CPUs.