Analyzing diffusion as serial reproduction
Authors: Raja Marjieh
Ilia Sucholutsky
Thomas A Langlois
Nori Jacoby
Thomas L. Griffiths
Venue: Submitted to ICLR 2023 (2023)
Keywords: diffusion models
cognitive science
serial reproduction
generative models

Abstract: Diffusion models are a class of generative models that learn to synthesize samples by inverting a diffusion process that gradually maps data into noise. While these models have enjoyed great success recently, a full theoretical understanding of their observed properties is still lacking, in particular, their weak sensitivity to the choice of noise family and the role of adequate scheduling of noise levels for good synthesis. By identifying a correspondence between diffusion models and a well-known paradigm in cognitive science known as serial reproduction, whereby human agents iteratively observe and reproduce stimuli from memory, we show how the aforementioned properties of diffusion models can be explained as a natural consequence of this correspondence. We then complement our theoretical analysis with simulations that exhibit these key features. Our work highlights how classic paradigms in cognitive science can shed light on state-of-the-art machine learning problems.

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ANALYZING DIFFUSION AS SERIAL REPRODUCTION
Anonymous authors
Paper under double-blind review
ABSTRACT
Diffusion models are a class of generative models that learn to synthesize sam- ples by inverting a diffusion process that gradually maps data into noise. While these models have enjoyed great success recently, full theoretical understanding of their observed properties is still lacking, in particular, their weak sensitivity the choice of noise family and the role of adequate scheduling of noise levels for good synthesis. By identifying a correspondence between diffusion models and a well-known paradigm in cognitive science known as serial reproduction, whereby human agents iteratively observe and reproduce stimuli from memory, we show how the aforementioned properties of diffusion models can be explained nat- ural consequence of this correspondence. We then complement our thcoretical analysis with simulations that exhibit these key features. Our work highlights how classic paradigms in cognitive science can shed light on state-of-ther of-the-rr machine learning problems.
INTRODUCTION
Diffusion models are a class of deep generative models that have enjoyed great success recently in the context of image generation (Sohl-Dickstein et al. 2015) Ho et al. 2020 Song & Ermon
2019 RRmbbcc et al. 2022, Ramesh et al. (022), with some particularly impressive text-to- to-image
applications such as DALL-E 2 (Ramesh et al. (202)) and Stable Diffusion (Rombach et al. 2022).
The idea behind diffusion models is learn data distribution by training model to invert diffusion process that gradually destroys data by adding noise (Sohl-Dickstein et al., 2015 Given the trained model, sampling is then done using a sequential procedure whereby an input signal (c.g., a noisy image) is iteratively denoised at different noise levels which, in turn, are successively made finer until sharp sample generated. Initially, the noise family was restricted to the Gaussian class (Sohl-Dickstein et al. 2013 [Song& & Ermon, 2019: Ho et al. 2020) and the process was understood as a form of Langevin dynamics (Song & Ermon 2019). However, recent work showed that this assumption can be relaxed substantially (Bansal et al. 2022 Daras et al. 2022) by training diffusion models with a wide array of degradation families. One feature of this work is that highlights the idea that sampling (i.e. synthesis) can be thought of more generally
alternating process
between degradation and restoration operators (Bansal et al. 2022). This in turn calls into question the theoretical understanding of these models and necessitates new approaches.
A hint at a strategy for understanding diffusion models comes from noting that the structure of the sampling procedure in these generalized models (i.e., as cascade of -eesing-eeniiiigg units), as well as its robustness to the choice of noise model, bears striking resemblance to a classic paradigm in cognitive science known as serial reproduction (Bartlett & Bartlett] 1995] Xu & Griffiths) 2010, Jacoby & McDermoti 2017 Langlois et al. 2021). In a serial reproduction task, participants ob- serve a certain stimulus, e.g., a drawing or piece of text, and then are asked to reproduce it from memory (Figure [T]. The reproduction then gets passed on to new participant who in turn repeats the process and so on. The idea is that as people repeatedly observe (i.e., encode) a stimulus and then reproduce (i.e., decode) it from memory, their internal biases build up so that the asymptotic dynamics of this process end up revealing their inductive biases (or prior beliefs) with regard to that stimulus domain. By modeling this process using Bayesian agents, Xu & Griffiths (2010) showed that the process can be interpreted as a Gibbs sampler, and more so, that the stationary behavior of this process is in fact independent of the nature of cognitive noise involved, making serial reproduc- tion a particularly attractive tool for studying human priors (Figure [B.. The main contribution of the present paper is to make the correspondence between diffusion and serial reproduction precise

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Figure 1: Serial reproduction paradigm. A. Participants observe (encode) a stimulus and then try reproduce (decode) from memory. As the process unfolds, the generated samples gradually change until they become consistent with people's ' priors. Drawings are reproduced from Bartlett & Bartlett (1995). B. Data from real serial reproduction task reproduced from Langlois et al. (2021). Participants observed red dot placed on background image (here triangle) and were instructed to reproduce the location of that dot from memory. The new dot location then gets passed to new participant who in turn repeats the task. The initial uniform distribution gets transformed into highly concentrated distribution around the triangle's corners, capturing people's visual priors.
and show how the observed properties of diffusion models, namely, their robustness to the choice of noise family, and the role of adequate noise level scheduling for good sampling, jointly arise as natural consequence. The paper proceeds as follows. In Section 2 we review the mathematical formulation of serial reproduction and diffusion models and set the ground for the analysis that fol- lows. In Section 3 we establish correspondence between sampling in diffusion models and serial reproduction and show how it explains key properties of these models. In Section 4 we complement theoretical analysis with simulations, and then conclude with discussion in Section 5
BACKGROUND
2.1 SERIAL REPRODUCTION
We begin with brief exposition of the mathematical formulation of the serial reproduction paradigm Xu & Griffiths 2010) Jacoby & McDermoti] (017). A serial reproduction process Markov chain over sequence of stimuli (images, sounds, text, etc.) Z0 + 21 + - 2G - ... where the dynamics are specified by the incoding-decoditg
cascade of Bayesian agent with some prior
(x) and a likelihood model p()]x). The prior captures the previous experiences of the agent with the domain (i.e., their inductive bias), and the likelihood specifies how input stimuli map into noisy percepts (e.g.. due to perceptual, production or cognitive noise). Specifically, given an input stimulus xx, the agent encodes new stimulus 7++1 by sampling from the Bayesian posterior (a phenomenon known as probability matching) (Griffiths & Kalish 2007).
noisy percept dt, and then reproduction decodes it into
p(z++1l0;) = p(e, (1+41 (ole(/(41))(((( (I(+1)
(I)
( 1) (141)ddT441
The generated stimulus I is then passed on to a new Bayesian agent with similar prior and likelihood who in turn repeats the process and so on. From here, we that the transition kernel of
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the process can be derived by integrating over all the intermediate noise values against the posterior
= / p( (1e+ ( )p (xa)d
p( (1x))
= -f
p(
J (1 ( ddū T (xz+
Crucially, by noting that žt+1 a dummy integration variable we see that the prior 7 (xx satisfies the detailed-balance condition with respect to this kernel
(())+1/2))T((
p(æ, (1++))((E+1))
(3)
This in turn implies that the prior π (z) is the stationary distribution the serial reproduction process irrespective of the noise model p( x), long it allows for finite transition probabilities between any pair of stimuli to preserve ergodicity (Xu & Griffiths 2010) Griffiths & Kalish, 2007]. This insensitivity noise is what makes serial reproduction particularly attractive tool for studying inductive biases in humans (Figure (B.. It is also worth noting that another way to derive these results by observing that the full process over stimulus-percept pairs (z,,)) whereby one alternates between samples from the likelihood p() (z) and samples from the posterior p(z) (2] implements Gibbs sampler from the joint distribution p(z, 2) p())z)T(x).
2.2 DIFFUSION MODELS
We next review the basics of diffusion models and set the ground for the analysis in the next section. Following DSchl-Dickstein al., 2015 Ho et al. a diffusion model is generative model that learns to sample data out of noise by inverting some specified forward process g(=o, ..,,, .T) that ....,...........................
gradually maps data 2o ~ qa(mo) to noise sr ~ (n(xT), where qa(x) and gn (IT) are given data and noise distributions, respectively. Such forward process can be implemented as Markov chain
(( ((00....,33 (=0)) II ([5a/(e))))
with some pre-specified transition probabilities ( Ta. (1e( B.) where T2 is noise (diffusion) kernel and 8, being some diffusion parameter for which the noise distribution qn is stationary, i.c., T. (g]=))n, 9 dz (3)). This ensures that for a sufficiently large time T we are guaranteed to transform (x(x) into qr, (z). A common explicit example of this is a Gaussian kernel T.. (2+ 1: 34) = N(r v1 3, (8T) where is the identity matrix, however we will not assume that. The inversion is then done by solving a variational problem with respect to trainable reverse process pe (Zo...... .T) which itself is assumed to be Markov
T
po po........ = p(xr) II po (xx-1 1
(5)
where p(æT) = (xT), that is, the reverse process starts from noise and iteratively builds its way back data. Since we are interested in the optimal structure pa we will suppress 0 what follows. The reverse process induces a probability distribution over data p(zo) by marginalizing Equation 5 over all zt>0- The variational objective is then given by bound K on the log-likelhood of the data under the reverse process L = - gu(2z) log (((o)dda and can be written as (Sohl-Dickstein et al. 2015
L2K Σl
/
g(zo......37 ((z) log "p( (1)e)) drn ... der Han ( [1)
where Hg,, is the entropy of the noise distribution On which constant. Finally, by defining the forward posterior
ql ((cg(ox-1)q(0--1))
a(xx (10))11))(((-))) ((x:))
where q(xx-1) and q(xx)) are the marginals of the forward process (Equation at steps t 1 and f, respectively, we can rewrite as (see Appendix A)
T
K =
En, n DeL 1200:=- + C
(8)
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where DKL is the Kullback-Leibler divergence and Cg is constant. While Equation g is not necessarily the most tractable form of the bound K, it will prove useful in the next section when we make the connection to serial reproduction (see Appendix A, Eqs. 17-26 in Ho et al. 2020 for
similar derivation).
Before proceeding the next section, it is worth pausing for moment review the existing theo- retical interpretations of diffusion models and the challenges they face. Two general formulations of diffusion models exist in the literature, namely, Denoising Diffusion Probabilistic Models (DDPMs) et al. 2020 Sohl-Dickstein et al., 2013) which adopt a formulation similar to the one used here, and Score- Based Models (Song & Ermon 2019: Daras al., 2022) which learn approximate the gradient of the log- ligelikei of the data distribution under different degradations, also known as the score of distribution, and then incorporate that gradient stochastic process that samples from the data distribution. These formulations are not entirely independent and in fact have been shown to arise from certain class of stochastic differential equations (Song et al., (2000). Importantly, these analyses often assume that the structure of noise is Gaussian, either to allow for tractable ex-
pressions for variational loss functions (Sohl-Dickstein et al.) 2015), or as a way to link sampling to well-known processes such Langevin dynamics (Song & Ermon 2019). This theoretical reliance
on Gaussian noise has been recently called into question as successful applications diffusion mod-
els with wide variety of noise classes were demonstrated empirically (Bansal al. 2022| Daras let al. 2022). We seek to remedy this issue in the next section.
3 DIFFUSION SAMPLING AS SERIAL REPRODUCTION
3.1 MATHEMATICAL DERIVATION
As noted in the introduction, the sampling procedure for diffusion models is strikingly similar to the process of serial reproduction. In what follows we will make this statement more precise and show how it allows to explain key features of diffusion models. First, observe that the non-negativity of the KL divergence implies that the bound in Equation is maximized by the solution
p(x+-1|<;) = q( = J g (ā:- (0) 1));;
In other words, what the diffusion model is approximating at each step t - is simply a Bayesian posterior with the diffusion kernel ((x,) Tq (z 12 B,) serving as likelihood and the forward marginal at step / - 1, namely (1xx-1), serving as a prior. For clarity, in what follows we will denote the optimal posterior distribution at step t (Equation 9 as Pt-1 (xly) yy) and the marginal at step t - as (x).
Next, make contact with the recent literature that extends diffusion models generalized noise families which decompose sampling into a process that alternates between restoration and degrada- tion (Bansal al. 2022, Daras et al., (022) we define the sampling process for a diffusion model as a Markov sequence IT -> 2T - ZT + + Ie > 2, - Ft-1 - ... - To- where ar ~ qn (IT), & a noisy version a, under the noise kernel T. (&x (zr;; Br), and the transition it + Iz-1 is done by denoising using the posterior 1t- While this might seem at first to be at odds with the standard definition of sampling in Gaussian DDPMs whereby sampling is done by iterative applications of the posterior in Equation 9 we will show in Section 3.2 that our definition is in fact equivalent. From here, the transition kernel for the sampling process at step t - which
we denote by ps,t- (x--1/2;) is given by
/ P ( Be d,
Ps.t Pa.-1(21-1
(10)
= / T (i. ( B (a: Be) (i: -::1:: 1 1
qt
This kernel has an identical structure to the serial reproduction kernel in Equation 2 and such, the forward marginal 9t-1 satisfies detailed balance with respect to it. In other words, we have a set of eetailed-balanee conditions given by
Pstt- ( 1)ge (zt -P..t- (2))) (xz)-
(11)
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Xu
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Figure 2: Simulation results for the optimal sampling process. Data was generated by sampling two dimensional vectors from Swiss-roll distribution. We considered different diffusion noise families and computed their corresponding reverse sampling processes. Each image corresponds to the distribution of samples a given iteration in process. A. Forward process for Gaussian noise kernel. B. The corresponding reverse sampling process. C. Reverse sampling process for a bimodal kernel. D. Reverse sampling for fade-type noise with uniform stationary distribution. E. Reverse sampling for fade-type noise with mixture stationary distribution.
In particular, the last of these Ps.0 satisfies this condition for the true data distribution since by definition (x(x) ga(x). Now, observe that unlike the case in serial reproduction where had a single distribution π (xx) satisfying detailed balance at all steps, here we have sequence of such distributions. Nevertheless, can still analyze the induced sampling distribution ps (2r) in light of Equation Indeed, by substituting the Ettailed-balanee conditions we have
(=() = (p.0(20l(11) ....n.....PPP.lill PA.T-1(TT IT (p)))/g (1
(12)
=qu(zo) / Pe pe.0(21120) ga(x1) P..T ( 11
(11)
On (n(xT) dx dI....T
4T-1(xT)
From here we see that the performance of (xo) as good approximator the true data distribution ga(zo) critically depends whether the integral the right-hand-side of Equation 12 sums up to one. Observe that the integral can be evaluated step-by by-step by first integrating over Er, then TT-1 and down to 21. For the 2r integral we have
I P 1 (xT-))
On (n(x)) dxr
T(Gler-1:81) (T
/ T(Hlor; Br) (xr)drrdd. (13)
@r (xT)
Now, using the fact that gn stationary with respect to Tú. and that by construction (x) qn (2r), the rightmost intergral and the denominator cancel out and the remaining integral over & integrates to 1. Going one step further to the integral over ØT-1 (and its own latent 4) we have
f PP.T-2(0T-11 - 22) 4T QT-1(ITT (I) dxy = / qr (
oT 2;
/ T( 1i -
qr qr-2(IT-1)
(14)
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Unlike before, we no longer guaranteed stationarity for 9r-1- One trivial solution for this is to use a very strong (abrupt) diffusion schedule {3t} such that the marginals behave like qo = qu and 9t-0 = 9n- that is, within step we pretty much at noise level. From the perspective of the Bayesian posterior in Equation [ this limit corresponds to the case where the Bayesian inversion simply ignores the very noisy input and instead relies its learned prior which happens to be the data distribution for Po- Such solution, however, is not really feasible in practice because it means that the denoising network that approximates the final Bayesian posterior po must learn map pure noise to true data samples in one step, but this is precisely the problem that we'r 're trying to solve. We therefore exclude this solution.
Luckily. there is another way around this problem, namely, if the diffusion parameter 3r-1 is chosen such that it changes qr-1 only by little so that it approximately stationary, that is,
-r- (0) R3 d Br-1)r------
(15)
then the integral in Equation 14 will again be close to 1, the denominator cancels out with the rightmost integral. By induction, we can repeat this process for all lower levels and conclude that
Pa (mo) = (d(o))
(16)
Crucially, under this schedule denoising networks only need to learn to invert between successive levels of noise. This suggests that the structure of the optimal sampling processes allows one to trade a hard abrupt noise schedule with a feasible smooth one by spreading the reconstruction complexity along the diffusion path in a divide-and-conquer fashion.
To summarize, we have shown that the sampling process of an optimal diffusion model approximates the true data distribution irrespective of the choice of noise family, so long as the noise schedule {..)) } is chosen such that it alters the input distribution gradually. This result is consistent with recent work suggesting that a good scheduling heuristic minimizes the sum of Wasserstein distances between pairs of distributions along the diffusion path of the forward process (Daras et al. (2022); see also Dhariwal& & Nichol (202I). It also explains the necessary thinning in 3, for later stages of the synthesis where the marginal becomes more and more structured. In the Section 4 we provide empirical support for this theoretical analysis.
3.2 CLARIFYING THE CONNECTION TO PREVIOUS WORK
Before moving on to the empirical analysis, we complete our theoretical derivation by showing that our definition of the sampling process is equivalent to the one used in the well-studied Gaussian DDPM formalism of Ho et al. 2020). As noted in that paper (Section 3.2 of Ho et al. (2200). the posterior is taken to of the form px=) = N (2 p(x,, t), E(z,,) t)) with E(r,,)) I. where u( 4) is trainable function and / is the identity matrix. This can be also written as 1/-1 p(x,,)) 1) + dyz where ~ No. 7). In other words, the posterior is given by Gaussian distribution around function of the input p(=z,() I) with some diagonal covariance matrix with variance of- Intuitively, equivalence then follows from the fact that introducing an additional noising step in the sampling process is simply adding Gaussian noise to Gaussian posterior which corresponds to redefinition of the mean and variance parameters (which are design parameters: Dhariwal & Nichol (202T). More explicitly, our sampling process is defined as FT -> fr - TT-1 that is, we start from an initial sample IT and then successively add noise to it and then denoise it with the posterior. Now, since the initial IT ~ (a(x)) is sampled from the stationary noise distribution, adding noise to it (i.e., transitioning to irr) does not change its distribution so we can cquivalently start by denoising FT using the posterior (as in DDPM) and then adding noise and so on. Mathematically, this corresponds to applying to the generated posterior sample (i.c., the denoised .T)) a generic noisy transformation of the form x -> az + az' where z' A 7) where and a are some scaling and variance parameters (similar to Equation 2 in Ho et al. (2020). Now, when combined with the Gaussian posterior above this yields IT = ap(xT,,1 T) + aotz + az' -N(TT 1 cu([T, T), (a"of + o)) but this is equivalent to the formula used in Ho et al. (2200) up to a redefinition of the mean and variance, namely, pr pr(2r, T) - au(rr, T') and of + -o + ²². The same holds for all subsequent steps which, as in the case of DDPM, terminate with a final application of the posterior denoiser. Thus, we see that the two definitions are equivalent up to redefinition of the trainable posterior parameters.
-> To.
which corresponds to p(IT-1 (TT
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Figure 3: Effect of noise schedule on reconstruction error. A. Different noise injection schedules for a Gaussian noise kernel. B. Reconstruction the generated distribution and the true data distribution. C. A measure of inversion "complexity" as quantified by the maximum KL divergence between two consecutive marginals along the forward diffusion path.
measured by the KI. divergence between
SIMULATIONS
To test the theoretical findings of the previous section, we selected simulation setup which the Bayesian posteriors are tractable and can be computed analytically. Specifically, similar to (Soll- Dickstein et al. (2015). we considered case where the data is given by two-dimensional vectors (a, y) in the range 0.5 & y, S 0.5 that sampled from Swiss-roll-like data distribution. For tractability, we discretized the space into a finite grid of size 41 x 41 (1,681 bins) with wrapped boundaries (to avoid edge artifacts). We then analytically computed the forward (noising) distribu- tions using Equation (Figure [A] and the reverse sampling distributions using Equation I0 (Figure 2B) (we did not train any neural networks). As for the noise families, considered multiple op- tions to test whether the process is indeed robust to noise type. Specifically, we considered Gaussian noise with a linear variance schedule o, 0.03 0.04 (t/T)) (Figure AB- a bimodal noise kernel consisting of a mixture with two modes symmetrically positioned 0.07 units away from the center resulting in diagonal smearing (Figure 2 fade-type noise where there is an increasing probability
0.01 + 0.99 (t/T) to sample from a different pre-specified distribution, namely, uniform (Figure 2 and Gaussian mixture (Figure 2). T denotes the total number of steps as be- fore (See Supplementary Section B for additional details about the simulations). As can be clearly seen, in all these cases, whether we changed the noise kernel or the stationary noise distribution, the sampler able to converge
good approximation for the underlying Swiss-roll distribution (see Supplementary Figure SI for additional simulations with the posterior-only sampling scheme and Supplementary Figure S2 for examples of bad samplers due to inadequate noise schedules).

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Next, we wanted to test how the approximation accuracy of the sampler depends on the graduality of the noise injection in the forward process as captured by the number of diffusion steps. To that end, restricted ourselves to the Gaussian case and considered different noise schedules by varying the number of steps interpolating between two fixed levels of variance, that is, oz = 0.01 + 0.04 (t/T) for different values of the number of steps T (Figure (3.). To quantify accuracy, we computed the reconstruction error as measured by the KL divergence between the generated data distribution and the true data distribution, i.e., D (ga(xo)llp(<xo) for each of the chosen schedules (Figure BB). We also added measure of inversion "complexity" which measures the biggest change between consecutive distributions along the forward diffusion path, that is, max; D (9t1 (()e The idea is that the bigger this number is the less gradual (more abrupt) the noise injection is, making the learning of the denoiser harder in practice, i.e., recovering a clean sample from noisy data (Figure BC). As predicted, we see that adding more steps results in lower reconstruction error and lower complexity. In addition, we also see that beyond a certain number of steps the accuracy of the sampler saturates, presumably because the forward process has enough time to reach stationarity and to do that in a smooth way (see also Figure S3 for another similar simulation with a different noise type).
Finally, we also performed diffusion experiments with deep neural networks to validate that this dependence on the number of steps indeed occurs. Specifically, we trained Denoising Diffusion Prohabilistic Model (Ho et al. 2020) to denoise MNIST (LeCun & Cortes (005). FMNIST (Xiao) al. 2017). KMNIST (Clanuwat al. 2018), and CIFARIO (Krizhevsky et al. 2009 images For this model, the noise at step & depends on the diffusion parameter 3r = Bmin + (amaz- 3minn )mmn)+t//Zl *t//T. To investigate the effect of the noise schedule, we set Symin = 0.0001, B Amaz 0.02 and retrained the model multiple times with different number of total steps each time (T E [50, 500)). To evaluate the sample quality from each trained model, we generated 6,000 images using the same number of steps as the model was trained on, and computed the Fréchet inception distance (FID; HHeusll et al. (2017) between each set of generated images and the training set. Smaller FID values correspond to better image generation quality. We report the results in Figure A alongside sample of the resulting images for different values of T (Figure 4B). We see that similar to the idcalized case, gradual noise schedules with bigger number of steps tend improve sample quality and reach some saturation level beyond a certain number of steps. We should note that the number of steps needed for high quality samples may be reduced by an appropriate choice of sampling method, however, the overall dependence on the number of steps remains the same (see e.g., Tables 1-4 and Figures 1, A.1-3 of Watson al. 2222
5 DISCUSSION
In this work, we derived a correspondence between diffusion models and an experimental paradigm used in cognitive science known as serial reproduction, whereby Bayesian agents iteratively encode and decode stimuli in a fashion similar to the game of telephone. This was achieved by analyzing the structure of the underlying optimal inversion that the model was attempting to approximate and integrating it in a diffusion sampling process. Crucially, this allowed us to provide a new theoretical understanding for key features of diffusion models that challenged previous formulations, specifi- cally, their robustness to the choice of noise family and the role of noise scheduling. In addition, validated these theoretical findings by simulating the optimal process well by training real diffusion models.
We conclude with some remarks regarding promising future directions. First, in this work we fo- cused on probabilistic diffusion models, as these are the most common. Nevertheless, a new class of models suggests that it is possible to implement completely deterministic diffusion processes (Bansal et al. 2022). A related formulation of serial reproduction (Griffiths & Kalish (007)) where the assumption of probability matching is replaced with deterministic estimation such that the Gibbs sampling process becomes an Expectation Exppetation-MMxxiizzztooo algorithm (EM) could provide a suitable setup for reinterpreting such diffusion models. Second, in the present work we assumed asymptotic stationarity (Sohl-Dicksted et al.) 2015) as we wanted to focus on the simplest theoretical setup that would allow addressing the question of robustness to noise types.
(MAP)
We used Brian Pulfer's PyTorch re-implementation of DDPM AriinPulfer/Papee aetereeeaaaeeellemetttttaaataaaa ementat
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Figure 4: Reconstruction error of a DDPM trained on various datasets for different noise schedules (as captured by the total number of steps interpolating between two noise levels the forward process). A. Performance quantified using FID score. B. Example samples from different models.
However, recent work suggests that this assumption too can be relaxed (Watson et al. 2021 l); follow-up study could extend our work to include such setups. Third, future work could use this new perspective as an inspiration for defining better noise scheduling metrics by directly trading off intermediate stationarity, convergence rate, and sample accuracy. Finally, this interpretation also suggests that it may be possible to come up with multi-threaded sampling procedures that incor- porate multiple serial processes with different learned 'priors' and selection strategies as a way of generating samples that possess collection of desired qualities. This is inspired by the idea that serial reproduction can be interpreted as a process of cumulative cultural evolution whereby a het- erogeneous group of agents jointly reproduce and mutate stimuli so optimize them for different
properties (Xu et al., [2133] Thompson et al. 2022 van Rijn et al. 2022). We hope to explore these avenues in future research.
Reproducibility Statement. The Matlab code for reproducing all simulation analyses of the op- timal diffusion sampler discussed in Section 4 is provided with the Supplementary Material at- tached to this submission. As for necessary code for reproducing the training of the DDPM model, we simply used Brian Pulfer's PyTorch re-implementation of DDPM that can be found Further details
https: pattps://qle / com. http://laaaP..feeeeapppeeeeeeeep can be found in Appendix
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Giannis Daras, Mauricio Delbracio, Hossein Talebi, Alexandros G Dimakis, and Peyman Milanfar. Soft diffusion: Score matching for general corruptions. arXiv preprint arXiv:2209.05444 2022.
Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis. Advances in Neural Information Processing Systems, 34:8780-8794, 2021.

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Thomas L Griffiths and Michael L Kalish. Language evolution by iterated learning with bayesian agents. Cognitive science, 31(3):441 1400440040 2007.
Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge local nash equilibrium. Advances in neural information processing systems, 30, 2017.
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Nori Jacoby and Josh H McDermott. Integer ratio priors on musical rhythm revealed cross-culturally by iterated reproduction. Current Biology, 27(3):359-370, 2017.
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Thomas A Langlois, Nori Jacoby, Jordan W Suchow, and Thomas Griffiths. Serial reproduction reveals the geometry of visuospatial representations. Proceedings of the National Academy of Sciences, (8(13):e2012938118 2021.
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Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen. Hierarchical text- conditional image generation with clip latents. arXiv preprint arXiv:2204.06125, 2022.
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Under review as a conference paper at ICLR 2023
APPENDIX
A DERIVATION OF EQUATION 8
To get Equation g simply plug Equation 7 into Equation 6 which yields
((()-110))) dxa... der Hqn g
K q(2o..... :..0................. xr) log
p( dro dro... ...r Hgn
q(50: q(FO...., .......,,)" xr) log
(17)
((2)-11xz))(6
=> / (.., .T) log
T
p [wx) (
dr l.o...dayy der + Cg
where we defined Cg Et=, ( .r) log(g(=2=1)9((z))) H., which is simply a con- stant with respect to p. Next, observe that the Markovian decomposition in Equation implies that
/ " æT)dro ... ... d 1x dry = -(xx(==--)9((x--)
(18)
where ((x--1) I / (0( ... (1 Iz Im (2o)d ... dx is the marginal step t 1. By combing this with Bayes formula in Equation we can write
p(1.- 12x))
Σ
g(14 b4 (000))))))((1-1111 log
drt-1dr4 + Cg
(
g(xt=11201) p(1(~1|<1)
(19)
Σ
log
dx + Cg
2m1
Ex
+ C
which is the desired result. It's worth noting that similar result can be found in Ho et al. (2020).
B ADDITIONAL SIMULATION DETAILS
B.I SIMULATIONS WITH IDEALIZED MODEL
We ran idealized simulations in two dimensions (z.y) in the range -0.5 S v, S 0.5. We used 41 by 41 finite grid (1,681 bins) with bin width of 0.025. For the data distribution we used Swiss-roll distribution similar to Sohl-Dickstein et al. (2015). We made training a bit more challenging by avoiding bins that have zero density. This was done by interpolating the Swiss-roll distribution p with uniform distribution over the finite grid Pa so in practice we used p' 0.9 p + 0.1 Pu- We computed marginal distributions vectors over the 1,681 bins, and conditional distributions as 1,681 by 1,681 matrices. In all cases, we computed noise terms with wrapped boundaries
avoided. We used three types of noise: (1) Gaussian noise, (2) bimodal noise and (3) fade noise. The Gaussian noise over (x, y) was de- the
that boundary artifacts
fined as p(y, .r) C exp (- _ (yy--) xx)
- E(y x)), where Σ noise parameter, and C is a normalization constant. The bimodal distribution was defined as fol- I(y x + +e)+ ( I E x p))),
In diagonal matrix,
lows: p(y, x C-(exp (-1 _ (g where μ = (0.05, 0.05) and Σ = a 12. The fade noise depended on a parameter f between 0 and 1, and linearly interpolated between not changing the distribution and completely changing
some final distribution pr. It was defined by the following formula: p(y, x)) = (1 - -5y x) + fpr- pr- Where 6(z) is the Dirac two-dimensional delta function. For stationary distribution we chose either uniform (Fig. 2D) or mixture with three modes (Fig. 2D. The three equally weighted modes were located in points D3 = (0.1, 0.2), x2 = (-0.2, 0.1), 23 = (0.2, -0.2) and had covariance matrix of 0.0128 I2, 0.0128 22, and 0.0128 Iy, where In is the 2 by identity matrix and Ig is the matrix [2, 1; 1,2). We provide the code reproducing all figures and simulation as part of the Supplemental materials.
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Under review as a conference paper at ICLR 2023
A. Gaussian Noise: Sampling with Noise Injection
B. Gaussian Noise: Sampling without Noise Injection
X
X2
X
X
X;
Xg
X10
Xy
Bimodal Noise: Sampling with Noise Injection
X, X₂
Xg
X₁
X;
X
X;
Xg
Xg
X/o
Xy
D. Bimodal Noise: Sampling without Noise Injection
X,
X2
X3
X
X;
X
X;
X5
X₂
X1o
Xy
Fade Noise with Non-uniform Stationary Distribution: Sampling with Noise Injection
X1
X2
X
X
Xs
X
X;
Xa
X;
Xio
Xr
40
+
++
4
+
Fade Noise with Non-uniform Stationary Distribution: Sampling without Noise Injection
X,
X2
X4
X5
X
X₂
X₁o
Xy
Figure S1: Comparison between sampling with noise injection and without noise injection for dif- ferent noise types.
B.2 SIMULATIONS WITH DDPM
For experiments with deep neural networks, trained Denoising Diffusion Probabilistic Model (Ho et al., 2020) to denoise MNIST (LeCun Cortes] 2005). FMNIST (Xiao et al.) 2017).
KMNIST (Clanuwat al. 2018), and CIFARTO (Krizhevsky al. 2009) images. We used Brian Pulfer's PyTorch re-implementation of DDPM -https:: thub. com wwt.:ccc///raaullaccccccc anrilnnullfr
PapersRe www.mmmmleeetatmmmmmmmmmmmmmm.aa ions)) The UNet (Ronneberger et al. 2015) used by this implemen- lation is designed to be compatible with single-channel images of size 28x28 (which is standard for MNIST variants) CIFARIO images first had be resized and transformed to grayscale. Our focus is not on the technical implementation of DDPM, so we direct interested readers to Brian contains helpful documentation and commentary. The key detail is that for this model, the noise at step / depends on the diffusion parameter Br Bmin + (8maz Bmin) +1/T where is the total number of steps and 0, 1,2, 2,.... To investigate
Pulfer's repository
effect of the noise schedule, set 8min 0.0001, 3max 0.02 and retrained the model multiple times with dif- ferent number of total steps each time (T E [50, 500).. For each dataset, this process took less than 2 hours to run model, we generated 6,000 images using the same number steps as the model was trained on, and computed the Fréchet inception distance (FID; Heusel et al. (2017)) between each set of generated images and the training set of the respective dataset.
single RTX 3080 Laptop GPU. To evaluate the sample quality from each trained
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Under review as a conference paper at ICLR 2023
A Gaussian noise
1
X
X
X
X;
X;
X
Xg
X
X10
Xr
-.
-
++
*#
B. Bimodal noise
X1
X2
X
X10
Xr
C. Fade noise with a different stationary distribution
X1
X2
X
X
X;
X;
X₁
Xa
Xa
X10
Xr
4B
+
-
+
-
+01
Figure S2: Examples of bad noise schedules. Similar the other simulations had T 10 steps, but unlike before the schedules here have fixed noise parameter that did not change over the 10 steps. Gaussian noise. We used constant
0.01 in all steps. Bimodal noise. We used constant a 0.001 in all steps. C. Fade noise. We used constant f 0.0001 in all steps. (see Supplementary Section B for the definitions of these parameters).
A.
Noise schedule
0.08
Noke (signne
0.06
0.04
7x12
7.00
0.02
B.
Steps
Deseince detibbuton (b18) (b98)
0.25
Error (DKL)
0.2
0.15
10
20
30
40
50 steps
60
70
80
90
100
C.
0.32
Complexity
Complexiy (als)
0.3
0.28
0.24
10
20
30
40
50
60
70
80
90
100
steps
Figure S3: Effect of noise schedule on reconstruction error for an additional bimodal noise type.
13


Reviews:
[{"review_id": "wY_bW9GGUw", "reviewer": "Reviewer_xQWb", "summary": "", "questions": "", "limitations": "", "rating": 5, "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 paper establishes a connection between diffusion models and serial reproduction. The main result is that as long as the distributions of images along the diffusion time remain approximately stationary, then the perfect solution for minimizing the KL between the sampling process and the real process leads to samples from the correct (clean) distribution. This holds independent of the corruption type, which explains to a certain degree the success of recent diffusion models with more general corruptions.", "summary_of_the_review": "I think this is a paper in an interesting direction. However, I am concerned because of the lack of clarity in the mathematical statements.", "strength_and_weaknesses": "I think this is an interesting paper. The main take-away (which is fairly intuitive) is that the corruption should be smooth -- and that's pretty much all that matters to sample from the correct distribution at the end. The main issue I have is that the mathematical statements yielding this result are not very precise, i.e. they do not consider how error propagates in Eq. 15 and the result only holds when the learning is perfect. \n\nStrengths:\n* the paper is well-written.\n* The simulation experiments are really interesting. They show that for toy-distributions (that we can characterize their density all the way), the corruption type is not really important to reconstruct the proper distribution. I am wondering if that's true for *any* corruption with smooth transitions.\n* The topic of the paper is relevant since there have been many recent attempts to generalize diffusion models.\n\n\nWeaknesses:\n* the mathematical statements in the paper are not precise. While I appreciate the fact that the authors try to build intuition, it would be cool to have an analysis of how the errors propagate in Equation 15. This \"approximately stationary\" statement might completely blow-up the distribution we are sampling from once you integrate over all diffusion steps. How \"approximately\" is needed to get a reasonable bound?\n* the analysis seems to be held only in the case where the solution gives 0 KL. How do learning errors propagate when that's not the case?\n* There is no characterization of the conditions under which \"approximately stationary\" is not too far off. How is this connected to the smoothness of corruption? Are there toy models that satisfy this property? Are there toy models that don't satisfy this property and for them the learning fails?\n* the connection to serial reproduction is interesting, yet kind of indirect. For me, the main point is that independent of the corruption type, you still sample from the correct final distribution as long as the condition of Eq. 15 is satisfied. It is true that in serial reproduction you also sample from the priors, independent of the transition kernel and I do appreciate the connection. However, I think it is an overstatement to say that this connection yields many insights on how diffusion models work.\n", "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": "ntwmmcZE9N1", "reviewer": "Reviewer_Q4TQ", "summary": "", "questions": "", "limitations": "", "rating": 8, "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": "Inspired by a cognitive science paradigm, known as \"serial reproduction\", the authors show how the diffusion model approximates a Bayesian posterior at every denoising step, with the diffusion kernel as likelihood and the forward marginal as prior. This re-interpretation is enough to explain two properties of diffusion models that have been empirically observed: i) a diffusion model can approximate\nthe true data distribution irrespective of the choice of noise family for a fine enough noise schedule; and ii) the noise parameter has to diminish in order to reduce reconstruction error. The paper supports the theoretical insights with illustrative experiments, first on a setup where the noising and denoising distributions can be computed analytically and different noise schemes; and second, with MNIST images. One final insight provided by these experiments is that the sample accuracy saturates after some number of steps, and based on the theoretical insights, the authors suggest this is due to the forward process having converged to its stationary distribution.", "summary_of_the_review": "#### Meta-review\nThe authors have addressed my concern on the definition of the denoising process, explaining that it follows the latest research on more effective sampling for the more general cold-diffusion processes, and showing an equivalence with the original DDPM derivation.\n\n---\n#### First review\nThe paper provides a novel and insightful formulation of diffusion models, shedding light on two empirical observations that had not clear explanation with previous formulations. This is an interesting contribution that improves our understanding of these powerful models. \nMy current recommendation is based on the understanding that my comment on quality is just a matter of clarity, and not a misunderstanding or variation of the denoising process.", "strength_and_weaknesses": "**Strength**\n- The paper establishes a novel and insightful connection with cognitive neuroscience, making it accessible and hopefully inspire other researchers.\n- The insights derived from this connection are relevant, as they explain two empirically observed properties, especially the robustness against the noise family, which made previous formulations that assumed Gaussian noise incomplete.\n\n**Weaknesses**\n- See comments on clarity below.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "OvCzhEvJLh", "reviewer": "Reviewer_fE3L", "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": "This paper builds connections between existing diffusion models such as DDPM with serial reproduction paradigm. One major conclusion is that, current diffusion models can be explained as a natural consequence of this connection correspondence. Simulations on the MNIST dataset shows that the connection can be utilized for richer noise distribution family such as bimodal and fade.", "summary_of_the_review": "Given current strengthes and weaknesses, it is difficult to score a high recommendation to this paper. Some equations can be updated, richer datasets are preferred and more sampling algorithms can be used to perform rich ablation study.\n\n--\n\nI would like to rank this paper higher, after reading their detailed responses.\nMost of my questions were well answered.\nAgain, if any larger scale datasets' results are reported or shared, this paper worth an even higher score.", "strength_and_weaknesses": "Strong:\n1. The analysis and connection of between diffusion models and serial reproduction paradigm;\n\n2. Simulations on datasets to show the robustness of applying different types of noise distributions.\n\n\nWeak:\n1. Larger scale datasets and richer experiments can help improve the quality of this paper;\n\n2. The connection between diffusion models and serial reproduction paradigm seems to have less novelty in terms of methodology and it is difficult to say that new knowledge is obtained after reading this article.\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": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "Dfal48A9kQ", "reviewer": "Reviewer_g85r", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "This work discusses the connection between the sampling process of diffusion probabilistic models and a cognitive paradigm known as serial reproduction. By building the correspondence, the work tries to provide new theoretical understandings of DPMs and intuitions of improvement of key features of DPMs.", "summary_of_the_review": "In conclusion, I have raised several concerns regarding the contribution and technical soundness of the proposed method, which prevents me from giving this paper a higher rating now. I will consider increasing my rating if the authors address my concerns and clarify my misunderstandings.", "strength_and_weaknesses": "Strength:\n\n- Theoretical understanding of diffusion probabilistic models is of great research interest recently. \n- The connection between the concept of serial reproduction and DPMs is also interesting.\n\nWeakness:\n- Contribution: It is not very clear what exactly is the contribution of the paper. The paper introduces a connection between the sampling process of DPMs and serial reproduction. Drawing inspiration from this connection, some key features of diffusion models such as noise scheduling can be principally improved. However, the conclusion that \"the noise schedule should be chosen such that it alters the input distribution gradually\" is not new (Dhariwal & Nichol, 2021 https://arxiv.org/abs/2105.05233).\n- Technical soundness:\n    * On page 4, between Eq.(9) and Eq.(10), the paper describes the sampling process of a DPM as $x_t \\rightarrow \\hat{x}_t  \\rightarrow  \\cdots$. Why a noisy version of $x_t$, i.e., $\\hat{x}_t$ is needed in the sampling process? This seems not true for the well-understood DPMs where the sampling process involves only the optimal posterior distribution (as in Eq.(5)).\n    * Latter derivations all depend on the above-mentioned assumption of the sampling process. So it should be crucial to clarify whether the standard sampling process resembles serial reproduction and what Eq.(10) actually means.\n\nMinors:\n- In section 2.2, the statement \"the noise distribution $q_n$ is stationary\" is not always true. For variance exploding diffusion (Song et al. 2020), the noise distribution is not stationary since the variance does not converge.", "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": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject