L, learned diagonal Σ
L, ﬁxed isotropic Σ
(cid:107) ˜µ − ˜µθ(cid:107)2
(cid:15) prediction (ours)

L, learned diagonal Σ
L, ﬁxed isotropic Σ
(cid:107)˜(cid:15) − (cid:15)θ(cid:107)2 (Lsimple)

7.28±0.10
8.06±0.09

23.69
13.22

–

–

–

–

7.67±0.13
9.46±0.11

13.51
3.17

training. However, we found it beneﬁcial to sample quality (and simpler to implement) to train on the
following variant of the variational bound:
Lsimple(θ) := Et,x0,(cid:15)

(cid:104)(cid:13)(cid:13)(cid:15) − (cid:15)θ(√¯αtx0 + √1 − ¯αt(cid:15), t)(cid:13)(cid:13)2(cid:105)

(14)
Denoising Diffusion Probabilistic Models