Understanding Diffusion Models by Feynman's Path Integral
Yuji Hirono Akinori Tanaka Kenji Fukushima
Score-based diffusion models have proven effective in image generation and
have gained widespread usage; however, the underlying factors contributing to
the performance disparity between stochastic and deterministic (i.e., the
probability flow ODEs) sampling schemes remain unclear. We introduce a novel
formulation of diffusion models using Feynman's path integral, which is a
formulation originally developed for quantum physics. We find this formulation
providing comprehensive descriptions of score-based generative models, and
demonstrate the derivation of backward stochastic differential equations and
loss functions.The formulation accommodates an interpolating parameter
connecting stochastic and deterministic sampling schemes, and we identify this
parameter as a counterpart of Planck's constant in quantum physics. This
analogy enables us to apply the Wentzel-Kramers-Brillouin (WKB) expansion, a
well-established technique in quantum physics, for evaluating the negative
log-likelihood to assess the performance disparity between stochastic and
deterministic sampling schemes.