  Diffusion models, which convert noise into new data instances by learning to
reverse a diffusion process, have become a cornerstone in contemporary
generative modeling. In this work, we develop non-asymptotic convergence theory
for a popular diffusion-based sampler (i.e., the probability flow ODE sampler)
in discrete time, assuming access to $\ell_2$-accurate estimates of the (Stein)
score functions. For distributions in $\mathbb{R}^d$, we prove that
$d/\varepsilon$ iterations -- modulo some logarithmic and lower-order terms --
are sufficient to approximate the target distribution to within $\varepsilon$
total-variation distance. This is the first result establishing nearly linear
dimension-dependency (in $d$) for the probability flow ODE sampler. Imposing
only minimal assumptions on the target data distribution (e.g., no smoothness
assumption is imposed), our results also characterize how $\ell_2$ score
estimation errors affect the quality of the data generation processes. In
contrast to prior works, our theory is developed based on an elementary yet
versatile non-asymptotic approach without the need of resorting to SDE and ODE
toolboxes.
