Conditional Diffusion Models are Minimax-Optimal and Manifold-Adaptive for Conditional Distribution Estimation
Authors: Rong Tang
Lizhen Lin
Yun Yang
Venue: ICLR 2025 Poster (2025)
Keywords: conditional distribution estimation
diffusion models
distribution regression
generative models
manifold
minimax rate
Primary area: learning theory

Abstract: We consider a class of conditional forward-backward diffusion models for conditional generative modeling, that is, generating new data given a covariate (or control variable). To formally study the theoretical properties of these conditional generative models, we adopt a statistical framework of distribution regression to characterize the large sample properties of the conditional distribution estimators induced by these conditional forward-backward diffusion models. Here, the conditional distribution of data is assumed to smoothly change over the covariate. In particular, our derived convergence rate is minimax-optimal under the total variation metric within the regimes covered by the existing literature. Additionally, we extend our theory by allowing both the data and the covariate variable to potentially admit a low-dimensional manifold structure. In this scenario, we demonstrate that the conditional forward-backward diffusion model can adapt to both manifold structures, meaning that the derived estimation error bound (under the Wasserstein metric) depends only on the intrinsic dimensionalities of the data and the covariate.

Published as a conference paper at ICLR 2025
CONDITIONAL DIFFUSION MODELS ARE MINIMAX- OPTIMAL AND MANIFOLD-ADAPTIVE FOR CONDI- TIONAL DISTRIBUTION ESTIMATION
Rong Tang
Lizhen Lin& Yun Yang
The Hong Kong University of Science and Technology martangeust...
University of Maryland, College Park (lizhenC1, yy84) 4 @umd. edu
ABSTRACT
We consider a class of conditional forward- diffusion models for con- ditional generative modeling, that is, generating new data given covariate (or control variable). To formally study the theoretical properties of these conditional generative models, adopt statistical framework of distribution regression characterize the large sample properties of the conditional distribution estimators induced by these conditional forward-backward diffusion models. Here, the ditional distribution of data is assumed to smoothly change over the covariate. In particular, our derived convergence rate is minimax-optimal under the total varia- tion metric within the regimes covered by the existing literature. Additionally, we extend our theory by allowing both the data and the covariate variable potentially admit www-dimesiionl manifold structure. In this scenario, we demonstrate that the conditional howaadd-bckwaan diffusion model can adapt to both manifold structures, meaning that the derived estimation error bound (under the Wasserstein metric) depends only on the intrinsic dimensionalities of the data and the covariate.
INTRODUCTION
Conditional distribution estimation aims to estimate the distribution (or its density if exists) of a response variable Y given some covariate or predictor variable X. which is fundamental problem in statistics with wide applicability in finance, economics (Li & Racine| (2077), biology Krishnaswamy al. (2014) and social science, to name just a few. The conditional distribution provides a full characterization of the dependence structure of the response variable on the predictors, which allows one gain deeper insights about the data characteristics beyond those from a simple mean regression model, such as capturing uncertainty and addressing multiple-modality. Conditional density estimation has received significant attention from both statistics and machine learning community with proposed estimators ranging from classical nonparametric estimates such those based smoothing techniques (Rosenblatt) 1969 Fan & Yim 2004] Holmes et al.] 2007 Bashtannyk & Hyndman 2001), Bayesian nonparametric estimates (Norets & Pati 2017). to recent methods
that utilize deep neural networks (Rothfuss et al. 2019).
Although there is rich literature conditional distribution estimation, many existing methods, such as the classical nonparametric estimators based on kernel smoothing (Bashtannyk & Hyndman] 2001 Izbicki & Lee 2016: Li al., (022), suffer from some limitations. One notable drawback of these classical methods is the requirement for the existence of a conditional density function, which is often violated when the response variable Y contains discrete components or itself a dige-diiensional object with low wiw-dimensional structures. As result, most classical methods can only deal with data with small dimensions, and their performance deteriorates quickly when the dimension increases, thus they suffer from the curse of dimensionality. In addition, these classical estimators generally do not have the ability to adapt to any potential intrinsic structure, such as the manifold structure of the data, characteristic of many modern high- nigh-diimnssioal datasets. For distribution or density estimation in the unconditional setting, estimators based on deep generative models appear to overcome the aforementioned challenges. Constructing a distribution estimator implicitly by specifying its data- generating process naturally allows singular structures in the data. Additionally, an emerging body of literature on the theoretical understanding of deep generative models, including diffusion-bast -osded

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models (Chae et al., 2023 Dahal al. 2022 Chen et al. 2023a Tang & Yang 2024). demonstrates that such models provide an estimator of the underlying distribution with convergence rates dependent only the intrinsic dimension of the data.
Motivated by advancements in deep generative modeling, in this work we explore conditional diffission models based deep neural networks for conditional distribution estimation, accounting for possible oow-dimensional manifold structures on either (or both) the covariate X and response Y. Unlike other generative model estimation procedures, such as GANs (Gooddellow al. 2014) and variational auto-encoders (Kingma & Welling 2013), which explicitly incorporate low-dimensional structures by operating in or maintaining a dowedimensional latent space, diffusion models operate directly in the original ambient data space. Therefore, it is both interesting and important to formally study whether they can still adapt to any low-dimensional structure, if present. Towards these goals, we consider conditional distribution estimators implicitly defined through class of conditional diffusion models with conditional score matching. We investigate theoretical properties of such estimators through finite-sample analysis of their statistical error bounds with respect to various metrics and examine their dependence on the intrinsic dimension and certain smoothness characteristies of the data. The key findings and contributions of our work can be summarized in the following:
Our rates are minimax-optime optimmll under the total variation metric in the classical setting when the conditional distribution admits smooth density function that also varies smoothly across different covariate values.
Our models encompass unconditional distribution estimation and nonparametric mean regression as special cases. When restricted the former, our derived estimation error bounds achieve the minimax rate under both the total variation and the Wasserstein metrics. For the latter, our rates recover the classical minimax rate of nonparametric regression under the L2 risk.
Our results show that conditional diffusion estimators are adaptive to intrinsic manifold structures when either (or both) the covariate X and response Y are concentrated around some lower- dimensional manifold; thus, our model can handle high-dimensional distribution regression with covariates exhibiting low-dimensional structures.
Other related works. There is vast literature nonparametric conditional distribution estimation. In addition to smoothing- methods such as the ones employ kernel smoothing or local polynomial regression (Fan & fim/2004), there are other approached based on mixture model (Bishop (2066)
Gaussian processes (Payne et al., 2019) Dutordoir et al., 2018), and nonparametric Bayes (Chung &
Dunson 2009 Dunson et al. 2007), among others. There has been recent line of work that utilizes deep generative approach for conditional sampling such as Zhou et al. (2022) and et al. (021).
Zhou al. (2022) utilizes conditional GAN based approach and derived a consistent conditional density estimator but no convergence rates or error bounds are provided. There is also a growing body of theoretical research on diffusion generative models, though most have not considered the
conditional setting as we do, such as Oko et al.] (2023); Chen et al. (2023a); Wang et al. (2024);
Li Yan (2024); ; De Bortoli et al. (2021): Lee et al. (2022); Chen et al. (2022); Lee et al. 2023 )
Chen et al. (2023b) Tang & Yang 2024 Li et al. (2024b) and Li et al. (2024a). Among these, Oko ct al. (2023); y Chen et al. (2023a) and lang & Yang (024)) study the approximation error and gencralization ability of diffusion models for estimating (unconditional) distributions when data exhibits cow-dimensional structures, which are shown to attain the minimax optimality in the 1- Wasserstein metric for distributions supported on low-dinensional hyperplanes (Oko et al. 2023) and general submanifolds (Tang & Yang 2024). Some works, such as Li & Yan (2024) and De Bortoli (2022), explicitly leverage low-dimensional data structures during the generative sampling phase of implementing diffusion models. In scenarios where a covariate is available, some recent works (Chen) al. 2024, Fu al., 2024) explore the theoretical properties of conditional diffusion models; however, they do not account for manifold structures when deriving their convergence rates.
2 FORWARD-BACKWARD DIFFUSION MODEL AND ITS CONDITIONAL VARIANT
In this section, will begin by reviewing the herward-bacwwart diffusion model for (unconditional) distribution estimation. After that, we will introduce its adaptations for estimating conditional distributions under the statistical framework of distribution regression.
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2.1 BACKWAR DIFFUSION MODEL WITH SCORE MATCHING
Forward-backward diffusion models (see, e.g., Ho et al. (200); Song et al. 2020); Nichol & Dhariwal (202T); Song & Ermon (2019) have emerged a new state-of-the-art class of generative models for estimating and generating samples from an underlying data distribution je' on a data space My с Rr, In typical forward- warkdadd diffusion model, two diffusion processes are utilized collaboratively: one process is designed for the estimation of time-dependent score function describing the direction towards a high data probability region, while the other is for generating samples through hime-inhomogeneous process, based on the estimated score functions. Consequently, this model overcomes the slow convergence issues (e.g., due to the multimodality of μ²) commonly observed in models that rely solely single diffusion process, such as Langevin diffusion. Throughout the remainder of this paper, the term "diffusion model" specifically refers to the waardddd diffusion model.
More concretely, the first diffusion process in the diffusion model, often referred to as the forward diffusion, employs simple diffusion starting from μ that admits closed-form solution and converges exponentially quickly to its limiting distribution. In this paper, we focus on the commonly used Ornstein- Uhlenbeck (OU) process as the forward process, which gradually injects Gaussian noise into the data and is described as a stochastic differntial equation (SDE)
dy - Y, dt 25, dB, Y PO~p"
where {B, : t> 0} denotes the standard Brownian motion in Dy and {ō; 2 0} is some (possibly time-dependent) drift coefficient. Note that the OU process admits the closed form solu- tion y.= m, Y. So mi m. ((m, y, I. where my exp -o 5g ds) and of distribution of Y.. denoted as pr, converges exponentially quickly to its limiting distribution Pso = (0, (Dy) under the Kullback-Leibler divergence.
220, dB,; thus, the conditional distribution of Y, given Y, = y is
m Therefore, the marginal
The second diffusion process in the diffusion model, usually called the backward diffusion, reverses the forward diffusion and can be written as the following SDE,
a4. = i + 28T V log Y x)] de + , Yo Pr-
(2)
Under mild conditions on μ4 (Song et al. 2020 Haussmann & Pardoux 1986) (valid for our setting),
the distribution of Y, 7 is pr so that YT Po n²" Since pr is close to po: = (0, Dy one can instead initialize the backward diffusion using the easy-to-sample distribution Pro. i.e. set
Yo NO, D. The drift term of the backward diffusion depends on the time dependent score function log Pt defined through the forward diffusion; therefore, the forward and the backward diffusions together constitutes generative model for sampling from ja"
Equations ① and (2) define the barward-backwand diffusion model at the population-level. a standard statistical setting, we utilize independent and identically distributed (i.i.d) samples (Y;) ;= from μ² to estimate the time-dependent score function V log ps in the backward diffusion. The estimation is achieved by the so-called score matching (Song & Ermon 2019 Vincent 2011). Specifically, one first numerically simulates for some sufficiently large time horizon T from a sample- level forward process {y, E [0. T]}. which is SDE (I) initialized at the empirical distribution of the data, that is, yo ~ Pa = n = Sy, with 8, denoting the point mass (Dirac) measure at point y. One then uses score approximating map So t) over space and time, indexed by parameter f, e.g., (deep) neural networks with controlled depth and number of non-zero parameters, estimate the true underlying score function log Pt (y). by minimizing the following (L2-)score matching risk (over 0):
Σx=
[" Egr-pi 10)) 2o,
R [1] t) log [0] X(t) dt,
where pr y) denotes the distribution of Y: : in forward diffusion I initialized at Y, y for any y E My. Here, X(t) is a weighting function (over time), and t is an early-stopping threshold for preventing the explosion (singularity) of the score function as - 0 commonly employed in practice (Song & Ermon| 2020 Oko et al. 2023). Equivalently. this score estimation step can be efficiently carried out in practice by simulating trajectory {vr 2 0} from SDE (I) starting from each data
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point Y,, that is. Y1 P)). One then uses So(y.t) t) to match the ensemble of all sample score
Y,).
functions V log Pr (ye. Y;) over all n simulated trajectories, by minimizing the following empirical risk function (over 0):
Berpp( [[So(();))) V log YOII X(1) dt.
We will adopt this statistical formulation of score matching to facilitate our theoretical analysis, leveraging tools from statistical learning theory.
Finally, let S(xr, 1) Sy(r; 1) denote the resulting score estimator. The distribution estimator of Jr based on the forward-backward diffusion model is then Pr-r, where pa represents the distribution of Y for & [0, T 7]. and Y follows the SDE below with plugged-in score,
d Y = 1 Yi + 26
dt + V2br-: Yi NO00 IDY).
2.2 CONDITIONAL KORWARD-BACKWARY DIFFUSION MODEL WITH CONDITIONAL SCORE MATCHING
notable characteristic of diffusion models is their flexibility in incorporating a covariate or control variable, denoted as X Mx Dxx guide the generation of new data Y E .My RDDr This can be equivalently formulated as the statistical problem of generating samples from the conditional distribution 1= of Y given X x for any covariate value x E .Mx. To facilitate the borrowing of information across different covariate values, commonly assumed that the conditional distribution My varies smoothly with E Mxx This assumption underlies statistical framework often referred to distribution (density) regression in the literature (Bashtannyk & Hyndman [001 Izbicki & Lee, 20166; Li et al.] 2022). Distribution regression expands the classical (nonparametric) mean regression by estimating not only the conditional expectation F[Y| X = ær] as smooth function of at but also the entire conditional distribution 1 that varies smoothly with .r. Compared to classical distribution regression methods based on kernel smoothing, which require Hy to admit density function (thus termed density regression in the early literature, see Bashtannyk & Hyndman (200T); Izbicki Lee (2016)), conditional diffusion model-based methods are more flexible. They can be more generally applicable to cases where py is supported therefore singular; see Section 3.2 for details.
low-dimensional manifold and is
A natural way to convert a diffusion model into a conditional diffusion model for sampling from 1x is to replace the (unconditional) score function V log Pt in the backward diffusion with some conditional score function V Plgge(( (: satisfying po (>) "y oa and pr xx) = Poo (-). Earlier literature considers the so-called classifier guidance method for estimating the conditional score using Bayes' rule (especially when covariate is discrete categorical; see, e.g.. Dhariwal & Nichol (002 Song et al. (2020)):
V log x:) = V log + V log (x (v)),
Here, the first term. log Pr (v:), is the unconditional score function defined in the forward diffu- sion (T) in the unconditional diffusion model, which can be estimated via score matching. The second term, V log (xr w/.,, is the likelihood function of an external "classifier" trained predict x from St- In other words, the classifier guidance method incorporates information from the covariate value x into the unconditional score function through the gradient of an external classifier to guide the backward diffusion for generating samples from 1Lyz"'
In this work, for the sake of theoretical simplicity and to avoid the need analyze and quantify the statistical accuracy of an external classifier, we consider a classifier-fres method, which at the population level directly applying conditional score matching method (Hyvärinen & Dayan 2005
Vincent 2011: Batzolis et al. 2021 Tashiro et al. 2021) based on simulating Y, from the same (marginal) forward diffusion
- dt 25 dB,, Y ny.
(5)
Here, μy denotes the marginal distribution of Y. Consider a generic conditional score approximating map Soly, xr 1) over space, covariate value and time, indexed by parameter 0. In the conditional
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score matching step, minimize the following Joo-iiinntl score matching risk over 8:
J. By (-)) (4.00)=wx,, [1] So 2, f) V log Po ( bb)) X(t) dr,
(6)
where 1x,, denotes the joint distribution of (X, Y), and pe(: l is the distribution of Y, in forward diffusion initialized Yo y for any y E Myy Here, the early stopping threshold and the weighting function A(t) are defined as before. It is straightforward to show (see Lemma C212 in Appendix C.2) that if So (y,,r, can range over all possible conditional score functions, then the global minimizer of the preceding risk function is precisely the true underlying conditional score function V log Pr (yt z). Here, Pr( z² Byo [pr(ya yo)] denotes the conditional
distribution of Y, given X = after marginalizing out Y o. where (X, Yo) ~ Mxxy and Y, follows forward diffusion starting from Y.. 0.
The corresponding conditional backward diffusion for sampling from 21 is then given by
dł f/r Y 1/= + 2 log log d/ [x) dt + 22ō dB Y ~ prppp-1a (7)
Here, we added a subscript æ in the notation f t/2 to indicate that, unlike the forward process 5p. the backward diffusion process is z-dependent. Similar to the (unconditional) diffusion model, choosing a sufficiently large T can guarantee pr(- [x) = (.o(.) No, IDy), which is independent of x. We will refer to equations (5) and (7) as the conditional (forward backadadbb diffusion model for the remainder of the paper.
To estimate the conditional score Vogg (- ar) using i.i.d. observations {(x, Y:) sampled from the joint distribution ('xyy of (X, Y) under the statistical framework of distribution regression, one can again minimize the following empirical version of conditional score matching risk (6).
Σ
E Eu- (II (ve
Pog X.)I A(t) dt.
(8)
Finally, let S(y, x, L) Sz(y, x, 1) denote the corresponding conditional score estimator. For each Mxx, the conditional distribution estimator of FY based the conditional forward-backward
diffusion model is then p- (-| 2)), where (r I) is the distribution of Y 4/2
for t E [0, r]. and
fi f/r follows the SDE below with plugged-in conditional score,
dY! UE [ōT Fi 1/w 26 ( 2, t)] dt + 2 YI Yf12 ~((, (9)
2.3 NEURAL NETWORK CLASS FOR CONDITIONAL SCORE FUNCTION APPROXIMATION
Definition (neural network class): A class of neural networks (HH, W, R, B, V) with height H. width vector W = (W1, W WH 11).). sparsity Rt norm constraint B, and function norm con- straint V is defined as Φ H, W, R, B, V) = {f(.) = (A(H) ReLU(-) ()(H)) ((H))....... ................................ (A(2) ReI.U(-)+ +
b(2)) (A b()),, so that A W.. 5() RW++; Σ" (1 llo 1100)) <
R; max; || V [| S B; || S V.. where ReLU(x) max (0, x} the rectified linear unit activation function and the max function is applied elementwise to a vector.
According to Oko et al. (023) and Tang & Yang 2024), the smoothing effect of gradually injecting Gaussian noise into the data distribution during the forward diffusion (5) suggests that the optimal size of the neural network for effectively approximating V log pe(:-1.20 x) should decrease as f increases. This observation motivates us to consider a neural network class whose size diminishes over time. For technical convenience, we discretize the time and adopt the following piece-wise constant complexity neural network class, as utilized in Tang & Yang 2024):
SNN S(,, x, ΣT Scy, x) (t,- St (4) S E $ (H, W, R,, B, V. E }
where 7 4 l T. 1et1 for any [Z]. and 7 2 2-IT for some I to be determined later. We have also conducted simulation study (see Appendix A) to demonstrate
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the effectiveness of this theoretically guided neural network architecture compared to standard single ReLU neural network (across both space and time). In this experiments, we consider cases where, given the covariate X, the response Y is supported on different (tilted) ellipses depending on the values of the covariate. Consistent with our theoretical findings, the simulations show that incorporating the piecewise structure into the neural network results in more accurate estimation of the conditional distribution. Recall that Syy. z, t) denotes the conditional score estimator, defined as the minimizer of the conditional score matching risk (8) over the class SNN with the weight function X() = 1 (although any other weights such as X(1) = would also suffice). We define a (truncated)
estimator [y |a for 12 ze as the distribution of Y T-rlæ 1[1 T-r|u oollo I L), where (L, T') are large enough constants so that My C. B (0, L/2) and pr ( x) R (0, bDy). Here, we truncate the random variable Y T-rlz to guarantee bounded support for the induced distribution estimator Pa which is solely for technical reasons.
3 MAIN THEORETICAL RESULTS
In this section, we present our main theoretical results characterizing the statistical accuracy of the conditional diffusion model for conditional distribution estimation (or distribution regression) under two scenarios. In the first scenario, we consider the classical density regression setting where the conditional distribution 1 admits a density function relative to the Lebesgue measure of the data
space My C Ry We derive the convergence rate of the estimator under both the total variation and the Wasserstein metrics. In particular, our derived convergence rate is minimax-optimg the total variation metric within the regime covered by the existing literature Li al. (002) (see Remark forr further details), and extends to broader regime. In the second scenario, we consider a nigh-dimensional distribution regression setting where both the response variable Y E Py and the covariate variable X E Px reside in high-dimensional ambient spaces characterized by large Dy and Dx- However, the covariate space Mxx of X has an intrinsic (or effective) dimension dx that is significantly smaller than Dx Furthermore, given any E Mxx the corresponding data space of Y. denoted as My can be x-dependent and also has small intrinsic dimension dy < Dy We demonstrate that the conditional diffusion model effectively adapts to the underlying manifold structures of both the data and the covariate variable. Specifically, we show that the convergence rate of the estimator depends solely the intrinsic dimensions (dy, dx), rather than the ambient dimensions (Dy,Dx),
under
In the following, we denote drv (μ, v) and W1 Wi(,,,) 2) the respective total variation distance and the 1- Wasserstein distance between two distributions μ and v. We denote M = {(xx y) x E Mx y E My the joint space of (X, Y) and My U My as the (marginal) data space. We the notation V b and a b to denote the respectively shorthand of max(a, b) and min b). For a sequence {ax n 2 1}. we use ((a,) to indicate the order of an to a multiplicative constant as + and (aa,) to indicate the order of an up to a multiplicative constant and logarithmic terms of n. Similarly, we use (axx) and Ö(a.) to indicate most of order an..
3.1 CLASSICAL DENSITY REGRESSION IN EUCLIDEAN SPACE
In this subsection, we consider the classical density regression setting where both the covariate space Mx C Dx and the data space My C Rv are compact subsets (with open interiors) of the Euclidean spaces, and the conditional distribution pLyz admits a density function, denoted as 4² .)) relative the Lebesgue measure of Dy, For simplicity, we assume My = [] and Mxx = 1] In order to derive a non-asymptotic bound to the expected total variation distance and Wasserstein distance between the conditional distribution estimator f and the target " with the expectation taken over X ,x, we impose certain smoothness condition to the condition density function μr (y/ x:) relative to x) as in the classical density regression literature (Li et al. 2022 Bilodeau et al., 2023). Specifically, we assume that μ (y, x). as a function of (y. x). is Cay -smooth, where ay and oxx quantify the respective smoothness in the response variable and the covariate x. Note that a function fly. x)) being C -smooth implies that, around any point (Yo, Zo). there exists local polynomial approximation of f. with an approximation error of order O(1 - yo | + || - Zp [[xx ); rigorous definition can be found in Appendix B. Formally, make the following assumptions.

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Assumption A (smoothness and lower boundness of py 2):: For each z E Mx,, the conditional distribution Ja admits a density function 1 [x) that Car -moooh in (y, x). Moreover, there exists a positive constant e so that (² (r) 2 c holds for any E EMx,y E My.
Assumption B (regularity of the drift coefficient): The drift coefficient 6, is infinitely differentiable and there exist positive constants C1. C2 so that C1 S 6, S 0g for any t 2 0.
Here the lower bound requirement of 1² x)) is a commonly made assumption for distribution estimation in the classical density regression literature, and is also imposed in Oko et al. (2023): Tang
& Yang 2024) for analyzing (unconditional) diffusion models. In practical applications, the drift coefficient óg typically chosen positive constant independent of t, thus naturally satisfying
Assumption B.
Theorem 1 (Density regression in Euclidean space). Suppose Assumptions A and B are satisfied. Let E1 = - DDay+Dy+ + ex Dx). If we take T = ( ( ((((++1))))
), T = *(log n), and neural xx
Dx
Dx -D ax
network sizes satisfying H1, = (log'" n), ||Will = 66n 2ax +Dx 2nx ;
Bax 12x
), R.
(n x xxx+xxx
-x Dx
- -D
Dx Dx
) B, Oxp(O(1og))) )) and V @((
log tAIi
for i E [
with I = logz( ()), then it holds with probability at least n- that
24 Dx
Exwuy (pr) [3) Ö
my
(10)
and E Wa 1 13) = 0 n
2x a
n
px By oY
(11)
Remark 1. In the special case when ax E [0, 1), et al. based estimator can achieve the same convergence rate Top in the total variation metric as our conditional diffusion model-based estimator; furthermore, this rate is shown to be minimax-optimal. Therefore, our result implies the minimax- cotiialatyy of the conditional diffusion model for density regression in the regime where a E [o, 1], although our upper bound is also applicable to ax > 1.
2022) shows that a well-designed kernel-
Remark 2. When specializing to the unconditional case with no covariate (that is, taking Dx = in Theorem D. our derived estimation error bounds (TU) and (IT) reduce respectively to the minimax rate of (unconditional) distribution estimation under the total variation metric and the Wasserstein metric (Liang 2021- Tang & Yangl 2023). However, unlike the upper bound proofs in [iang (2027); Tang& & Yang (2023), which rely on generative adversarial network (GAN) type estimators, our proof demonstrates that the diffusion model is also minimax-optimul for distribution estimation. In particular, our results recover those from Oko et al. (2023) as a special case (by taking Dx = 0).
Remark 3. The derived W, error bound 11b comprises two terms n. n= (2xxx / +Dx) and
-ay +1)/2yy Pxay +Dy) The first term resembles the classical minimax rate of nonparametric ax
regression under the L2 risk and can be interpreted as mainly capturing the estimation error related to learning the dependence of the response variable Y on the covariate X, so that it only depends on the smoothness and intrinsic dimension of X. Technically, this term arises from the approximation of the conditional score function for large time t, where finer details of the conditional distribution in Y have been smoothed out and only the global dependence on X matters. The second term reflects the estimation error of recovering the entire conditional distribution of Y given X, and depends on characteristics related to the response variable Y, such as the smoothness ay of the conditional density function and the dimension Y. Interestingly, the derived rate suggests a phase transition phenomenon: if the dimension of the response variable Dy satisfies Dy S 2 - Dx then the estimation error under the W1 metric remains of order n ax smoothness level ay, and the Wi estimation error is dominated by the error of capturing the global dependence of the response variable Y on the covariate variable X; otherwise, the W1 estimation error is influenced by both the smoothness dy of conditional density on Y and the smoothness ax. which captures the finer details of the conditional distribution of Y given X.
ax regardless of the
Remark 4. A recent related work Fu et al. (2024) also explores theoretical properties of conditional diffusion model, and show the minimax optimality of diffusion model under the total variation distance.
Herre we the notation [Z] (1,2,...,I)
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In our work, we allow the conditional distribution of given X to have difference smoothness levels ar and x on the response Y and covariate X; in comparison, Fuu et al. (2024) assumes the two smoothness levels are the same. The varying smoothness levels can allow for the applicability of the results in more general settings. For instance, when specializing to the mean regression case, where the conditional distribution is a Gaussian distribution centered the evaluation of an ax smooth regression function over RDxx, our derived estimation error bound (10) under drv (that is,
taking ay + DO in Theorem 1 can recover the classical minimax rate 12 2 DDx of nonparametric regression under the La risk.
3.2 HIGH-DIMENSIONAI DISTRIBUTION REGRESSION WITH LOW-DIMENSIONAL MANIFOLD STRUCTURES
In this subsection, consider the case where both the covariate space Mx and the response space My may have low-dimensional structures in their respective ambient spaces R Dx and Ry For the covariate space MMx, the low-dimensional structure imposed in terms of its upper Minkowski dimension, which is related to the growth of its packing number (see Assumption C). This low- dimensional structure is notably less stringent than typical manifold assumption, as it does not require any smoothness properties of Mx. For the response space My, since we allow the con- ditional distribution 1r have different supports, denoted as My]z. for each E Mxx, we can decompose My as UnEMy My In our theory, we require each "section" MMyj to be a smooth submanifold in F Py; additionally, we require My to vary smoothly with (sce Assumption D). We list the concrete assumptions as follows.
Assumption C (intrinsic dimension of Mx): .Mx is compact set in R Dx and there exist constants (Ci, (2) that for any e >> O, any e1 sD, and any x E M, we have M( (2, e), el)1 == max(m: 3 -packing of BM (x,=) =) of size m] I C2(
This assumption naturally holds if Mx is a compact subset of dx- di--immesioona hyperplane, or more generally, a compact dx dimmennnio submanifold embedded in Rx with its react bounded away from zero. Therefore, the constant dx in the assumption can be interpreted as the intrinsic dimension of Mx. Next, we introduce our assumption on the conditional distribution - For easy understanding, we present an informal assumption here and postpone the more rigorous and detailed version to Appendix in the supplement. Recall that M denotes the joint space of (X, Y).
Assumption D (smoothness of Mrix. informal version): For any x E Mx. My is a dy- dimensional submanifold in Pr, and p admits a density with respect to the volume measure of My which is uniformly lower bounded away from zero Moreover, for any w = (o).30) E M and x E Bax (Iu, ro), there exists an encoder-decoder pair (Q+ (v), (z))), such that (- maps y E My (yo, ro) to a low-dimensional latent variable 2 E d and G%)()) reconstructs the data y through the latent variable 2. Here, the decoder G (z) 8 smooth in (2, and the induced (local) conditional density function 8td (2|ar) of the latent variable 2, as the pushforward measure through the encoder (v) of the restriction of the measure 1z7 onto B By (yo, ro). is
Çay,ax -smooth in (z,x).
Constants (3y, Bx) in Assumption D quantify the smoothness of the submanifold My which is the image (range) of the decoder Cz((), Specifically, index By characterizes the smoothness level of the manifold Mylz supporting the response variable for any fixed x E MM: and index Ax characterizes the smoothness level of the section manifold |r in x, that is, how similar My and My when is close to z' in MMx. In contrast, constants (ay, , axx in Assumption D quantify the smoothness of the conditional distribution PA].. or more precisely, the corresponding conditional density function on its supporting manifold My Specifically, the index ay characterizes the smoothness level of the conditional density function the response variable for any fixed x E Mxx. while the index ax captures how smoothly the conditional density function changes with z. The
A P C S is a e-packing of S if for every x' E P we have ||æ x' 'The reach of a closed subset A C De is defined as TA = inf dist(p, (e)( inf () di A), where dist(z, A) infpe ||p - 2|l denotes the distance function to A, and Med(A) (A) the medial axis of A consisting of the points that have least two nearest neighbors.
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constant dy in the assumption can be viewed as the intrinsic dimension of the response variable Y. Similar to Tang & Yang 2024), the requirements on the conditional distribution | in Assumption D are stated in a local manner since a manifold, as a topological space, is only locally defined. In fact, many common manifolds, such spheres, do not admit global parameterization (or encoder-decoder representation).
Theorem 2 (Distribution regression manifolds). Suppose Assumptions B. C and D satis- fied with By 2 ay +1+1 and 3x 2 ax + a². Let E =
=n== nn/ (2ar +dx + ax dx) If we
take t = ( 2lar+1), T ((log n), and neural network sizes satisfying Hi = A(log" n). xay R₁ = O(n xux t2 xxx+"x
-oxdy || = ( 2n xxxxxxx t, xxx="x A -dy oydx B, exp(O(log n))) and V @
A -dy xx
ayax
ox
logn tAI
for i E [ with I logz( ()), then it holds with
probability at least 1 - n= that
Ex-uy [Wi (1xp My
2+ ax
Vn
ax
ay
Remark 5. Since fr and 12 are almost surely mutually singular measures (supporting on differ- ent submanifolds), the total variation metric is always 1 and, therefore, not suitable for quantifying their closeness. Consequently, our error bound is stated only in terms of the Wasserstein metric. In fact, even in the (unconditional) distribution estimation case, Tang & Yang (203) shows that estimator can achieve estimation consistency under the total variation metric.
Remark 6. Theorem 2 demonstrates that the statistical accuracy of the conditional diffusion model depends solely on the intrinsic dimensions (dx, dy) rather than the ambient dimensions xDx, Dy) modulo multiplicative constants and logarithmic terms. This indicates that the conditional diffusion model can adapt to the low-dimensional manifold structures in both the response and the covariate variables. In particular, when there is no lindeinnsional manifold structures (i.e., Dx dx and Dy = dy ), the W1 error bound in Theorem 2 recovers the W1 error bound in Theorem 7 in the classical density regression.
Remark The same remarks after Theorem Iin the previous subsection also apply: when specializ- ing to the unconditional case with no covariate (that is, taking Dx O in Theorem error bound reduces to the minimax rate of (unconditional) distribution estimation under the Wasserstein metric (Tang & Yang (023) with (sufficiently smooth) manifold structure; when specializing to the mean regression case, our error bound can recover the classical convergence rate n 22 tax of nonparametric regression when the covariate X is supported on a dx-dimensional submanifold
Yang & Dunson 2016 Jiao et al.) (023).
Remark 8. Several works (Chen al. 2023a Oko al. 2023) have also studied the unconditional
diffusion model with a dineeiieai structure, where the data lies in a subspace. However, our work addresses more general setting in which the manifold is unknown and can be highly nonlinear. In particular, for a linear subspace, the (unconditional) diffusion process can be decomposed into the tangent part and orthogonal part, so that the subspace estimation error and the estimation ervor of the distribution on the subspace can be decoupled and analyzed separately. In comparison, for a nonlinear manifold, such a decomposition does not exist, and the manifold estimation error and distribution estimation error are coupled in complicated manner. Furthermore, due to the nonlin- earity, our approximation error analysis using neural networks, we have locally approximate class of projection operators of the nonlinear manifold that changes cross the manifold, rather than approximating a single global projection operator onto a linear subspace.
3.3 PROOF HIGHLIGHTS
Since Theorem 2 extends Theorem b incorporating manifold structures, will only outline the proof for the former in this subsection. All missing definitions, formal assumptions, and detailed proofs provided in the appendices of the supplementary material for this paper.
Our strategy for bounding the distribution estimation error mainly follows the pipeline of Oko et al. (2023): ; Tang & Yang (2024). First, construct specific neural network within the class SN to approximate the true conditional score function V log Pe(,, I. with controlled error, which is summarized in the following lemma.

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Lemma 1 (Score approximation error by neural network class). Under the same neural network sizes {H,, W,, R., B, V.) and choices of T, T as in Theorem 2 there exists neural network x,, E ((H,, W,, R2, B;, Vi) for any i E [I] that
(ds(w,,,,) V Vo (-2))(())))))) pe(- (x) dw
14-1
2ay
2 ox
20ytdyt oy 4x + t
20y or 4x
20y toy ay
11
if S 4 < n
2ax
ax dy -xx
n1 20x +dx
to
in 2 dyy tdx
S T.
The primary technical challenge in proving Lemma T Marises from the fact that the space MMyl 1z is a general (possibly nonlinear) manifold that depends x. To address this, we partition the joint space M of (X,, Y) into small pieces with varying resolution levels in and Y. tailored to the smoothness levels (ay, axx). dimensions (dy, dx), and times 4, Within each pieces,
carefully construct local polynomials to approximate the local charts (i.e., the decoders G (z) defined in Assumption D) of My The actual proof is much more involved and delicate in order to optimally balance between the approximating neural network size and the approximate error; see Appendix C for details. Based on Lemma !
we can now utilize the complexity of SNN to control the generalization error for our conditional score estimator S, which minimizes the empirical score matching risk (8). The result is summarized as follows.
Lemma 2 (Score matching generalization error). It holds with probability at least 1 - n that,
[1
/ JOY
Epp
(,(,,,,) V log ( (-]) pr (w) dt
I min SES,
I)((,,,,t V lo ( (z) pp(::ppppp p((+la)(w)) dt du
RDY
R2H, log { R.H Will v 1) (logn)² (logn) : for each i E [Z].
Oko et al. (2023) derived a similar result in the context of (unconditioned) distribution estimation without x; in addition, their error bound is not a high probability bound but instead takes another expectation with respect to the randomness of S. In contrast, our proof for the conditional distribution estimation requires a high probability bound. We utilize more technical tools in empirical process theory, such as the localization and peeling techniques Wainwright (2019). to derive such a high probability bound in Lemma 2 The rest of the analysis is similar to a standard analysis for score-basee based diffusion models Song & Ermon (2019): Chen et al. (2022); Oko et al. (2023), where we apply Girsanov's 's theorem to relate the distribution estimation error with the obtained L2 score estimation error.
4 CONCLUSION
In this study, we investigate the theoretical properties of conditional forward-hackward diffusion estimators within the statistical framework of distribution regression. Our results identify the primary sources of error in conditional distribution estimation using conditional diffusion models and include earlier results on unconditional distribution estimation and nonparametric mean regression as special cases. Notably, our findings demonstrate that although (conditional) diffusion models operate directly in the original ambient data space and do not explicitly incorporate diwdeeensional structures, the resulting conditional distribution estimators can still adapt to intrinsic manifold structures when either (or both) the covariate X and response Y are concentrated around a liwer-dimensional manifold. Our analysis also offers practical guidance for designing the neural network approximation family to optimally control different types of errors. This includes recommendations for the architecture of the neural network, well as how the network's size (depth, width, sparsity, etc.) should depend on various problem characteristics, such as sample size, smoothness levels, and intrinsic dimensions.
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13


Reviews:
[{"review_id": "pYaQYZ7yda", "reviewer": "Reviewer_x4xs", "summary": "Previous work has studied the min-max optimality of score-based diffusion models in estimating the density of an unknown distribution, and in estimating the intrinsic dimension of data lying on a submanifold of a sample space. \n\nThis work extends these results to the conditional modelling setting, where a conditioning variate changes the distribution being modeled. The distribution is assumed to vary smoothly with the conditioning variate.\n\nThese results show that very similar rates of convergence of the total variation and Wasserstein distances between the true and  estimated data distributions hold to the unconditional modeling setting, with additional terms depending on the dimensionality of the conditioning variate. In the limit of the conditioning variate having dimension zero, the same rates as previous works on the unconditional setting are obtained.", "questions": "Please see the weaknesses section.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The contributions are, while possibly not surprising, are meaningful and interesting results extending the theoretical analysis of diffusion models.\n- The paper is generally very well written and easy to follow, given the technical nature of the content. In particular the remarks on the theorems make understanding the implications of them more accessible.", "weaknesses": "- The practical implications of the theoretical work presented are not completely clear to me. The authors mention that the results suggest how to design networks, but from my understanding on theorem 2, this suggests simply that larger networks produce better bounds?\n- The implications of assumption D are unclear to me. The variable $r_0$ is undefined. Is this condition meant to hold for any $r_0$? If so then this implies a need for the decoding to be smooth globally, with I believe would limit topological structure of the low-dimension data manifold. If this is local condition, the is this not a consequence of the smooth manifold structure of $\\mathcal{M}$?", "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": "", "extra_scores": {}}, {"review_id": "QvEqJtCGhd", "reviewer": "Reviewer_2asE", "summary": "- The paper presents a derivation of the estimation error bound for the diffusion models with sampling guided by the covariate regressor. The authors closely build upon the methodology from Oko et al 2023 https://proceedings.mlr.press/v202/oko23a/oko23a.pdf who explored the statistical learning theory for diffusion models via score matching estimators and proved nearly minimax optimal estimation rates in total variation and Wasserstein metric of order 1 for fully connected neural-networks (including suggestions on generalisation to U-nets and ConvNets). \n\n- The presented paper aims to introduce the statistical learning theory for conditional diffusion models using the distribution regression of the covariate structure guiding the sampling. Authors investigate the properties of the estimators using the finite sample analysis of the statistical error bound under various metrics, the relations of latent lower dimensional data structures and the smoothness properties of the data. \n\n- The main contribution is: \n   1) Given the assumptions on the size of the architecture,  the estimation error bound of the conditional diffusion estimator is minimax optimal under total variation and Wasserstein distance of order 1, authors include the discussion on the generalization to the unconditional case as a special case, \n   2) if there is a low-dimensional latent space structure within the high-dimensional data manifold (say within the feature space or feature and label space), the resulting conditional distribution estimators adapts to the lower-dimensional latent space.", "questions": "1) Are the covariates X also located in the Besov space support?\n2) The marginals M_X and M_Y are assumed to be  [-1,1]  D-X, resp. D_Y dimensional cube. How does the scaling of the data to the [-1, 1] cube impact some other intrinsic features, e.g. latent translations? What type of scaling transformations would you recommend?\n3) Can you provide guidelines on how you estimate $\\alpha_X$ and $\\alpha_Y$ or do you assume these values?\n4) How does presented results in Theorem 2 relate to latent diffusion models?\n\n5) p.7 Remark 1: Does this remark imply that the kernel estimator can well approximate the performance of the conditional diffusion based estimator?\n\n6) It would be great if authors can  include experiments for different embedded lower dimensional structures, and demonstrate how the presented theory supports and navigates the experimentally obtained results. Using even simple synthetic geometric objects would be great.\n\n7) Can you provide more discussion on the implications of the different regimes discussed in Remark 3? In particular how would you select the smoothness parameters $\\alpha_x$ to investigate the transition phases?\n\n8 )   How does Lemma 2 helps to assess the generalization error?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- The paper is mathematically well written. The notation is consistent and precise. Definitions and proofs are provided in the extensive Appendix. These can be useful for deriving and investigating other novel properties and algorithms.\n- The problem is relevant and well chosen and the previous work is well mapped. \n- There is an interesting potential applications for measuring the quality of the conditional diffusion estimators and guided  sampling. The paper also  provides a background for the reasoning about the selection of the model architecture.\n- The interesting contribution is in analyzing the latent dimension representations of the data and the adaptability of the higher dimensional estimator.\n- The results provide insight into the impact of the smoothness of the underlying data manifold on the performance of the estimator.", "weaknesses": "1) Paper is missing motivational examples and practical experiments to demonstrate how the presented theory can be used on real-life problems. The results are theoretically interesting and there is no issue with the soundness of the arguments. Nevertheless, without a practical demonstration on how to use the derived results, the main contribution of the paper may be harder to access to the wider machine learning community.\n\n2) While it is great to see the methodology related to the proofs as a part of the main paper, it would be great to see particular direct applications in the main body of the paper.\n\n3) The usual approach to estimate the gradients of log p_t(y_t|y) (see line 200) is to use the gradient of unconditional model and add gradient of log-likelihood of the ‘reverse predictor' which detects ‘the covariate values’ x based on the features constructed from whom the unconditional values ‘y_t’. Authors decided to use marginal forward diffusion which results in empirical estimation of the joint density between the covariate X (the guiding covariates) and Y (the unconditional diffusion). The discussion on incorporating computational efficiency of this approach in comparison to the 'reverse predictor' approach is missing.\n4) The proven Theorems assume the initial constraints of the sizes of the architecture such as depth, width, norm of the parameters, etc. These constraints are set as functionals of the order of magnitude of the data sample $n$. It is not well discussed what happens if these constraints are not satisfied.", "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": "", "extra_scores": {}}, {"review_id": "eMNHMtaq3E", "reviewer": "Reviewer_YRRd", "summary": "This paper uses distribution regression to estimate the conditional distribution induced by conditional forward-backward diffusion models. Authors utilize neural network to estimate the key points and also allow for high-dimensional X and Y with low-dimensional structures. Solid theoretical results are provided.", "questions": "What is p_data on Line 122?\n\nAuthors need to check all the parenthetical and narrative citations throughout the whole paper. Many were mis-used. For example, Line 316 should be parenthetical citation.\n\nDefinition of neural network class is already given in Tang and Yang, 2024 (AISTATS). Authors need to reference it or directly call it from the citation.\n\nIn remark 1, can authors discuss the case when alpha_X>1? What does this mean to the minimax rate?\n\nIn Remark 2, can the authors clarify why D_Y need to satisfy this condition to have the first term dominate the second one? \n\n\n\n\nRong Tang and Yun Yang. Adaptivity of diffusion models to manifold structures. 27th International Conference on Artificial Intelligence and Statistics, 2024.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Authors introduce SDE in a simple way and made it easy to connect SDE to distribution regression.\n\n2. The paper extends the unconditional distribution estimator to the conditional one and provides theoretical results. Theories are given in an elegant way that they clearly explain the connections to the theories of unconditional distribution estimator, mean regression problems, and nonparametric regression under l2 error. Theoretical results are interesting and inspiring.", "weaknesses": "The paper mentions that it provides practical guidance to design the neural network, including the network size, smoothness level and so on. Though it is a theoretical work, some empirical results would significantly demonstrate the usefulness of the paper. I would recommend authors run some simulations to show how the theoretical results guide the design of neural networks. For example, under what scenarios, the errors are optimally controlled?\n\nThe theoretical results are interesting.  Can the authors highlight the difficulties and summarize the new ways of proof if there are? This could be a huge contribution to the statistical learning community. I didn't see the challenges from my end and all seem to be standard.", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)