Blurring Diffusion Models
Authors: Emiel Hoogeboom
Tim Salimans
Venue: ICLR 2023 poster (2023)
Keywords: blurring
diffusion
generative model

Abstract: Recently, Rissanen et al., (2022) have presented a new type of diffusion process for generative modeling based on heat dissipation, or blurring, as an alternative to isotropic Gaussian diffusion. Here, we show that blurring can equivalently be defined through a Gaussian diffusion process with non-isotropic noise. In making this connection, we bridge the gap between inverse heat dissipation and denoising diffusion, and we shed light on the inductive bias that results from this modeling choice. Finally, we propose a generalized class of diffusion models that offers the best of both standard Gaussian denoising diffusion and inverse heat dissipation, which we call Blurring Diffusion Models. 

Published as a conference paper at ICLR 2023
BLURRING DIFFUSION MODELS
Emiel Hoogeboom Google Research, Brain Team, Amsterdam, Netherlands
Tim Salimans
Google Research, Brain Team, Amsterdam, Netherlands
ABSTRACT
Recently, Rissanen et al. (2022) have presented a new type of diffusion process for generative modeling based on heat dissipation, or blurring, as an alternative to isotropic Gaussian diffusion. Here, we show that blurring can equivalently be defined through Gaussian diffusion process with non-isotropic noise. In making this connection, we bridge the gap between inverse heat dissipation and denoising diffusion, and we shed light on the inductive bias that results from this modeling choice. Finally, we propose a generalized class of diffusion models that offers the best of both standard Gaussian denoising diffusion and inverse heat dissipation, which call Blurring Diffusion Models.
INTRODUCTION
Diffusion models are becoming increasingly successful for image generation, audio synthesis and video generation. Diffusion models define (stochastic) process that destroys signal such image. In general, this process adds Gaussian noise to each dimension independently. However, data such images clearly exhibit multi-scale properties which such a diffusion process ignores.
Recently, the community is looking at new destruction processes which are referred to as determin- istic or 'cold' diffusion (Rissanen et al. 2022, Bansal al.) 2022). In these works, the diffusion process is either deterministic or close to deterministic. For example, in (Rissanen et al.) 2022) diffusion model that incorporates heat dissipation is proposed, which can be seen as a form of blur- ring. Blurring is natural destruction for images, because it retains low frequencies over higher frequencies.
However, there still exists considerable gap between the visual quality of standard denoising dif- fusion models and these new deterministic diffusion models. This difference cannot be explained away by limited computational budget: A standard diffusion model can be trained with relative little compute (about one to four GPUs) with high visual quality on a task such as unconditional CIFAR10 generation!] In contrast, the visual quality of deterministic diffusion models have been
example of denoising diffusion implementation htt:::/hhhhabbbomm///8777777777
(a) Diffusion (Sohl-Dicksteee Dickstein al. 2015 Ho et al. 2020))
(b) Heat Dissipation [Rissanen ct al.
2022
(c) Blurring Diffusion
Figure 1: Comparison between standard diffusion, heat dissipation and blurring diffusion.

---

Published as a conference paper at ICLR 2023
much worse far. In addition, fundamental questions remain around the justification of determin- istic diffusion models: Does their specification offer any guarantees about being able to model the data distribution?
In this work, we aim to resolve the gap in quality between models using blurring and additive noise. We present Blurring Diffusion Models, which combine blurring (or heat dissipation) and additive Gaussian noise. We show that the given process can have Markov transitions and that the denoising process can be written with diagonal covariance in frequeney space. As result, we use modern techniques from denoising diffusion. Our model generates samples with higher visual quality, which evidenced by better FID scores.
2 BACKGROUND
2.1 DIFFUSION MODELS
Diffusion models SSohl-Dicsstenn et al. 2013 Song & Ermon 2019 Ho et al. 2020) learn to gen- erate data by denoising pre-defined destruction process which is named the diffusion process. Commonly, the diffusion process starts with a datapoint and gradually adds Gaussian noise to the datapoint. Before defining the generative process, this diffusion process needs to be defined. Fol- lowing the definition of (Kingma et al. 2021 the diffusion process can be written as:
(z .)) ( [1,0, o,
(1)
where æ represents the data and z; are the noisy latent variables. Since ar is monotonically decreas- ing and o monotonically increasing, the information from x in 2 will be gradually destroyed as increases. Assuming that the above process defined by Equation T is Markov, it has transition distributions for z, given Za where 0 S <<k
a(z1 = =( 1.
(2)
where 0tjs = az/a, and 0//- = of - - of A convenient property is that the grid of timesteps
can be defined arbitrarily and does not depend on the specific spacing of and t. We let T = denote the last diffusion step where g(zr|æ) RJ N/(2 0, I). standard normal distribution. Unless otherwise specified, a time step lies in the unit interval [0, 1].
The Denoising Process Another important distribution is the true denoising distribution (2,, at) given datapoint æ. Using that (22,121,,) 2)) q(2)|z))((a one can derive that:
g(2. Zr, x) = -N2z, ([ I.
(3)
where
0/1
auja n1 +
Os
and p4-.4 o++s
(4)
03
o3
To generate data, the true denoising process is approximated by learned denoising process p(z,/2z), where the datapoint æ is replaced by prediction from a learned model. The model distribution is then given by
(5) where a(zz) a prediction provided by neural network. As shown by Song al. (0220), the true ((z,12,) + (z,,/=,, x [[)) as + i, which justifies this choice of model: If the generative model takes sufficiently small steps, and if ((x)) is sufficiently expressive, the model can learn the data distribution exactly.
p(z, 22) q(=.|=e, o(zz)),
Instead of directly predicting æ, diffusson models can also model é; fo t), where fo a neural net, so that:
ã = 21/0, //////--0//... 01/ozē.
which is inspired by the reparametrization to sample from Equation I which is 2, axæ + di This parametrization is called the epsilon parametrization and empiriallly leads to better sample quality than predicting directly (Ho et al. 2020).
2

---

Published as a conference paper at ICLR 2023
Optimization As shown in (Kingma et al. 2021), continuous-time variational lower bound on the model log likelihood log p(z) is given by the following expectation over squared reconstruction errors:
C = E, (wt)) - eell.),
(7)
where 2, a, G.C When these terms weighted appropriately with a particular weight w(t), this objective corresponds to a variational lowerbound on the model likelihood log p(z). However, empirically constant weighting w(t) has been found be superior for sample quality.
2.2 INVERSE HEAT DISSIPATION
Instead of adding increasing amounts of Gaussian noise, Inverse Heat Dissipation Models (IHDMs) use heat dissipation to destroy information (Rissanen et al. 2022). They observe that the Laplace partial differential equation for heat dissipation
i Ot
= Az(i,j)) j.t)
(8)
can be solved by diagonal matrix in the frequency domain of the cosine transform if the signal is discretized to grid. Letting& 2, denote the solution to the Laplace equation at time-step this can be efficiently computed by:
A,50 VD,VT=
(9)
where VT denotes Discrete Cosine Transform (DCT) and V denotes the Inverse DCT and zo. Ze should be considered vectorized over spatial dimensions to allow for matrix multiplication. The diagonal matrix D, is the exponent of weighting matrix for frequencies A and the dissipation time
t so that D = exp( cxp(-At). At). For the specific definition of A see Appendix A In (Rissanen et al. 2022) marginal distribution of the diffusion process defined
<<=1(x) = (z, I").
(10)
The intermediate diffusion state is thus constructed by adding fixed amount of noise to an increasingly blurred data point, rather than adding an increasing amount of noise as in the DDPMs described in Section 2.1] The generative process in (Rissanen et al. 2022) approximately inverts the heat dissipation process with a learned generative model:
p(z--1 2x) (V 6"1,
(11)
where the mean for ø;-1 directly learned with a neural network and has fixed scalar variance 82. Similar to DDPMs, the IHDM model is learned by sampling from the forward process (z (æ) for a random timestep t, and then minimizing the squared reconstruction error between the model fa(z,) and a ground truth target, which in this case is given by B(z, (x) A, yielding the training loss L = (.t.,uu(.....BBB.-ooe. [ fz 1)
Arbitrary Dissipation Schedule There is no reason why the conceptual time-steps of the model should match perfectly with the dissipation time. Therefore, in (Rissanen al.) 2022) D, cxp(-Ar)) is redefined where 7, monotonically increases with t. The variable Ti has very similar function og and or in noise diffusion: it allows for arbitrary dissipation schedules with respect to the conceptual time-steps of the model.
To avoid confusion, note that in (Rissanen al. 2022) is is used as the conceptual time for the diffusion process, & is the dissipation time and us denotes the latent variables. In this paper, t is the conceptual time and 2; denotes the latent variables in pixel space. Then Tr is used denote dissipation time.
Open Questions Certain questions remain: (1) Can the heat dissipation process be Markov and what is g(2,) )z (2) Is the true inverse heating process also isotropic, the generative process in Equation II (3) Finally, are there alternatives to predicting the mean the previous time-step?
In the following section it will turn that: (1) Yes, the process be Markov. As result. de- noising equations similar to the ones for standard diffusion can derived. (2) No, the generative process is not isotropic, although it is diagonal in the frequency domain. As consequence, the
3

---

Published as a conference paper at ICLR 2023
correct amount of noise (per-dimension) can be derived analytically instead of choosing it heuristi- cally. This also guarantees that the model p(zg] 21) can actually express the true q 22) as + t, because it is known tend towards g( ( Zt, E|a[| (Song et al. 2020). (3) Yes, processes like heat dissipation can be parametrized similar to the epsilon parametrization in standard diffusion models.
3 HEAT DISSIPATION AS GAUSSIAN DIFFUSION
Here we reinterpret the heat dissipation process as a form of Gaussian diffusion similar to that used in (Ho et al. 2020; Sohl-Dickstein al., 2015 Song & Ermon 2019) and others. Throughout this paper, multiplication and division between two vectors is defined to be elementwise. We start with
the definition of the marginal distribution from (Rissanen et al. 2022):
(z [x) (z, I²I)
(12)
where A,
VT denotes the blurring or dissipation operation as defined in the previous sec- tion. Throughout this section we let VT denote the orthogonal DCT, which is specific normal- ization setting of the DCT. Under the change of variables u, = V žz we can write the diffusion process in frequency space for ue: g(ug)uz) = -u Uw ITI)
(13)
where u, = V x is the frequency response of æ, d, is the diagonal of D, and vector multiplication is done elementwise. Whereas we defined D, = exp(-Arj we let À denote the diagonal of A so that de (xp(-Arr). Essentially, d, multiplies higher frequencies with smaller values.
Equation shows that the marginal distribution of the frequencies u, is fully factorized over its scalar elements (s) for each dimension i. Similarly, the inverse heat dissipation model po(u,/ut /u.) is also fully factorized. We can thus equivalently describe the heat dissipation process (and its inverse) in scalar form for each dimension i: (i) q(u," (6) uu (()) = (( ( |a "n 0²)
( -(" up + oe2, with G ~ N( 1).
(3)
(14)
special case of the standard Gaussian diffusion process introduced in Section 2.1 Let 8t denote a standard diffusion process in frequency space, so d(e) and of = d². As shown by Kingma et al. 2022]) from probabilistic perspective only the ratio at/ oe matters here, not the particular choice of the individual ar, o,, This is true because all values can simply be re-scaled without changing the distributions in a meaningful way.
This equation can be recognized
(() 0r
(6) 0.c We can see that Rissanen et al. (022) have chosen
This means that, rather than performing blurring and adding fixed noise, the heat dissipation process can be equivalently defined as a relatively standard Gaussian diffusion process, albeit in frequency space. The non-standard aspect here is that the diffusion process in (Rissanen et al.) 2022) is defined in the frequency space u. and that it uses a separate noise schedule a, at for each of the scalar elements of i.e. the noise in this process is non-isotropic. That the marginal variance r² is shared between all scalars (+) under their specification does not reduce its generality: the ratio az/a can be freely determined per dimension, and this is all that matters.
Markov transition distributions An open question in the formulation of heat dissipation models by Rissanen et al, (2022) was whether or not there exists a Markov process quu (u.) that corresponds to their chosen marginal distribution g(2, (z). Through its equivalence to Gaussian diffusion shown above, we can now answer this question affirmatively. Using the results summarized in Section 2.1 have that this process is given by
q(ue u,) = Nuulayy Us aip))
(15)
where Cejs 04/a, and o of atj o3 Substituting in the choices of Rissanen et al.
2022
a; = d, and of (i) = a. then gives
Otjs d'/d. and (1-
(16)
Note that if d, is chosen that it contains lower values for higher frequencies, then dr|s will add more noise on the higher frequencies per timestep. The heat dissipation model thus destroys information more quickly for those frequencies compared standard diffusion.
4

---

Published as a conference paper at ICLR 2023
g x)
ac
z
2
2T
5Fz F.)
g u Mz)
Uh
Us
Ut
...
UT
Q (..)
Figure 2: A blurring diffusion process with latent variable 20: .0..... 21 is diagonal (meaning can be factorized over dimensions) in frequency space, under the change of variable u, V zz. This results in corresponding diffusion process in frequency space up. u1
Denoising Process Using again the results from Section 2.1 find an analytic expression for the inverse heat dissipation process:
q(u, (e, x)) = NNu,, Petto: oZoo):
(17)
where
oi
1 o
and Me-+s oi Oxtla the
(18)
2/+
Except for Uz, we can again plug in the expressions derived above in terms of d, 0². The analysis in Section 2.T then allows predicting e, using neural network complete the model, as is done in standard denoising diffusion models. In comparison (Rissanen et al. 2022) predict pt-+ directly, which is theoretically equally general but has been found lead to inferior sample quality. Further- more, they instead chose to single scalar value for of for all time-steps: the downside of this is that loses the guarantee of correctness as + as described in Section
4 BLURRING DIFFUSION MODELS
In this section we propose Blurring Diffusion Models. Using the analysis from Section 3 we can define this model frequency space as Gaussian diffusion model, with different schedules for dimensions. Blurring diffusion places more on emphasis low frequencies which are visually more important, and it may also avoid over-fitting to high frequencies. is important how the model is parametrized and what the specific schedules for a and or are. Different from traditional models, the diffusion process is defined a frequency space:
g( ((ui/u) ( = N(ulaau I²I)
(19)
and different frequencies may diffuse at a different rate, which is controlled by the values in the vec- tors Cke, de (although will end up picking the same scalar value for all dimensions in at). Recall that the denoising distribution then given by q(u, u. x) = N(usluuts ?r as specified in Equation 17
Learning and Parametrization An important reason for the performance of modern diffusion models is the parametrization. Learning μt-ss directly turns out to be difficult for neural networks and instead an approximation for is learned which is plugged into the denoising distributions, often indirectly via an epsilon parametrization (Ho et al. 2020). Studying the re-parametrization of Equation 19
U = a,u, + uru, where u V and u.,t = V
(20)
and take that as inspiration for the way we parametrize our model:
Ub -,ūr,)) / = ūz:
(21)
5

---

Published as a conference paper at ICLR 2023
Algorithm 1 Generating Samples
Algorithm 2 Optimizing Blurring Diffusion
Sample 2T I)
Sample (400
for l in ( (T...., 7 where 8 =-1-1/1 do
Sample ~ (N(00,1)
U V and Ue,t = Vo
Minimize fooaaooVa toc
Compute t- and fot+o with Eq.
18 23
Sample c IO,
2 t ( + C
which is the blurring diffusion counterpart of Equation 6 from standard diffusion models, Although is convenient express our diffusion and denoising processes in frequency space, neural networks have been optimized to work well in standard pixel space. It is for this reason that the neural network So takes as input 2, Vu, and predicts é, After prediction we can always easily transition back and forth between frequency space if needed using the DCT matrix VT and inverse DCT matrix V. This is how tt VT is obtained. Using this parametrization for è and after transforming frequency space ú V à we can compute fi, using Equation 18 where u, is replaced by the prediction úg to give:
p(u,/u)) q(u, ue, ún)) NN (i Ti-ss
(22)
for which Pr-s can be simplified further in terms of iee instead of úz!
Oktla oii
u. +
(u: o.ū...)
(23)
asjoo?
Optimization Following the literature (Ho et al. 2020) we optimize an unweighted squared error in pixel space:
L Ee, (llfool t) c,,, where 2, V(a, Va,VV', +o, c.).
(24)
Alternatively, one can derive a variational bound objective which corresponds to a different weight- ing explained in section 2.1
However, it is known that such objectives tend to result in inferior
sample quality (Ho et al. 2020 Nichol & Dhariwal 202T)
Noise and Blurring Schedules To specify the blurring process precisely, the schedules for an, o need to be defined for t E [0, 1]. For o,, we choose the same value for all frequencies, so it suffices give schedule for scalar value dr- The schedules are constructed by combining typical Gaussian noise diffusion schedule (specified by scalars Ot, 0t) with blurring schedule (specified by the vectors de).
For the noise schedule, following (Nichol & Dhariwal 2021 we choose a variance preserving cosine schedule meaning that of 1- af, where a; = cos (T/2 for & E [0, 1]. To avoid instabilities when t + 0 and t + 1, the log signal to noise ratio (log a² /a() is at maximum +10 for / 0 and least 10 for = See (Kingma et al., 202T) for more details regarding the relation between the signal to noise ratio and dg, dr. For the blurring schedule, we use the relation from (Rissanen et al.) 2022) that a Gaussian blur with scale B corresponds to dissipation with time 7 = / Empiricaly we found the blurring schedule:
B = 0 B,max sin sin(tt// /2)²
(25)
to work well, where BB tune-able hyperparameter that corresponds to the maximum blur that will be applied to the image. This schedule in turn defines the dissipation time via 7. / As described in Equation 23 the denoising process divides elementwise by the term 02 0x. o If one would naively d, exp(-Ar,) for a and equivalently for step s. then the term d, dd/d, could contain very small values for high frequencies. As a result, an undesired side-effect is that small errors may be amplified by many steps of the denoising process. Therefore, modify the procedure slightly and let:
de (1- exp(-ATr) + dmii:
(26)
where we set dmin = 0.001. This blurring transformation damps frequencies to a small value dmin and at the same time the denoising process amplifies high frequencies less aggressively. Because

---

Published as a conference paper at ICLR 2023
Rissanen al. 2022) did not the denoising process, this modification not necessary for their model. Combining the Gaussian noise schedule (at, oi) with the blurring schedule (dz) we obtain:
04 = Gr d. and or = l
(27)
where 1 is a vector of ones. See Appendix A for more details on the implementation and specific settings.
4.1 A NOTE ON THE GENERALITY
In general, an orthogonal base uy = VTx x that has a diagonal diffusion process g(u (ur)
=
(u II corresponds to the following process in pixel space:
<<=x(==>= N(2x VV=i/Viia((::) Vdiag(of)VV*) where te ===Vi V
(28)
where diag transforms vector to a diagonal matrix. More generally, diffusion process defined
in any invertible basis change u = P corresponds to the following diffusion process in pixel space:
(<x1<<) = (z, Pdiag((a)) J æ; Pdiag(o()PP)) where
(29)
As such, this framework enables larger class of diffusion models, with the guarantees of standard diffusion models.
RELATED WORK
Score-based diffusion models -ickstDin al. 2013 Song & Ermon 2019 Ho et al. 2020) have become increasingly successfully in modelling different types of data, such as images (Dhari wal & Nichol 202T), audio (Kong al. 202T), and steady states of physical systems (Xu al., 2022). Most diffusion processes are diagonal, meaning that they can be factorized over dimensions. The vast majority relies independent additive isotropic Gaussian noise diffusion process.
Several diffusion models use a form of super-resolution to account for the multi-scale properties in images (Ho al., 2022 Jing et al. 2022). These methods still rely on additive isotropic Gaussian noise, but have explicit transitions between different resolutions. In other works (Serrà et al. 2022] Kawar et al. 2022) diffusion models are used restore predefined corruptions on image or audio data, although these models do not generate data from scratch. (heis et al. (2022) discuss non- isotropic Gaussian diffusion processes the context of lossy compression. They find that non- isotropic Gaussian diffusion, such as our blurring diffusion models, can lead to improved results if the goal is to encode data with minimal mean-squared reconstruction loss under a reconstruction model that is constrained to obey the ground truth marginal data distribution, though the benefit over standard isotropic diffusion is greater for different objectives.
Recently, several works introduce other destruction processes as alternative to Gaussian diffusion with little to no noise. Although pre-existing works invert fixed amounts of blur (Kupyn et al.) 2018 Whang et al. 2022). in (Rissanen et al., (002) blurring is directly built into the diffusion process via heat dissipation. Similarly, in (Bansal et al. 2022 several (possibly deterministic) destruction mechanisms proposed which are referred
"cold diffusion'. However, the generative pro- cesses of these approaches may not be able to properly Iearn the reveres process if they do not satisfy the condition discussed section 2.1] Furthermore (Lee al. 2022) a process is introduced that combines blurring and noise and is variance preserving in frequency space, which may not be the ideal inductive bias for images. Concurrently, in (Daras al., [202)] a method introduced that can incorporate blurring with noise, although sampling is done differently. For all these approaches, there is still a considerably gap performance compared to standard denoising diffusion.
EXPERIMENTS
6.1 COMPARISON WITH DETERMINISTIC AND DENOISING DIFFUSION MODELS
In this section our proposed Blurring Diffusion Models are compared to their closest competitor in literature, IHDMs (Rissanen et al. 2022), and to Cold Diffusion Models (Bansal et al. 2022 In addition, they are also compared to denoising diffusion baseline similar to DDPMs (Ho et al. 2020) which refer to Denoising Diffusion.
7

---

Published as a conference paper at ICLR 2023
Table 1: Sample quality CIFARIO mea- sured in FID score, lower is better.
Table 2: Sample quality LSUN churches 128 x 128 measured in FID score.
CIFAR10
FID
Model
FID
Cold Diffusion (Blur)"
80.08
IHDM (Rissanen et al. 2022
45.1
IHDM [Rissanen al., 2022)
18.96
Denoising Diffusion
4.68
Soft Diffusion (Daras et al. 2022)
4.64
Blurring Diffusion (ours)
3.88
Denoising Diffusion
3.58
Blurring Diffusion (ours)
3.17
Not unconditional, starts from blurred image.
The first generation task is generating im- ages when trained on the CIFARIO dataset (Krizhevsky al. 2009). For this task, we run the blurring diffu- sion model and the denoising diffusion baseline both us- ing the same UNet architecture as their noise predictor fo. Specifically, the UNet operates at resolutions 32 x 32, 16 16 and x with 256 channels each level. At ev- ery resolution, the UNet has 3 residual blocks associated with the down-sampling section and another blocks for the up-sampling section. Furthermore, the UNet has self- attention at resolutions 16 x 16 and head. Although IHDMs used only 128 channels the 32 x 32 resolutions, they use 256 channels on all other resolutions, they include the blocks instead of 3 blocks. Also see Appendix
CIFARIO
with single
resolution and use 4
Figure 3: Samples from Blurring Dif- fusion Model trained on CIFARIO.
To measure the visual quality of the generated samples we use the FID score measured on 50000 samples drawn from the models, after million steps of training. As can be seen from these scores (Table D. the blurring diffusion models are able to generate images with considerable higher quality than IHDMs, as well as other similar approaches outperform standard denoising diffusion models, although the difference in performance is less pronounced that case. Random samples drawn from the model are depicted in Figure
literature. Our blurring diffusion models also
LSUN Churches Secondly, we test the perfor- mance of the model when trained on LSUN Churches with resolution of 128 x 128. Again, used for the noise predic-
a UNet architecture tion network fe. This time the UNet operates 64 channels for the 128 x 128 resolution, 128 chan- nels for the 64 x 64 resolution, 256 channels for the 32 x 32 resolution, 384 channels for the 16 x 16 res- olution, and 512 channels for the 8 x resolution. At each resolution there are two sections with 3 residual blocks, with self-attention on the resolutions 32 32, 16 x 16, and 8 x 8. The models in (Rissanen et al. 2022) use more channels each resolution level but only 2 residual blocks (see Appendix A.2).
The visual quality is measured by computing the FID score on 10000 samples drawn from trained mod- els. From these scores (Table 2 again we see that the blurring diffusion models generate higher qual- ity images than IHDMs. Furthermore, Blurring Dif- fusion models also outperform denoising diffusion models, although again the difference in performance is smaller in that comparison. See Appendix for more experiments.
Figure 4: Samples from a Blurring Diffusion model trained LSUN churches 128 128.
8

---

Published as a conference paper at ICLR 2023
Table 3: Blurring Diffusion Models with dif- ferent maximum noise values
Table 4: Different maximum noise levels and schedules on CIFARIO
B.max
CIFARIO
LSUN Churches (128x)
maaax
0 Hax sin (t (2)
0 B,max sin
0
3.60
4.68
0
3.60
3.58
1 3.49
4.42
3.49
3.37
3.26
3.65
10
3.26
4.24
20
3.17
3,88
20
3.17
6.54
6.2 COMPARISON BETWEEN DIFFERENT NOISE LEVELS AND SCHEDULES
In this section analyze the models from above, but with different settings terms of maximum blur max) and two different noise schedule (sin² 2 and sin). The models where mam 0 are equivalent to a standard denoising diffusion model. For CIFARIO, the best performing model uses a blur of B B.max = 20 which has an FID of 3.17 over 3.60 when no blur is applied, as can be seen in Table 3 The difference compared to the model with O B,max 10 relatively small, with an FII of 3.26. For L.SUN Churches, the the best performing model uses little less blur B,max = 10 although performance is again relatively close to the model with B.max 20. When comparing the sin? schedule with a sin schedule, the visual quality measured by FID score seems to be much better for the sin schedule (Table 4). In fact, for higher maximum blur the sin' schedule performs much better. Our hypothesis is that the sin schedule blurs too aggressively, whereas the graph of a sin" adds blur more gradually at the beginning of the diffusion process near = 0.
Interesting behaviour of blurring diffusion models that models with higher maximum blur (o converge more slowly, but when trained long enough outperform models with less blur. When comparing two blurring models with BB set to either or 20, the model with B, = 20 has better visual quality only after roughly 200K training steps for CIFAR 10 and 1M training steps for LSUN churches. It seems that higher blur takes more time to train, but then learns to fit the data better. Note that an exception was made for the evaluation of the CIFAR10 models where o max is 0 and 1, as those models show over- wett-fitinn behaviour and have better FID at million steps than at million steps. Regardless of this selection advantage, they are outperformed by blurring di diffusion models with higher d maxx.
7 LIMITATIONS AND CONCLUSION
In this paper we introduced blurring diffusion models, class of generative models generalizing over the Denoising Diffusion Probabilistic Models (DDPM) of Ho et al. (2020) and the Inverse Heat Dissipation Models (IHDM) of Rissanen et al. 2022). In doing so, showed that blurring data, and several other such deterministic transformations with addition fixed variance Gaussian noise, equivalently be defined through Gaussian diffusion process with non-isotropic noise. This allowed us to make connections to the literature on non-isotropic diffusion models (e.g. (Theis al. 2022), which allows to better understand the inductive bias imposed by this model class. Using our proposed model class, we were able to generate images with improved perceptual quality compared to both DDPM and IHDM baselines.
A limitation of blurring diffusion models is that the use of blur has a regularizing effect: When using blur it takes longer to train generative model convergence. Such regularizing effect is often beneficial, and can lead to improved sample quality we showed in Section 6 but may not be desirable when very large quantities of training data are available. As we discuss in Section 4 the expected benefit of blurring is also dependent on our particular objective, and will differ for different ways of measuring sample quality: We briefly explored this in Section 6 but we leave a more exhaustive exploration of the tradeoffs in this model class for future work.
9

---

Published as a conference paper at ICLR 2023
REFERENCES
Arpit Bansal, Fitan Borgnia, Hong-Min Chu, Jie S. Li, Hamid Kazemi, Furong Huang, Micah Gold- blum, Jonas Geiping, and Tom Goldstein. Cold diffusion: Inverting arbitrary image transforms without noise. CoRR, abs/2208.09) 09399, 2022.
Giannis Daras, Mauricio Delbracio, Hossein Talebi, Alexandros G Dimakis, and Peyman Milanfar. Soft diffusion: Score matching for general corruptions. arXiv preprint urXiv:2209.05442 2022.
Prafulla Dhariwal and Alex Nichol. Diffusion models beat gans on image synthesis. CoRR, abs/2105.05233 2021.
Jonathan Ho. Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In Hugo Larochelle. Mare' Aurelio Ranzato, Raia Hadsell, Maria-Floring Balcan, and Hsuan-Tien Lin (eds.), Advances in Neural Information Processing Systems 33: Anmual Conference on Neural Information Processing Systems 2020, NeurIPS, 2020.
Jonathan Ho, Chitwan Saharia, William Chan, David J. Fleet, Mohammad Norouzi, and Tim Sali- mans. Cascaded diffusion models for high fidelity image generation. J. Mach. Learn. Res., 23: 47:1-47:33, 2022.
Bowen Jing, Gabriele Corso, Renato Berlinghieri, and Tommi S. Jaakkola. Subspace diffusion generative models. CoRR, abs/2205.01490, 2022.
Bahjat Kawar, Michael Elad, Stefano Ermon, and Jiaming Song. Denoising diffusion restoration models. CoRR, abs/2201.11799. 2022.
Diederik P. Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. Variational diffusion models. CoRR, 0iss21077006300 2021.
Zhifeng Kong, Wei Ping, Jiaji Huang, Kexin Zhao, and Bryan Catanzaro. DiffWave: A versatile dif- fusion model for audio synthesis. In 9th International Conference on Learning Representations, ICLR, 2021.
Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.
Orest Kupyn, Volodymyr Budzan, Mykola Mykhailych, Dmytro Mishkin, and Jiri Matas. Deblur- gan: Blind motion deblurring using conditional adversarial networks. In 2018 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2018, Salt Lake City, UT, USA, June 18- 22, 2018, pp. 1183-8192 Computer Vision Foundation IEEE Computer Society, 2018. doi: 10.1109/CVPR.20 2018 00854. URL httt:://ppenaccesss thecvf. wtw. coom///oteet_cpp...... 2018 Dew.ukupryneebuurAA. BlinddMotionn____eeieeee CV____VRR__________ 2PPPR_208___0 101088_pper.htt.
Sangyun Lee, Hyungjin Chung, Jachyeon Kim, and Jong Chul Ye. Progressive deblurring of
diffusion models for coarse-to-fine image synthesis. CoRR, abs/2207..1119.. 2022. doi: 10.488550/rxXi.2227..111 1992. https::tt: wtt..//.../...///......... 2207 11192
Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. In Marina Meila and Tong Zhang (eds.), Proceedings of the 38th International Conference on Machine Learning, ICML., 2021.
Severi Rissanen, Markus Heinonen, and Arno Solin. Generative modelling with inverse heat dissi- pation. CoRR, abs/2206. 13397, 2022.
Joan Serrà, Santiago Pascual, Jordi Pons, R. Oguz Araz, and Davide Scaini. Universal speech enhancement with score-based diffusion. CoRR, abs/2206.03065. 2022.
Jascha Sohl- Dickstiic,,,,,, Eric A. Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsuper- vised learning using nonequilibrium thermodynamics. In Francis R. Bach and David M. Blei (eds.), Proceedings of the 32nd International Conference Machine Learning, ICML, 2015.
Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribu- tion. In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS, 2019.
10

---

Published as a conference paper at ICLR 2023
Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In Interna- tional Conference on Learning Representations, 2020.
Lucas Theis, Tim Salimans, Matthew D Hoffman, and Fabian Mentzer. Lossy compression with gaussian diffusion. arXiv preprint 2788892200.008889, 2022.
Jay Whang,
Mauricio Delbracio, Hossein Talebi, Chitwan Saharia, Alexandros G. Dimakis, and Peyman Milanfar. Deblurring via stochastic refinement. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, CVPR 2022, New Orleans, LA, USA, June 18-24, 2022, pp- 16272-16282. IEEE, 2022. doi: 10. 111199CVVRR2228222222..8888 URL https://ddo org/
10. 1111/110R29/666S0 2022. 01581
Minkai Xu, Lantao Yu, Yang Song, Chence Shi, Stefano Ermon, and Jian Tang. Geodiff: A geomet- ric diffusion model for molecular conformation generation. In The Tenth International Confer- ence on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022. OvenReview.net. 2022.

---

Published as a conference paper at ICLR 2023
A ADDITIONAL DETAILS ON BLURRING DIFFUSION
In this section we provide additional details for blurring diffusion models. In particular, we provide some pseudo-code to show the essential steps that are needed to compute the variables associated with the diffusion process.
A.1 PSEUDO-CODE OF DIFFUSION AND DENOISING PROCESS
Firstly, the procedure to compute the frequency scaling (d,) is given below:
def get_frreuencc__cale
(t, min_scale=0.000m
# compute dissipation time sigma_blur = sigma_blur_mx, sin(t pi / ~2^2 dissipation_tian = sigma_t^2 /
# compute frequencies freq = pi linspace img_dim
dim 1, ang_dim) aig_dim
labda =
None, No freqs [None. None, :, None ] *2
compute scaling for frequencies
scaling expbalabda dissipation_.ime) (1 min_scale)
scaling = scaling + min_scale return scaling
Note here the computation A is from (Rissanen et al. 2022 and the variable 'scaling' refers to d, in equations. Next, define wrapper function return the required at, of values.
def get_alpha_siima (t):
freq_scaling get_ sec.euueccccsccllnnggcc (t)
a, sigma get__oost se__noise_scalinngeeeeeeeseeesee (t)
alpha = freq_scaling Combine dissipation and scaling. return alpha, sigma
Which also requires a function to obtain the noise parameters. We use a typical cosine schedule for which the pseudo-code is given below:
def get__noiss se_ schal caahalinng_csiiii (t, Logsnr_min=-De logsnr_max=10)=
limit_max arctan (-0. * logsnr_max))
limit_min = arctan (exp (-0.5 (-xp logsnr_min) limit_max limit_max))
logsnr = -2 log (tan (limmitm_min
Transform logsnr a, sigma. return sqrt (sigmoid (logsnr)) sqrt (sigmoid(-1ogsn)))
To train the model we desire samples from (u, (u,) In the pseudo-code below, the inputs (œ) and outputs (Z,, 6,) are defined in pixel space. Recall that z, = Vu, IDCT(u,) and then:
def diffuse (x, t): x_freq DCT (x)
alpha, sigma = get_alpha_sigma (t)
eps = random normal_like
# Since chose sigma to be scalar, eps does need to be # passed through DCT/IDCT in this case.
z_t (alpha x_freq) + sigma * eps
return z_t,
Given samples 2, from the diffusion process one can now directly define the mean squared error loss epsilon defined below:
12

---

Published as a conference paper at ICLR 2023
def :
t = random. www.uniffo@m. z_t, eps diffuse (x, t) error (eps neural_ned (z_t, t)))a
random_uniform (0, 1)
return mean (error)
Finally, to sample from the model repeatedly sample from p/2 1/T(2())))))))))))))) for the grid of timesteps I t= T,T 1/TT-1/T.... 1/T.
def denoise t, delta=le-8): alpha_s, sigma_s @et_alpha_sigmaCC / T)
alpha_t, sigma_t = get_alpha_sigmact.
# Compute helpful coefficients. alpha_ts alpha_t alpha_s alpha_st alpha_ts sigma2_ts = (sigma_t*2 alpha_ts*2 sigma_s^2)
Denoising variance.
sigma2_denoise
denoise / clip(
/ clip(sigma_stt s*2, min=delta)
1 clip(sigma_t22 alpha_ts sigma_s*2, min=delta), min=delta)
# The coefficients for u_t and u_eps coeff_terpe = alpha_ts sigma2_denoise / (sigma2_te ts + delta)
coeff_term2 = alpha_st sigma2_ts / s*2, min=delta)
Get neural net prediction. hat_eps neural,net t)
# Compute terms.
uut DCT(z_t)
terml = [DC((coff__terml u_t)
term2 = termm u_t sigma_t (CT(hht_)ps)))
mu denoise = terml + term2
Sample from the denoising distribution. eps handom_normal. return mu_denoise + IDCT (sqrt (sigma2_denoise
mmu_denosee)
eps)
More efficient implementations that use less DCT calls are also possible when the denoising function directly defined in frequency space. This is not really issue however, because compared the neural network the DCTs are relatively cheap. Additionally, several values are clipped to minimum of 10-8 avoid numerically unstable divisions.
In the sampling process of standard diffusion, before using the prediction the variable is trans- formed to ēt, clipped and then transformed back to é. This procedure is known to improve visual quality scores for standard denoising diffuuion, but it is not immediately clear how to apply the tech- nique in the case of blurring diffusion. For future research, finding a reliable technique to perform clipping without introducing frequency artifacts may be important.
A.2 HYPERPARAMETER SETTINGS
In the experiments, the neural network function (fo equations) is implemented as a UNet archi- tecture, is typical in modern diffusion models (Ho et al., (020). For the specific architecture details see Table Note that as is standard in UNet architectures, there an downsample and up- sample path. Following the common notation, the hyperparameter 'ResBlocks/ / Stage' denotes the blocks per stage per upsample/downsample path. Thus, a level with ResBlocks per stage as in ResBlocks, where the (3 1) originates from the upsample path which always uses an additional block. In addition, the downsample / upsample blocks also apply an additional ResBlock. All models where optimized with Adam, with learning rate of 2 10-4 and batch size
total 3 + (3 (3++)
13

---

Published as a conference paper at ICLR 2023
128 for CIFAR-10 and learning rate of 1 10 and batch size 256 for the LSUN models. All
methods are evaluated with an exponential moving average computed with a decay of 0.9999.
Table 5: Architecture Settings
Experiment
Channels
Artention Resolutions
Head dim
ReaBlocks Stage
Channel Multiplier
Dropout
CIFARIO
256
256
1,1,
0.2
LSUN Churches 64
128
64
1,2.3,4
0.2
LSUN Churches 128
64
8, 16,
1,2,4,6,8
0.1
B ADDITIONAL EXPERIMENTS
In this section, some additional information regarding the experiments are shown. Th FID score the eval set of CIFARIO and LSUN churches 128 128 is presented. The best performing models match with the results in the main text on train FID. For CIFAR1O, we also report the Inception Score which corresponds
the certainty the Inception classifier. Here the results are less clear, because all models have roughly similar scores. The best performing model uses B.max 10 and achieves 9.59. To confirm that the loss and parametrization are important, the best CIFAR10 model (with max 20) is trained using mean squared error on æ ã when predicting æ, but this only achieves 23.9 FID versus the 3.17 of the epsilon parametrization. This diminished performance is also observed for standard diffusion (Ho et al.) 2020). Furthermore, as an ablation study we trained the best performing model in the frequency domain (where the UNet takes input uz). This model only produced gray samples with some checkerboard artifacts, and had higher loss throughout training. This indicates that learning UNet directly in frequency space is not straightforward.
Table Blurring Diffusion Models with different maximum noise values (eval FID) and Inception Score (IS) for CIFARIO.
T max
CIFARIO (FID eval)
CIFAR1O (IS)
LSUN Churches (eval FID)
0
5.58
9.54
44.1
5,44
9.51
43.6
10
5.35
9.59
42.8
20
5.27
9.51
43.1
For completeness an additional experiment on LSUN churches 64 x 64. Results are similar to the higher resolution case, the Blurring Diffusion Model with B.m.ax whereas the baseline denoising model B, = 0) achieves 2.70.
20 achieves 2.62 FID train
Table 7: Results on LSUN 64 x 64
0 B.max
FID train
FID eval
0
2.70
44.1
20
2.62
43.1
14


Reviews:
[{"review_id": "i8PfyDRCSFG", "reviewer": "Reviewer_PRJZ", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes a Blurring Diffusion Model. It is a type of diffusion model based on heat dissipation, or blurring. The model obtains better performance compared with some SOTA models (e.g., Denoising Diffusion, IHDM).", "summary_of_the_review": "The idea of the blurring diffusion model is good. And the FID values show that the proposed model has better performance than previous models. While the paper needs to be polished, especially for the experiment part.", "strength_and_weaknesses": "Strength:\n(1) It makes sense for combining heat dissipation and blurring into the diffusion model.\n(2) The model can generate images with lower FID.\n\nWeaknesses:\n(1) The related work part needs to be polished. The previous works are not sufficient for heat dissipation, blurring, Gaussian noise, etc.\n(2) The experimental results are not stable.\n  a. I can not view the generated images with high resolution.\n  b. The FID is the only metric used in this paper.\n  c. There is no user study. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "WYK8KLkOZX", "reviewer": "Reviewer_oaCX", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The presented work establishes a theoretical link between standard diffusion models and Inverse heat dissipation models (IHDMs). They show that IHDM formulation can be equally represented as a standard diffusion model, albeit in the frequency domain. They use this theory to apparently get the best from both approaches: (a) They are able to work with blurring which standard diffusion models are not, and (b) use epsilon parametrization which has been empirically shown to outperform the basic representation for standard diffusion models. IHDM could not use it since it was not a diffusion model. \nThe utility of the presented approach is demonstrated on two datasets, improving the performance of heat dissipation models significantly. ", "summary_of_the_review": "A good quality paper which theoretically links heat dissipation models with diffusion models and use that to improve the heat dissipation model. Some more work can be done on ablation studies. \n", "strength_and_weaknesses": "Strength:\n1. The authors derive an equivalence between IHDMs and standard diffusion models.\n2. This is later used to significantly improve the performance of heat dissipation based models. \n3. Other theoretical contributions: a) heat dissipation can be a markov process by explicitly giving one formulation, and (b) inverse heat dissipation, unlike inverse diffusion,  is not an isotropic process (in the frequency domain).\n\nWeaknesses:\n1. In pixel space, it has been shown that epsilon formulation is better. It would be a good ablation to show this holds in the frequency domain as well.\n2. Page 5 last line: “ …it is convenient to express our diffusion and denoising processes in frequency space, neural networks tend to operate well on a standard pixel space”. Either there should be references supporting this or even better, an ablation study showing that neural networks perform better on pixels. \n3. The significance of the two theoretical contributions namely a) inverse heat dissipation not being isotropic and b) possibility of heat dissipation being a markov process is not discussed. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "T3WdM2Eh07", "reviewer": "Reviewer_HZtN", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper analyzes the connection between two diffusion models of DDPM and IHDM. Showing that the heat dissipation process is a Gaussian diffusion one in the frequency domain. Based on this insight, the authors propose the “blurring diffusion model”, which explicitly defines latent variables of the intermediate diffusion steps in the frequency space. Experiments show comparable results with the DDPM and improved ones compared with IHDM. ", "summary_of_the_review": "In general this is a piece of interesting insight but the presentation could have been better. The distinguishment to most related work is not clear to me and I wish that the authors could address my questions in the section above. ", "strength_and_weaknesses": "The major contribution, to the best of my understanding, lies in the analysis of IHDM drawing its connection to denoising diffusion in frequency space, as well as the proposed latent variable model. Another interesting insight is that other deterministic transformations may be also applicable for destroying the signal. \nHowever, I’m not completely sure about the significance of the contribution, especially in comparison with Rissanen et al. 2022. Perhaps I missed something important but it would be great if the authors could highlight the technical improvement / difference. \nI’m not sure if the experimental result is convincing enough either. Checking the Ho et al. 2020, I realized that their simple loss already yielded an FID of 3.17 on CIFAR10, which is exactly the same as the reported performance of the blurring diffusion model. \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "P9A-QjHdBRT", "reviewer": "Reviewer_6MNZ", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "A paper bringing together aspects from diffusion (noising) and heat dissipation (blurring)  in a best-of-both-worlds fashion. Proof-of-concept results deliver impressive image quality.", "summary_of_the_review": "This is a paper that is by no means perfect, but should be of interest to the audience of ICLR. The idea in itself is simple and extends the recent work by Rissanen, but it is a novel and impactful one.\n", "strength_and_weaknesses": "*Strengths:*\n\n1. This is an interesting paper that bridges the gap between inverse heat dissipation (recently proposed by Rissanen et al.) and denoising models. The novelty lies in formulating the (de)blurring model as a Gaussian diffusion process with non-isotropic noise. This formulation is novel and complements the work by Rissanen.\n\n2. The experiments are brief, but demonstrate the strengths of the proposed approach. The visual quality of the generated samples (as well as FID scores) is impressive---especially if compared to that given by vanilla IHDM.\n\n3. The paper is well-structured and easy to follow. \n\n*Weaknesses:*\n\n4. The paper builds directly on Rissanen et al. (2022) for the deblurring part and Song's and Ho's work on denoising. The paper is written almost more as a technical note or technical comment on these existing works than an actual full paper. This, however, makes it easy to position the paper and follow the idea. \n\n5. The paper focuses largely on technical details, and it would have benefited from expanding on discussion points (e.g., the regularisation effect briefly covered in Sec. 6 and 7). Points on the lack of 'guarantees of convergence to the data distribution' (Sec. 1) appear vague and could be either expanded, removed, or rephrased. Additional discussion on the role of the neural network architecture and its role would have been interesting.\n\n6. Even if the presentation itself is clear, it appears slightly sloppy in parts, or written in haste. I would the authors to carefully go through the paper and fix in-text citations to read well.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster