Wavelet Score-Based Generative Modeling
Authors: Florentin Guth
Simon Coste
Valentin De Bortoli
Stéphane Mallat
Venue: NeurIPS 2022 Accept (2022)
Keywords: score-based generative model
diffusion model
wavelet decomposition
cascading algorithm
acceleration

Abstract: Score-based generative models (SGMs) synthesize new data samples from Gaussian white noise by running a time-reversed Stochastic Differential Equation (SDE) whose drift coefficient depends on some probabilistic score. The discretization of such SDEs typically requires a large number of time steps and hence a high computational cost. This is because of ill-conditioning properties of the score that we analyze mathematically. Previous approaches have relied on multiscale generation to considerably accelerate SGMs. We explain how this acceleration results from an implicit factorization of the data distribution into a product of conditional probabilities of wavelet coefficients across scales. The resulting Wavelet Score-based Generative Model (WSGM) synthesizes wavelet coefficients with the same number of time steps at all scales, and its time complexity therefore grows linearly with the image size. This is proved mathematically for Gaussian distributions, and shown numerically for physical processes at phase transition and natural image datasets.

Wavelet Score-Based Generative Modeling
Florentin Guth
Simon Coste Computer Science Department, ENS, CNRS, PSL University
Computer Science Department, ENS, CNRS, PSL University
Valentin De Bortoli
Stéphane Mallat
Computer Science Department, ENS, CNRS, PSL University
Collège de France, Paris, France Flatiron Institute, New York, USA
Abstract
Score-bascd generative models (SGMs) synthesize data samples from Gaus- sian white noise by running a time-reversed Stochastic Differential Equation (SDE) whose drift coefficient depends such SDEs typically requires a large number of time steps and hence a high com- putational cost. This is because of ill-conditioning properties of the analyze mathematically. Previous approaches have relied on multiscale generation considerably accelerate SGMs. We explain how this acceleration results from implicit factorization of the data distribution into product of conditional proba- bilities of wavelet coefficients across scales. The resulting Wavelet Score-based Generative Model (WSGM) synthesizes wavelet coefficients with the same number of time steps at all scales, and its time complexity therefore grows linearly with the image size. This is proved mathematically for Gaussian distributions, and shown numerically for physical processes at phase transition and natural image datasets.
some probabilistic score. The discretization of
that we
Introduction
Score-based Generative Models (SGMs) have obtained remarkable results to learn and sample probability distributions of image and audio signals [44, 3, 24, 38, 39, 6]. They proceed follows: the data distribution is mapped to a Gaussian white distribution by evolving along a Stochastic Differential Equation (SDE), which progressively adds noise the data. The generation is implemented using the time-reversed SDE, which transforms Gaussian white noise into a data sample. At each time step, pushes samples along the gradient of the log probability, also called score function. This score estimated by leveraging tools from score-matching and deep neural networks [13, 47]. At sampling time, the computational complexity is therefore proportional to the number of time steps, i.e., the number of forward network evaluations. Early SGMs in [44, 46, 11] used thousands of time steps, and hence had a limited applicability.
Diffusion models map a Gaussian white distribution into a highly complex data distribution. We thus expect that this process will require large number of time steps. It then comes surprise that recent approaches have drastically reduced this time complexity. This is achieved by optimizing the discretization schedule by modifying the original SGM formulation [18, 17, 27. 53, 42, 37, 43, 23, 11, 29, 41, 51] High-quality score-based generative models have also been improved by cascading multiscale image generations [40, 12, 6] or with subspace decompositions [16]. We make explicit the reason of this improvement, which provably accelerates the sampling of SGMs.
A key idea is that typical high-dimensional probability distributions coming from physics natural images have complex multiscale properties. They can be simplified by factorizing them as a product of conditional probabilities of normalized wavelet coefficients across scales, shown in [33].
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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These conditional probabilities are more similar to Gaussian white noise than the original image distribution, and can thus be sampled more efficiently. On the physies side, this observation rooted in the renormalization group decomposition in statistical physics [49], and has been used to estimate physical energies from data [33]. In image processing, it relies on statistical observations of wavelet coefficient properties [48]. A Wavelet Score-based Generative Model (WSGM) generates normalized wavelet coefficients from coarse to fine scales, as illustrated in Figure 1. The conditional distribution of each set of wavelet coefficients, given coarse scale coefficients, is sampled with its own (conditional) SGM. The main result is that normalization of wavelet coefficients allows fixing the same discretization schedule at all scales. Remarkably, and opposed to existing algorithms,, implies that the total number of sampling iterations per image pixel does not depend on the image size.
After reviewing score-based generation models, Section 2 studies the mathematical properties of its time discretization, with a focus on Gaussian models and multiscale processes. Images and many physical processes are typically non-Gaussian, but do have singular covariance with long- and short-range correlations. In Section 3, we review how to factorize these processes into probability distributions which capture interactions across scales by introducing orthogonal wavelet transforms. We shall prove that it allows considering SGMs with the same time schedule at all scales, indepen- dently of the image size. In Section 4, present numerical results on Gaussian distributions, the y physical model at phase transition, and the CelebA-HQ image dataset [19]. The main contributions of the paper follows:
A Wavelet Score-based Generative Model (WSGM) which generates samples from the conditional distribution of normalized wavelet coefficients, with the same discretization schedule at all scales. The number of time steps per image pixel does not need to depend upon the image size to reach fixed error level.
Theorems controlling errors of time discretizations of SGMs, proving accelerations obtained by scale separation with wavelets. These results are empirically verified by showing that WSGM provides an acceleration for the synthesis of physical processes at phase transition and natural image datasets.
2 Sampling and Discretization of Score-Based Generative Models
2.1 Score-Based Generative Models
Diffusions and time reversal A Score-based Generative Model (SGM) [44, 46, 11] progressively maps the distribution of data into the normal distribution, with forward Stochastic Differential Equation (SDE) which iteratively adds Gaussian white noise. It is associated with a noising process (zz).. with zo distributed according to the data distribution p, and satisfying:
dz, -2, + V2dw,,
(1)
where (w:), is Brownian motion. The solution is an Omstein-Uhlenbeed Uhlenbeck process which admits the following representation for any > 0:
x4 The process (ag), is therefore an interpolation between a data sample xp and Gaussian white noise. The generative process inverts (1). Under mild assumptions p [2. 91, for any T > 0, the reverse- time process - satisfies:
26 2, + 1
2 ~ Id.
(2)
dzr-t= (xr-t + 2V (og dt + 2d
where P, is the probability density of x,, and V logp, P, is called the Stein score. Since IT is close to a white Gaussian random variable, one can approximately sample from By by sampling from the normal distribution. We generate Ty from Iy by solving this reverree SDE, if we can estimate an accurate approximation of the score V log p, at each time t, and if we can discretize the SDE without introducing large errors.
Efficient approximations of the Stein scores are the workhorse of SGM. [13] shows that the score v log Pt can be approximated with parametric functions sy which minimize the so-called implicit

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SGM WSGM
Gaussian white noise
image
Wavelet coefficients
Reverse diffusion
Conditionaal reverse diffusion
wavelet transform
Figure An SGM generates images by discretizing reverse diffusion, which progressively transforms white Gaussian noise into a natural image. A WSGM generates increasingly higher- resolution images by discretizing reverse diffusions on wavelet coefficients at each scale. It begins by generating a first low-resolution image. Renormalized wavelet coefficients are then generated conditionally to this low-resolution image. fast inverse wavelet transform reconstructs a higher- resolution image from these wavelet coefficients. This process is repeated at each scale. The number of steps the same each scale, and can orders of magnitude smaller than for SGM.
score matching loss:
8: = arg min E,, / [1 (av()
(4)
or, equivalently, the denoising score matching loss:
8; = arg min E ((0.id) [|s0(e "mp + V
20
(5)
For image generation, is calculated by neural network parameterized by 0. In statistical physics problems where the energy can be linearly expanded with coupling parameters, we obtain linear models so (x) 0 (U()). This the case for Gaussian processes where ((x)) = x. it also applies to non-Gaussian processes, using non-quadratic terms in Uxx).
Time discretization of generation An approximation of the generative process (3) is computed by approximating log by s, and discretizing time. It amounts to approximating the time-reversed SDE by Markov chain which is initialised by ΣT (O0 Id), and computed over times t, which decrease from ty = T to to O, at intervals dz. = ty t--1;
J.. Su 28,, (T, 220
idd. AO Id).
Ignoring the enror due to the score model, the minimum number of time steps is limited by the Lipschitz regularity of the score loggt, see [5, Theorem 1]. The overall complexity of this generation is N evaluations of the score (x)).

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2.2 Discretization of SGM and Score Regularity
We now study how the regularity of the score V log p affects the discretization of (6). Assuming that the score is known, i.e, that s = V log Pt, we prove that for Gaussian processes, the number of time steps reach fixed error E depends on the condition number of its covariance. This result is generalized to non-Gaussian processes by relating this error to the regularity of V Po
Gaussian distributions Suppose that the data distribution is a Gaussian p NO, Σ) with covari- ance matrix Σ, in dimension d. Let Pt be the distribution of 24. Using (2), we have:
V llggp(>)) - (Id +(E Idje -
Let Pt be the distribution of ã. obtained by the time discretization (6). The approximation error between the distribution Po obtained with the time-reversed SDE and the data distribution p stems from (i) the mismatch between the distributions of 2T and žy. and (ii) the time discretization. The following theorem relates these two errors to the covariance M of x in the particular case of a uniform time sampling intervals 62 5. We normalize the signal energy by imposing that Tr(>:) d. and we write x the condition number of Σ, which the ratio between its largest and smallest eigenvalues.
Theorem 1. If the data distribution p = (0, Σ). the distribution Po of io in (6) with a uniform discretization 02- = 8 satisfies (L Po) S Er + Egg + E with
ET ffe= | Id)=))))
-
(7)
Eg- = f(6)TT(E - E(E - 1d) Id)- log(E)/2 (Id -Id-E-1)33))
(8)
where f(t) = -t--ttl log(1 t) and ET,s a higher-order term with ET, = ol5 + c (T) when ō 0
and T + +00. Furthermore, for any > O, there exists T.6 2 0 such that:
(1/d)(Ex E + F.$) and N T // < Ce 2
with C > 0 a universal constant and the conditioning number of Σ.
This theorem specifies the dependence of the Kullback-Leibler error on the covariance matrix. computes upper bound on the number of time steps N = / to reach eπrоr as function of the condition number K of E. As expected, indicates that the number of time steps should increase with the condition number of the covariance. This theorem is proved in more general case in Appendix S5, which includes the case where p has non non-zero zero mean. An exact expansion of the
Kullback-Leible divergence is also given.
For stationary processes of images, the covariance eigenvalues are given by the power spectrum, which typically decays like | at a frequency w. results that x proportional to a power of the image size. Many physical phenomena produce such stationary images with a power spectrum having a power law decay. these typical cases, the number of time steps must increase with the image size. This is indeed what is observed in numerical SGM experiments, seen in Section 3.
General processes Theorem can be extended to non-Gaussian processes, The number time steps then depends on the regularity of the score V log Pt.
Theorem 2. Assume that (x is 8 in both and x, and that:
supz,t 1 (og < K and ||0 log S Me |
(10)
for some K, M, 0. Then 1p PO S ET + Es ET,s, where:
Ex /20 T KL( 12
(11)
E5 = 6 V (I 1/4 [1+ K + M(1 + 1/(2a)1/311
(12)
and EsT is a higher order term with E = ol -e when 8 and T + +
The proof of Theorem 2 is postponed to Appendix S5 and we show that the result can be strengthened by providing a quantitative upper bound on 1p - Poll Theorem improves on [5, Theorem 1] by proving explicit bounds exhibiting the dependencies on the regularity constants K and M of the

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score and by eliminating an exponential growth term in T in the upper bound. Theorem 2 is much more general but not as tight as Theorem
The first error term (11) is due to the fact that T is chosen to be finite. The second error term (12) controls the error depending upon the discretization time step 5. Since p: is obtained from p through a high-dimensional convolution with a Gaussian convolution of variance proportional to t, the regularity of V log Px (4)) typically increases with so || V/ log logpe(z)) (2:]l and ||0 V log Pt (a)] rather decrease when increases. This qualitatively explains why quadratic discretization schedule with non-uniform time steps 6x oc k are usually chosen in numerical implementations of SGMs [38. 45]. For simplicity, we focus on the uniform discretization schedule, but our result could be adapted to non-uniform time steps with no major difficulties. This remark also explains that it is mainly the regularity of the score at time t
log p which determines the error decay (12).
While Theorem 2 is more general than Theorem 1, the Gaussian case provides intuition about the speed of the error decay (12) through the value of the constants K and M. If p is Gaussian, then the Hessian 2 log p is the negative inverse of the covariance matrix. We verify in Appendix S5 that in this case, the assumptions of Theorem 2 are satisfied. Furthermore, the constants K and M, and hence the number of discretization steps, are controlled using the condition number of Σ. We thus conjecture that non-Gaussian processes with an ill-conditioned covariance matrix will require many discretization steps to have small error. This will be verified numerically. As we now explain, such processes are ubiquitous in physics and natural image datasets.
Multiscale processes Most images have variations on a wide range of scales. They require to use many time steps to sample using an SGM, because their score is not well-conditioned. This is also true for a wide range of phenomena encountered in physics, biology. or economies [22, 32]. We define a multiscale process as a stationary process whose power spectrum has power law decay. The stationarity implies that its covariance is diagonalized in a Fourier basis. Its eigenvalues, which then coincide with its power spectrum, have power law decay defined by:
P(w) (5" + |=
(13)
where r) >> O and 2 is the maximum correlation length. Physical processes near phase transitions have such power-law decay, but it is also the case of many disordered systems such as fluid and gas turbulence. Natural images also typically define stationary processes. Their power spectrum satisfy this property with 77 2 and 2 = L for images of size L x L. To efficiently synthesize images and more general multiscale signals, we must eliminate the ill-conditioning properties of the score. This is done by applying wavelet transform.
3 Wavelet Score-Based Generative Model
The numerical complexity of the SGM algorithm depends on the number of time steps, which itself depends upon the regularity of the score. We show that important acceleration is obtained by factorizing the data distribution into normalized wavelet conditional probability distributions, which closer to white Gaussian distribution, and so whose score is better-conditioned.
3.1 Wavelet Whitening and Cascaded SGMs
Normalized orthogonal wavelet coefficients Let x be the input signal of width L and dimension d dL", with
for images. We write 2, its low-frequency approximation subsampled at intervals 2', of size (2 (2"* with xi = x. At each scale 2'-1 > fast wavelet orthogonal transform decomposes æー- into (I,, 22)) where žj the wavelet coefficient which carries the higher frequency information over 2" - 1 signals of size (2-*L)" [30]. They are calculated with convolutional and subsampling operators G and G specified in Appendix S3:
Iy 7 G G and ī, 7, G Gzz-1
(14)
The normalization factor 7, guarantees that [|| || = (2" - (3(2~T)) We consider wavelet
orthonormal filters where (G, G) is unitary operator, i.e.:
GG GG. 0 and G ld.
5

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It results that - is recovered from (z), with:
Jz = G Ijj 23 G Tj-
The wavelet transform is computed over J = log, scales by iterating J times on (14). The last zJ has a size (2 D)" x 1. Appendix S3 contains a more detailed introduction to the wavelet transform. The choice of wavelet filters G and G specifies the properties of the wavelet transform and the number of vanishing moments of the wavelet, as explained in Appendix S4.
Renormalized probability distribution A conditional wavelet renormalization factorizes the distribution p(x) of signals x into conditional probabilities over wavelet coefficients:
p(x) = a (1
(15)
where (the Jacobian) depends upon all 7/.
Although p(a:) is typically highly non-Gaussian, the factorization (15) involves distributions that are closer to Gaussians. The largest scale distribution P, is usually close to a Gaussian when the image has independent structures, because æ is an averaging of over large domains of size 2". In images, the wavelet coefficients v, are usually sparse and thus have highly non- non-Gaussian distribution; however, it has been observed [48] that their conditional distributions p, (T 12,, become much more Gaussian, due to dependencies of wavelet coefficients across scales. Furthermore, because of the renormalization, the normalized wavelet coefficients Ty have white spectrum, opposed to power-law decay for Ty, which implies they are closer to white Gaussian distribution. In statistical physics, the analysis of high frequencies conditioned by lower frequencies have been studied in [50]. More recently, normalized wavelet factorizations (15) have been introduced in physics to implement renormalization group calculations, and model probability distributions with maximum likelihood estimators near phase transitions [33].
Wavelet Score-based Generative Model Instead of computing a Score-based Generative Model (SGM) of the distribution p(x), a Wavelet Score-based Generative Model (WSGM) applies an SGM at the coarsest scale (*,) and then on each conditional distribution Py (7, for j S J. It is thus a cascaded SGM, similarly to [12, 40], but calculated on Py ( (rr)) instead py (=(x5-111r) 1x,) The normalization of wavelet coefficients Ty effectively produces a whitening which can considerably accelerate the algorithm by reducing the number of time steps. This is not possible on I5-1 because its covariance is ill ill-condiiioned. It will be proved for Gaussian processes.
forward noising process is computed on each TJ for j S J and x.
dæ -,, di + V2dūz, and dær
J, di + v J.
where the wj.: W.t are Brownian motions. Since ž is nearly white and has Gaussian properties, this diffusion converges much more quickly than if applied directly x. Using (4) or (5), we compute a score function (T,r) which approximates the score V (). For each j I we also compute the conditional score 5,,e (Z,,,1zz)) which approximates V log P5,e(
The inverse generative process is computed from coarse to fine scales as follows. At the largest scale 2', we sample the low-dimensional x. by discretizing the inverse SDE. Similarly to (6), the generative process is given by:
Fritz JJH. 5x f. 28,,, (I),1,) + 25k
FJ Lid (O Id). (16)
For j going from J to we then generate the wavelet coefficients 5, conditionally the previously calculated j. by keeping the same time discretization schedule at all scales:
س EyH + tā + 2,, (Ieee (x,)) + /20 Zj,ki
2j L14 (O,
(17)
The inverse wavelet transform then approximately computes a sample of æ5-1 from (T ry)::
- = 7, G xx, 7, G Z
(18)
The generative process is illustrated in Figure/ 1 and its pseudocode is given in Algorithm 1 in Appendix $2, The appendix also verifies that if x of size d then the numerical complexity of the generation is O(Nd), where N is the number of time steps, which is the same at each scale. For multiscale processes, we shall see that the number of time steps N does not depend upon to reach a fixed error measured with a KL divergence.

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Related work Multi-scale representations. based on wavelets or not, have been incorporated in many generative modeling approaches in order increase generation quality and sampling efficiency. Specifically, they have been shown to improve results for auto-encoders [4], GANs [7] and normalizing flows [26]. Closer in spirit to our work, [52] introduces Wavelet Flow, normalizing flow with cascade of layers generating wavelet coefficients conditionally on lower-scales, then aggregating them with an inverse wavelet transform. This method yields training time acceleration
and high-resolution (1024 X 1024) generation.
WSGM closely related to other cascading diffusion algorithms, such as the ones introduced in [12, 40, 6]. The main difference lies in that earlier works on cascaded SGMs do not model the wavelet coefficients { } but the low-frequency coefficients (x, J As result, cascaded models do not explicitly exploit the whitening properties of the wavelet transform, nor the fact that conditional wavelet distributions are often nearly Gaussian, and the mechanisms behind the acceleration remain implicit. We also point out the recent work of [16] which, while not using the cascading framework, drop subspaces from the noising process at different times. This allows using only one SDE to sample approximately from the data distribution. However, the reconstruction is still computed with respect to { instead of the wavelet coefficients.
Finally, we highlight that our work could be combined with other acceleration techniques such as the ones of [17, 27, 53, 42, 37, 43, 11, 23, 29, 41, 51] in order to improve the empirical results of WSGM.
3.2 Discretization and Accuracy for Gaussian Processes
We now illustrate Theorem and the effectiveness of WSGM on Gaussian multiscale processes. We use the whitening properties of the wavelet transform to show that the time complexity required in order reach a given error is linear in the image dimension.
The following result proves that the normalization of wavelet coefficients performs a preconditioning of the covariance, whose cigenvalues then remain of the order of 1. This is consequence of theorem proved by [34] on the representation of classes of singular operators in wavelet bases, Appendix $4. As result, the number of iterations N = T/o required to reach error is independent of the dimension.
Theorem 3. Let x be a Gaussian stationary process of power spectrum P(w)= c(E" ((<"=+<oe + with y > 0 and & > O. If the wavelet has compact support, 2 vanishing moments and is '6', then the first-order terms Ey and Es in the sampling error of WSGM (p(pp))) are such that for any > 0, there exists C > O such that for any 6, T:
(1/d)(Er + E&) SE and N - S -2
(19)
To prove this result, we show that the conditioning number of the covariance matrix of the renormal- ized wavelet coefficients does not depend on the dimension, by using Sobolev norm equivalences [15, 34]. We conclude upon combining this result, the cascading property of the Kullback-Leii Leibler divergence and an extension of Theorem the setting with non-zero mean. The detailed proof is postponed to Appendix $6.
Numerical results We illustrate Theorem 3 on Gaussian field x, whose power spectrum P has power law decay (13). In Figure 2, we display the sup-norm between P and the power spectrum P of the samples obtained using either vanilla SGM or WSGM with uniform stepsize 6; = 8. In the case of vanilla SGM, the number (C)) of time steps needed to reach small error P PI with the size of the image L (Fig. right). Equation (9) suggests that N(e) scales like a power of the conditioning number x of Σ, which is for multiscale Gaussian processes λ ~ L", for images of size L. In the WSGM case, we sample from the conditional distributions p, of wavelet coefficients T given low frequencies v,. At scale the conditioning numbers K, of the conditional covariance become iinieesioon-nnepeeeeee (Appendix S4), removing the dependency of ((s) on the image size L as suggested by (19).
increases
7

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SGM
WSGM
10
10
WISUM
1. 16
SCM
10-1
101
10-2
10-1
109
Number of discretization stops of the backward SDE
imigge size L
Figure 2: Left and middle: evolution of the error the estimated covariance matrix using either SGM or WSGM w.r.t. the number of stepsizes used in the model (T 10 fixed). Right: number (E) of discretization steps required to reach given enror 0.1 using either SGM WSGM.
Acceleration with WSGM: Numerical Results
For multiscale Gaussian processes, we proved that with WSGMs, the number of time steps (E) to reach fixed eπrоr e does not depend on the signal size, opposed to SGMs. This section shows that this result applies oon-Gaussian multiscale processes. We consider physical process near phase transition and images from the CelebA-HQ database [19].
4.1 Physical Processes with Scalar Potentials
Gaussian stationary processes are maximum entropy processes conditioned by second order moments defined by a circulant matrix. More complex physical processes are modeled by imposing constraint on their marginal distribution, with so-called scalar potential. The marginal distribution of is the probability distribution of an image pixel æzu). which does not depend upon if x is stationary. Maximum entropy processes conditioned by second order moments and marginal distributions have
probability density which is Gibbs distribution p(x) Z e F(z) with:
E(x) = 1/20'' T Cr En V(a(u))
(20)
where is circulant matrix and V: R
scalar potential. Appendix S8 explains how to parameterize V linear combination of family of fixed elementary functions. The yp model is particular example where C' = - is the negative Laplacian and V is fourth-order polynomial, adjusted order to impose that s(u) x with high probability. For so-called critical values of these parameters, the resulting process becomes multiscale with long range interactions and a power law spectrum, see Figure 3-(c).
We train SGMs and WSGMs on critical y processes of different sizes; for the score model sg- we use a simple linear parameterization detailed in Appendix S8.2. To evaluate the quality of the generated samples, it is sufficient to verify that these samples have the same second order moment and marginals as x² We define the error metric the sum of the T, error on the power spectrum and the total-varian variataia distance between marginal distributions. Figure 3-(a) shows the decay of this error as function of the number of time steps used an SGM and WSGM with a uniform discretization. With vanilla SGM, the loss has strong dependency in L, but becomes almost independent of L WSGM. This empirically verifies the claim that an ill-conditioned covariance matrix leads to slow sampling of SGM, and that WSGM is unaffected by this issue by working with the conditional distributions of normalized wavelet coefficients.
4.2 Scale- Wise Time Reduction in Natural Images
Images highly non-Gaussian multiscale processes whose power spectrum has power law decay. We now show that WSGM also provides an acceleration over SGM in this case, by being independent of the image size.
We focus on the CelebA-HQ dataset [28] at the 128 x 128 resolution. Its power spectrum has power law decay, as shown in Figure 4, and thus suffers from ill-conditioning, even though it is

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S63M
WSSGIM
1-88 1=166
10%
true
po0e quedumm
1--22
301
synthesss
10P
10P
10P
10a
Number diseretization steps of the bockoward SDE
Figure 3: Left: error between ground-truth 4 datasets in various dimensions L, and the synthetized datasets with SGM and WSGM, for various number of discretization steps. Middle: realizations of xp" (top) and WSGM samples (bottom). Right: power spectrum of yp for L 256.
non-stationary process. We compare SGM [1I] samples at the 128 x 128 resolution with WSGM samples which start from the 32 x 32 resolution. Though smaller, the 32 x 32 resolution still suffers from power law decay of its spectrum over several orders of magnitude. The reason why limit this coarsest resolution because border effects become dominant at lower image sizes. To simplify the handling of border conditions, we Haar wavelets.
Following [38], the global scores (0(r)) are parameterized by a neural network with U-Net architec- ture. It has 3 residual blocks at each scale, and includes multi-head attention layers at lower scales. The conditional scores $g (ā, (z,, are parameterized in the same way, and the conditioning on the low frequencies z is done with simple input concatenation along channels [38, 40J. The details of the architecture are in Appendix S9. We use uniform discretization of the backward SDE to stay in the setting of Theorem 2, and show that WSGM still obtains satisfactory results in this case.
The generation results are given in Figure 4. With the same computational budget of 16 discretizations steps at the largest scale (iterations at smaller scales having negligible cost due to the exponential decrease image size), WSGM achieves much better perceptual generation quality. Notably, SGM generates noisy images due to discretization errors. This is confirmed quantitatively with the Fréchet Inception Distance (FID) [10]. The FID of the WSGM generations decreases with the number of steps, until it plateaus. This plateau is reached with at least 2 orders of magnitude less steps for WSGM than SGM. This number of steps also independent the image size for WSGM, thus confirming the intuition given in the Gaussian case by Theorems I and 3. Our results confirm that vanilla SGM on a wide range of multiscale processes, including natural images, suffers from ill-conditioning. in the sense that the number of discretization steps grows with the image size. WSGM, the contrary, leads to uniform discretization schemes whose number of steps at each scale does not depend on the image size.
We also stress that there exists many techniques [18, 17, 27, 53, 42, 37, 43, 23, 11, 29, 41, 51] to accelerate the sampling of vanilla SGMs, with sometimes better FID-time complexity tradeoff curves. Notably, the FID plateaus at a relatively high value of 20 because the coarsest resolution 32 x 32 is still ill-conditioned, and thus requires thousands of steps with non-uniform discretization schedule to achieve FIDs less than 10 with vanilla SGM [38]. Such improvements (including proper handling of border conditions) are beyond of the scope of this paper. The contribution of WSGM is rather to show the reason behind this sampling inefficiency and mathematically prove the Gaussian setting that wavelet decompositions of the probability distribution allows solving this problem. Extending this theoretical result to a wider class of non-Gaussian multiscale processes, and combining WSGM with other sampling accelerations, are interesting research directions.
5 Discussion
This paper introduces a Wavelet Score- based Generative Model (WSGM) which applies an SGM to normalized wavelet coefficients conditioned by lower frequencies. We prove that the number of steps SGMs controlled by the regularity the score of the target distribution. For multiscale processes such as images, it requires considerable number of time steps to achieve good accuracy, which increases quickly with the image size. We show that WSGM eliminates ill-conditioning issues by normalizing wavelet coefficients, As result, the number of steps in WSGM does not increase with

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107
107
142
3GM
W5GM
340
140
730
Figure 4: Top. (a): Generations from SGM with 16 discretization steps. (b): Generations from
WSGM with 16 discretization steps at each scale. (c): Power spectrum of CelebA-HQ. Bottom.
(a): Evolution of the FID w.r.t the number of diffusion steps for SGM and WSGM with L = 128.
(b): Evolution of the FID w.r.t. the number of diffusion steps for SGM at several image sizes (c) Evolution of the FID w.r.t. the number of diffusion steps for WSGM at several image sizes L.
the image size. We illustrated our results on Gaussian distributions, physical processes and image datasets.
One of the main limitations of WSGM that is limited to multiscale processes for which the conditional wavelet probabilities are nearly white. A promising direction for future work is to combine WSGM with other acceleration techniques such adaptive time discretizations handle such cases. In another direction, one could strengthen the theoretical study of SGM and extend our results beyond the Gaussian setting, in order fully describe SGM on physical processes that can be seen as perturbations of Gaussian distributions.
Acknowledgments
This work was supported by grant from the PRAIRIE 31A Institute of the French ANR-19 19-P3IA- 0001 program. We would like to thank the Scientific Computing Core at the Flatiron Institute for the use of their computing resources.
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Checklist
For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] See the end of Section 4
(c) Did you discuss any potential negative societal impacts of your work? [No] Like all generative models, WCRG can reproduce or amplify biases present in the data, or could be used to generate deceptive content. We stress that this work is theoretical in nature and should not be used for practical applications without a careful consideration of the aforementioned issues.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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(a) Did you state the full of assumptions of all theoretical results? [Yes] Assumptions are stated in the respective theorems.
(b) Did you include complete proofs of all theoretical results? [Yes] See Appendices $5 and $6.
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(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)? [Yes] All the code needed to reproduce the experiments is provided in the supplemental material.
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(c) Did you report error bars (e.g., with respect to the random seed after running exper- iments multiple times)? [No] Standard errors are negligible for the comparisons we perform.
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No] We estimate the total amount of compute used during the preparation of this paper, including preliminary experiments, at around 10k hours of NVIDIA A100 GPUs.
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(c) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No]
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14


Reviews:
[{"review_id": "tU6x0BYNORB", "reviewer": "Reviewer_6XjU", "summary": "In this paper, the authors attempted to alleviate the high computation cost in Score-based generative models (SGMs) by reducing the number of time steps. In particular, the authors first found that the number of times steps in SGMs to reach a fixed error depends on the condition number of the Gaussian covariance. To break this dependence, they proposed a novel SGMs with wavelet whitening. Solid theoretical and technical contributions have been witnessed in this paper.", "questions": "Is the proposed model competitive compared with other baselines to accelerate SGMs, such as the models in referred paper \"[16, 15, 23, 46, 36, 31, 37, 20, 10, 25, 35, 45] \"? ", "limitations": "None", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1.The authors provide a theoretical analysis of how the regularity of the stein score affects the discretization of generation. They found that, to reach the fixed error, the number of time steps increases with the condition number of the covariance. \n\n2.It is an interesting and novel idea to utilize the wavelet orthogonal transform to decompose and subsample the input image, which can reduce the image size.\n\n3.The authors prove that the proposed model can fix the same discretization schedule for all image scales and accelerate the generation in SGMs, where the time complexity grows linearly with the image size.\n\nWeakness:\n\n1.The authors should provide more descriptions of the wavelet transforms in this paper. It is hard for me to understand the major idea in this paper before learning some necessary knowledge about wavelet whitening, wavelet coefficient, and so on.\n\n2.It is better for authors to display the performance of accelerating SGMs by involving some other baselines with a different perspective, such as “optimizing the discretization schedule or by modifying the original SGM formulation” [16, 15, 23, 46, 36, 31, 37, 20, 10, 25, 35, 45] \n", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "hO7_aggLwB", "reviewer": "Reviewer_FAib", "summary": "This paper introduces the wavelet diffusion model, which is theoretically guaranteed to achieve $\\epsilon$ error, independent of the step size.", "questions": "-", "limitations": "-", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "$\\textbf{Strengths}$\n- This work provides theoretically and empirically sufficient evidence that the wavelet diffusion model has a fixed number of function evaluations to synthesize an image, independently of the resolution.\n- In particular, the theory is excellent, and the arguments are solid.\n\n$\\textbf{Weaknesses}$\n- In contrast to the theory, the experiments are disappointing. The authors only demonstrate their claim on the CelebA-HQ dataset, which significantly differs from the ImageNet or CIFAR-10 dataset in its diversity. If the suggested wavelet diffusion model is truly effective, then at least the authors should have proved its efficacy on various datasets.\n- The number of sampling iterations to reach an error $\\epsilon$ is independent of the dimension only for the case of the Gaussian process. I'm a fan of this paper, but the lack of the analogous proof of Theorem 2 to the wavelet diffusion model is a bit unsatisfactory, though I understand how difficult it is to prove.\n- Though the paper has a clear and solid lesson, it is still largely missing how to use this lesson in practice. For instance, it would be much better to compare the vanilla SGM and the WSGM when the number of discretization increases to $O(1000)$. I suspect that SGM outperforms WSGM as $L>1000$, is it right? If then, the authors should clearly point out that as their limitations.", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "1zK9o0zUTu5", "reviewer": "Reviewer_4Ru5", "summary": "This paper proposes Wavelet score-based generative model (WSGM), a multi-scale score-based generative model with an accelerated sampling by reducing the number of required iterations of the score estimator leveraged by renormalized wavelet coefficients. An important property of WSGM is a theoretical analysis on the required number of sampling iterations to reach the desired error which is not dependent on the dimension of the signal per scale.", "questions": "Understanding that the high-resolution image experiments may be challenging during the discussion period, do the authors have an opinion on expected outcome? Are there practical blockers of WSGM for the scaled-up evaluation? I believe that a high-impact work related to the diffusion based models often comes with the promising empirical studies using real-world high-resolution datasets.\n\nAccelerating SGM inference is one of the most actively studied area, which naturally deviates from the controlled experimental setup of WSGM (uniform discretization). Can the authors consider more realistic experimental setups using several popular acceleration methods? What will be the implication from such assessment?", "limitations": "The authors have addressed the limitation of WSGM, that is, using uniform discretization schedule for SGM sampling which does not represent modern SGMs.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\nThe paper presents a rigorous theoretical proof and mathematical analysis on the connection between the normalized wavelet coefficients and the sampling iterations of SGM. The proposed method, the renormalized wavelet decompositions, provides a principled approach to reduce the Lipschitz irregularity [De Bortoli+2021] from the score estimator.\n\nThe experimental results on three classes (Gaussian process, physical process, and natural images) support the theoretical foundation of WSGM, where it surpasses the baseline multi-scale SGM under the same number of discretized reverse diffusion steps. Although the considered baseline model is not based on recent state-of-the-art SGM (uniform discretization for sampling, for example), the controlled experiments in a principled way are convincing enough for me to justify the rigorous theoretical contribution of WSGM.\n\n### Weaknesses\nAlthough the wavelet decomposition-based approach is well motivated with mathematical connections, I would like to point out that such approach for the multi-scale generative model has also  been studied, such as Wavelet Flow [Yu+2020] on normalizing flows, or SWAGAN [Gal+2021], for example. Although the main scope of this paper is SGM, I suggest allocating paragraphs for a description of related work using wavelet approach for the multi-scale generative model, to present a complete picture on the considered approach.\n\nThe current scope of empirical experiments is sound, but I consider it a missed opportunity to show the result on WSGM on the high-resolution images. Considering the recent progress on SGM on high-fidelity synthesis of images, WSGM is expected to have a potential to achieve a high-resolution image modeling (1024x1024 for example) which is currently not easily approachable. In fact, [Yu+2020] showcased their method as the first to unlock model training and sampling on 1024x1024 images. I believe that additional experiments on the high-resolution images will strengthen the impression of WSGM also from the practical point of view. ", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept