Title: Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models
Venue: NeurIPS 2024
Accepted: yes
Keywords: diffusion models

score-based generative models

low-dimensional structure

coefficient design
Abstract: This paper investigates score-based diffusion models when the underlying target distribution is concentrated on or near low-dimensional manifolds within the higher-dimensional space in which they formally reside, a common characteristic of natural image distributions. Despite previous efforts to understand the data generation process of diffusion models, existing theoretical support remains highly suboptimal in the presence of low-dimensional structure, which we strengthen in this paper.  For the popular Denoising Diffusion Probabilistic Model (DDPM), we find that the dependency of the error incurred within each denoising step on the ambient dimension $d$ is in general unavoidable. We further identify a unique design of coefficients that yields a converges rate at the order of $O(k^{2}/\sqrt{T})$ (up to log factors), where $k$ is the intrinsic dimension of the target distribution and $T$ is the number of steps. This represents the first theoretical demonstration that the DDPM sampler can adapt to unknown low-dimensional structures in the target distribution, highlighting the critical importance of coefficient design. All of this is achieved by a novel set of analysis tools that characterize the algorithmic dynamics in a more deterministic manner.

Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models

Abstract

This paper investigates score-based diffusion models when the underlying target distribution is concentrated on or near low-dimensional manifolds within the higher-dimensional space in which they formally reside, a common characteristic of natural image distributions. Despite previous efforts to understand the data generation process of diffusion models, existing theoretical support remains highly suboptimal in the presence of low-dimensional structure, which we strengthen in this paper. For the popular Denoising Diffusion Probabilistic Model (DDPM), we find that the dependency of the error incurred within each denoising step on the ambient dimension d is in general unavoidable. We further identify a unique design of coefficients that yields a converges rate at the order of O ( k 2 / √ T ) (up to log factors), where k is the intrinsic dimension of the target distribution and T is the number of steps. This represents the first theoretical demonstration that the DDPM sampler can adapt to unknown low-dimensional structures in the target distribution, highlighting the critical importance of coefficient design. All of this is achieved by a novel set of analysis tools that characterize the algorithmic dynamics in a more deterministic manner.

1 Introduction

Score-based diffusion models are a class of generative models that have gained prominence in the field of machine learning and artificial intelligence for their ability to generate high-quality new data instances from complex distributions, such as images, audio, and text [23, 11, 25, 24, 9]. These models operate by gradually transforming noise into samples from the target distribution through a denoising process guided by pre-trained neural networks that approximate the score functions. In practice, score-based diffusion models have demonstrated remarkable performance in generating realistic and diverse content across various domains [19, 20, 21, 7], achieving state-of-the-art performance in generative AI.

1.1 Diffusion models

The development of score-based diffusion models is deeply rooted in the theory of stochastic processes. At a high level, we consider a forward process:

which draws a sample from the target data distribution (i.e., X 0 ∼ p data ), then progressively diffuses it to Gaussian noise over time. The key aspect of the diffusion model is to construct a reverse process:

Yuling Yan ∗

satisfying Y t d ≈ X t for all t , which starts with pure Gaussian noise (i.e., Y T ∼ N (0 , I d ) ) and gradually converts it back to a new sample Y 0 sharing a similar distribution to p data .

The classical results on time-reversal of SDEs [1, 10] provide the theoretical foundation for the above task. Consider a continuous time diffusion process:

for some function β : [0 , T ] → R + , where ( W t ) 0 ≤ t ≤ T is a standard Brownian motion. For a wide range of functions β , this process converges exponentially fast to a Gaussian distribution. Let p X t ( · ) be the density of X t . One can construct a reverse-time SDE:

where ( Z t ) 0 ≤ t ≤ T is another standard Brownian motion. Define Y t = ˜ Y T -t . It is well-known that X t d = Y t for all 0 ≤ t ≤ T . Here, ∇ log p X t is called the score function for the law of X t , which is not explicitly known.

The above result motivates the following paradigm: we can construct the forward process (1.1) by time-discretizing the diffusion process (1.3), and construct the reverse process (1.2) by discretizing the reverse-time SDE (1.4) and learning the score functions from the data. This approach leads to the popular DDPM sampler [11, 16]. Although the idea of the DDPM sampler is rooted in the theory of SDEs, the algorithm and analysis presented in this paper do not require any prior knowledge of SDEs.

This paper examines the accuracy of the DDPM sampler by establishing the proximity between the output distribution of the reverse process and the target data distribution. Since these two distributions are identical in the continuous time limit with perfect score estimation, the performance of the DDPM sampler is influenced by two sources of error: discretization error (due to a finite number of steps) and score estimation error (due to imperfect estimation of the scores). This paper views the score estimation step as a black box (often addressed by training a large neural network) and focuses on understanding how time discretization and imperfect score estimation affect the accuracy of the DDPM sampler.

1.2 Inadequacy of existing results

The past few years have witnessed a significant interest in studying the convergence guarantees for the DDPM sampler [4, 6, 3, 14, 13]. To facilitate discussion, we consider an ideal setting with perfect score estimation. In this context, existing results can be interpreted as follows: to achieve ε -accuracy (i.e., the total variation distance between the target and the output distribution is smaller than ε ), it suffices to take a number of steps exceeding the order of poly ( d ) /ε 2 (ignoring logarithm factors), where d is the problem dimension. Among these results, the state-of-the-art is given by [3], which achieved linear dependency on the dimension d .

However, there seems to be a significant gap between the practical performance of the DDPM sampler and the existing theory. For example, for two widely used image datasets, CIFAR-10 (dimension d = 32 × 32 × 3 ) and ImageNet (dimension d ≥ 64 × 64 × 3 ), it is known that 50 and 250 steps (also known as NFE, the number of function evaluations) are sufficient to generate good samples [16, 9]. This is in stark contrast with the existing theoretical guarantees discussed above, which suggest that the number of steps T should exceed the order of the dimension d to achieve good performance.

Empirical evidence suggests that the distributions of natural images are concentrated on or near low-dimensional manifolds within the higher-dimensional space in which they formally reside [22, 18]. In view of this, a reasonable conjecture is that the convergence rate of the DDPM sampler actually depends on the intrinsic dimension rather than the ambient dimension. However, the theoretical understanding of diffusion models when the support of the target data distribution has a low-dimensional structure remains vastly under-explored. As some recent attempts, [8] established the first convergence guarantee under the Wasserstein-1 metric. However, their error bound has linear dependence on the ambient dimension d and exponential dependence on the diameter of the low-dimensional manifold. Another line of works [5, 26, 17] focused mainly on score estimation with properly chosen neural networks that exploit the low-dimensional structure, which is also different from our main focus.

1.3 Our contributions

In light of the large theory-practice gap and the insufficiency of prior results, this paper takes a step towards understanding the performance of the DDPM sampler when the target data distribution has low-dimensional structure. Our main contributions can be summarized as follows:

We show that, with a particular coefficient design, the error of the DDPM sampler, evaluated by the total variation distance between the laws of X 1 and Y 1 , is upper bounded by

up to some logarithmic factors, where k is the intrinsic dimension of the target data distribution (which will be rigorously defined later), and s glyph[star] t (resp. s t ) is the true (resp. learned) score function at each step. The first term represents the discretization error (which vanishes as the number of steps T goes to infinity), while the second term should be interpreted as the score matching error. This bound is nearly dimension-free - the ambient dimension d only appears in logarithmic terms.

We also show that our choice of the coefficients is, in some sense, the unique schedule that does not incur discretization error proportional to the ambient dimension d at each step. This is in sharp contrast with the general setting without a low-dimensional structure, where a fairly wide range of coefficient designs can lead to convergence rates with polynomial dependence on d . Additionally, this confirms the observation that the performance of the DDPM sampler can be improved through carefully designing coefficients [2, 16].

As far as we know, this paper provides the first theory demonstrating the capability of the DDPM sampler in adapting to unknown low-dimensional structures.

2 Problem set-up

In this section, we introduce some preliminaries and key ingredients for the diffusion model and the DDPM sampler.

Forward process. We consider the forward process (1.1) of the form

where W 1 , . . . , W T i.i.d. ∼ N (0 , I d ) , and the learning rates β t ∈ (0 , 1) will be specified later. For each t ≥ 1 , X t has a probability density function (PDF) supported on R d , and we will use q t to denote the law or PDF of X t . Let α t := 1 -β t and α t := ∏ t i =1 α i . It is straightforward to check that

We will choose the learning rates β t to ensure that α T becomes vanishingly small, such that q T ≈ N (0 , I d ) .

Score functions. The key ingredients for constructing the reverse process with the DDPM sampler are the score functions s glyph[star] t : R d → R d associated with each q t , defined as

These score functions are not explicitly known. Here we assume access to an estimate s t ( · ) for each s glyph[star] t ( · ) , and we define the averaged glyph[lscript] 2 score estimation error as

This quantity captures the effect of imperfect score estimation in our theory.

The DDPM sampler. To construct the reverse process (1.2), we use the DDPM sampler

where Z 1 , . . . , Z T i.i.d. ∼ N (0 , I d ) . Here η t , σ t > 0 are the hyperparameters that play an important role in the performance of the DDPM sampler, especially when the target data distribution has low-dimensional structure. As we will see, our theory suggests the following choice

For each 1 ≤ t ≤ T , we will use p t to denote the law or PDF of Y t .

Target data distribution. Let X ⊆ R d be the support set of the target data distribution p data , i.e., the smallest closed set C ⊆ R d such that p data ( C ) = 1 . To allow for the greatest generality, we use the notion of ε -net and covering number (see e.g., [27]) to characterize the intrinsic dimension of X . For any ε > 0 , a set N ε ⊆ X is said to be an ε -net of X if for any x ∈ X , there exists some x ′ in N ε such that ‖ x -x ′ ‖ 2 ≤ ε . The covering number N ε ( X ) is defined as the smallest possible cardinality of an ε -net of X .

( Low-dimensionality ) Fix ε = T -c ε , where c ε > 0 is some sufficiently large universal constant. We define the intrinsic dimension of X to be some quantity k > 0 such that

for some constant C cover > 0 .

( Bounded support ) Suppose that there exists a universal constant c R > 0 such that

Namely we allow polynomial growth of the diameter of X in the number of steps T .

Our setting allows X to be concentrated on or near low-dimensional manifolds, which is less stringent than assuming an exact low-dimensional structure. In fact, our definition of the intrinsic dimension k is the metric entropy of X (see e.g., [28]), which is widely used in statistics and learning theory to characterize the complexity of a set or a class. The low-dimensionality is also a concept of complexity, therefore it is natural to use covering number, or metric entropy to characterize the intrinsic dimension. As a sanity check, when X resides in an r -dimensional subspace of R d , a standard volume argument (see e.g., [27, Section 4.2.1]) gives log N ε ( X ) glyph[equivasymptotic] r log( R/ε ) glyph[equivasymptotic] r log T , suggesting that the intrinsic dimension k is of order r in this case. In addition, in applications like image generation, the data is naturally bounded, as pixel values are typically normalized within the range [ -1 , 1] . For example, the glyph[lscript] 2 norm of an image from the CIFAR dataset is typically below 60 .

Learning rate schedule. Following [14], we adopt the following learning rate schedule

for some sufficiently large constants c 0 , c 1 > 0 . This schedule is not unique - any other schedule of β t satisfying the properties in Lemma 8 can lead to the same result in this paper.

3 Main results

We are now positioned to present our main theoretical guarantees for the DDPM sampler.

3.1 Convergence analysis

We first present the convergence theory for the DDPM sampler. The proof can be found in Section 4.

Theorem 1. Suppose that we take the coefficients for the DDPM sampler (2.3) to be η t = η glyph[star] t and σ t = σ glyph[star] t (cf. (2.4)), then there exists some universal constant C > 0 such that

Several implications of Theorem 1 follow immediately. The two terms in (3.1) correspond to discretization error and score matching error, respectively. Assuming perfect score estimation (i.e., ε score = 0 ) for the moment, our error bound (3.1) suggests an iteration complexity of order k 4 /ε 2 (ignoring logarithmic factors) for achieving ε -accuracy, for any nontrivial target accuracy level ε < 1 . In the absence of low-dimensional structure (i.e., k glyph[equivasymptotic] d ), our result also recovers the iteration complexity in [4, 6, 3, 14] of order poly ( d ) /ε 2 . 2 This suggests that our choice of coefficients (2.4) allows the DDPM sampler to adapt to any potential (unknown) low-dimensional structure in the target data distribution, and remains a valid criterion in the most general settings. The score matching error in (3.1) scales proportionally with ε score , suggesting that the DDPM sampler is stable to imperfect score estimation.

3.2 Uniqueness of coefficient design

In this section, we examine the importance of the coefficient design in the adaptivity of the DDPM sampler to intrinsic low-dimensional structure. Our goal is to show that, unless the coefficients η t , σ t of the DDPM sampler (2.3) are chosen according to (2.4), discretization errors proportional to the ambient dimension d will emerge in each denoising step.

In this paper, as well as in most previous DDPM literature, the analysis on the error TV ( q 1 , p 1 ) usually starts with the following decomposition

Here step (i) follows from Pinsker's inequality, step (ii) utilizes from the data-processing inequality, while step (iii) uses the chain rule of KL divergence. We may interpret each term in the above decomposition as the error incurred in each denoising step. In fact, this decomposition is also closely related to the variational bound on the negative log-likelihood of the reverse process, which is the optimization target for training DDPM [11, 2, 16].

We consider a target distribution p data = N (0 , I k ) , where I k ∈ R d × d is a diagonal matrix with I i,i = 1 for 1 ≤ i ≤ k and I i,i = 0 for k +1 ≤ i ≤ d . This is a simple distribution over R d that is supported on a k -dimensional subspace. 3 Our second theoretical result provides a lower bound for the error incurred in each denoising step for this target distribution. The proof can be found in Appendix B.

Theorem 2. Consider the target distribution p data = N (0 , I k ) and assume that k ≤ d/ 2 . For the DDPM sampler (2.3) with perfect score estimation (i.e., s t ( · ) = s glyph[star] t ( · ) for all t ) and arbitrary coefficients η t , σ t > 0 , we have

for each 2 ≤ t ≤ T . See (2.4) for the definitions of η glyph[star] t and σ glyph[star] t .

Theorem 2 shows that, unless we choose η t and σ 2 t to be identical (or exceedingly close) to η glyph[star] t and σ glyph[star] 2 t , the corresponding denoising step will incur an undesired error that is linear in the ambient dimension d . This highlights the critical importance of coefficient design for the DDPM sampler, especially when the target distribution exhibits a low-dimensional structure.

Finally, we would like to make note that the above argument only demonstrates the impact of coefficient design on an upper bound (3.2) of the error TV ( q 1 , p 1 ) , rather than the error itself. It might be possible that a broader range of coefficients can lead to dimension-independent error bound like (3.1), while the upper bound (3.2) remains dimension-dependent. This calls for new analysis tools (since we cannot use the loose upper bound (3.1) in the analysis), which we leave for future works.

4 Analysis for the DDPM sampler (Proof of Theorem 1)

This section is devoted to establishing Theorem 1. The idea is to bound the error incurred in each denoising step as characterized in the decomposition (3.2), namely for each 2 ≤ t ≤ T , we need to bound

This requires connecting the two conditional distributions p X t -1 | X t and p Y t -1 | Y t . It would be convenient to decouple the errors from time discretization and imperfect score estimation by introducing auxiliary random variables

On a high level, for each 2 ≤ t ≤ T , our proof consists of the following steps:

Identify a typical set A t ⊆ R d × R d such that ( X t , X t -1 ) ∈ A t with high probability.

Establish point-wise proximity p X t -1 | X t ( x t -1 | x t ) ≈ p Y glyph[star] t -1 | Y t ( x t -1 | x t ) for ( x t , x t -1 ) ∈ A t .

Characterize the deviation of p Y glyph[star] t -1 | Y t from p Y t -1 | Y t caused by imperfect score estimation.

4.1 Step 1: identifying high-probability sets

For simplicity of presentation, we assume without loss of generality that k ≥ log d throughout the proof. 4 Let { x glyph[star] i } 1 ≤ i ≤ N ε be an ε -net of X , and let {B i } 1 ≤ i ≤ N ε be a disjoint ε -cover for X such that x glyph[star] i ∈ B i . Let

where C 1 > 0 is some sufficiently large universal constants. Then ∪ i ∈I B i and G can be interpreted as high probability sets for the variable X 0 and a standard Gaussian random variable in R d . For each t = 1 , . . . T , we define a typical set for each X t as follows

and a typical set for ( X t , X t -1 ) jointly as follows

The following lemma shows that A t is indeed a high-probability set for ( X t , X t -1 ) .

Lemma 1. Suppose that C 1 glyph[greatermuch] C cover . Then for each 1 ≤ t ≤ T we have

4.2 Step 2: connecting conditional densities p X t -1 | X t and p Y glyph[star] t -1 | Y t

Given the definition of Y glyph[star] t -1 in (4.1), we can write down the conditional density p Y glyph[star] t -1 | Y t as follows

Next, we will investigate the conditional density p X t -1 | X t for the forward process. For each x 0 ∈ X , we define the shorthand notation

and define a function ∆ x t ,x t -1 : X → R as follows

The next lemma provides a characterization for p X t -1 | X t that shows an explicit connection with p Y glyph[star] t -1 | Y t .

Lemma 2. For any pair ( x t , x t -1 ) ∈ R d × R d , we have

Taking Lemma 2 and (4.2) collectively yields

which allows us to control the density ratio by the magnitude of ∆ x t ,x t -1 . By a careful analysis of the above integral for all ( x t , x t -1 ) ∈ A t , we show in the next lemma that the density ratio is uniformly close to 1 within the typical set A t .

Lemma 3. Suppose that T glyph[greatermuch] k 2 log 3 T . Then there exists some universal constant C 5 > 0 such that, for any 2 ≤ t ≤ T and any ( x t , x t -1 ) ∈ A t , we have

For ( x t , x t -1 ) outside the typical set A t , the following lemma gives a coarse uniform bound for the density ratio, which is already sufficient for our later analysis.

Lemma 4. Suppose that T glyph[greatermuch] 1 . Then for any 2 ≤ t ≤ T and any pair ( x t , x t -1 ) ∈ R d × R d , we have

Proof.

Armed with Lemmas 3 and 4, we are ready to bound the expected KL divergence between the two conditional distributions p X t -1 | X t and p Y glyph[star] t -1 | Y t .

4.3 Step 3: bounding the KL divergence between p X t -1 | X t and p Y glyph[star] t -1 | Y t

We first decompose the expected KL divergence between p X t -1 | X t and p Y glyph[star] t -1 | Y t into

where ∆ t, 1 and ∆ t, 2 are the integrals over A t and A c t . It boils down to bounding these two terms.

By a direct application of Lemma 3 together with the first-order Taylor expansion of log( x ) around x = 1 , one can easily show that | ∆ t, 1 | glyph[lessorsimilar] k 2 log 3 ( T ) /T . However this naive bound will lead to a vacuous final bound on TV ( q 1 , p 1 ) , which depends on the sum of ∆ t, 1 over all 2 ≤ t ≤ T according to (3.2). By a more careful analysis, we achieve a better bound for ∆ t, 1 , as shown in the following lemma.

Lemma 5. Suppose that T glyph[greatermuch] k 2 log 3 T . Then for each 2 ≤ t ≤ T , we have

For ∆ t, 2 , we can employ the course bound in Lemma 4 to show that it is exponentially small.

Lemma 6. Suppose that T glyph[greatermuch] 1 . Then for each 2 ≤ t ≤ T , we have

By putting together Lemma 5 and Lemma 6, we achieve

provided that T is sufficiently large.

4.4 Step 4: bounding the KL divergence between p X t -1 | X t and p Y t -1 | Y t

Since our goal is to bound the expected KL divergence between p X t -1 | X t and p Y t -1 | Y t , we also need to upper bound the following difference

where we define

It then boils down to bounding K t , which is presented in the following lemma.

Lemma 7. Suppose that T glyph[greatermuch] k 2 log 3 T . Then we have

Hence we know that for 2 ≤ t ≤ T ,

Here the first relation follows from Lemma 7 and (2.4); while the second relation follows from Lemma 8 and holds provided that T is sufficiently large.

4.5 Step 5: putting everything together

By taking (4.5) and (4.8) collectively, we have

Here the last relation follows from an application of the AM-GM inequality

Finally we conclude that

as claimed. Here the first relation follows from (3.2), while the second relation follows from the fact that KL ( p X T ‖ p Y T ) ≤ T -100 provided that T is sufficiently large (see Lemma 10).

5 Simulation study

We conducted a simple simulation to compare our coefficient design (2.4) with another design

which has been widely adopted in theoretical analysis of diffusion model (see e.g., [14, 15]). We consider the degenerated Gaussian distribution p data = N (0 , I k ) in Theorem 2 as a tractable example, and run the DDPM sampler with exact score functions (so that the error only comes from discretization). We fix the intrinsic dimension k = 8 , and let the ambient dimension d grow from 10 to 10 3 . We implement the experiment for four different number of steps T ∈ { 100 , 200 , 500 , 1000 } . Instead of using the learning rate schedule (2.5), which is chosen mainly to facilitate analysis, we use the schedule in [11] that is commonly used in practice. Figure 1 displays the error, in terms of both the TV distance TV ( q 1 , p 1 ) and KL divergence KL ( q 1 ‖ p 1 ) , as the ambient dimension d varies. As we can see, our design (2.4) leads to dimension-independent error while the other design (5.1) incures an error that grows as d increases. This provides empirical evidence that (2.4) represents a unique coefficient design for DDPM in achieving dimension-independent error.

10

1

10

2

10

3

ambient dimension d

10

-4

10

-2

10

0

T = 100

10

1

10

2

10

3

ambient dimension d

10

-2

10

-1

10

0

T = 100

10

1

10

2

10

3

ambient dimension d

10

-4

10

-2

10

0

T = 200

10

1

10

2

10

3

ambient dimension d

10

-2

10

-1

10

0

T = 200

10

1

10

2

10

3

ambient dimension d

10

-4

10

-2

10

0

T = 500

10

1

10

2

10

3

ambient dimension d

10

-2

10

-1

10

0

T = 500

10

1

10

2

10

3

ambient dimension d

10

-4

10

-2

10

0

T = 1000

10

1

10

2

10

3

ambient dimension d

10

-2

10

-1

10

0

T = 1000

6 Discussion

The present paper investigates the DDPM sampler when the target distribution is concentrated on or near low-dimensional manifolds. We identify a particular coefficient design that enables the adaptivity of the DDPM sampler to unknown low-dimensional structures and establish a dimensionfree convergence rate at the order of k 2 / √ T (up to logarithmic factors). We conclude this paper by pointing out several directions worthy of future investigation. To begin with, our theory yields an iteration complexity that scales quartically in the intrinsic dimension k , which is likely sub-optimal. Improving this dependency calls for more refined analysis tools. Recent work [15] achieved a convergence rate of order O ( d/T ) , suggesting the potential for enhancing the dependence on T . Furthermore, as we have discussed in the end of Section 3.2, it is not clear whether our coefficient design (2.4) is unique in terms of achieving dimension-independent error TV ( q 1 , p 1 ) . Finally, the analysis ideas and tools developed for the DDPM sampler might be extended to study another popular DDIM sampler.

Acknowledgments and Disclosure of Funding

Gen Li is supported in part by the Chinese University of Hong Kong Direct Grant for Research. Yuling Yan was supported in part by a Norbert Wiener Postdoctoral Fellowship from MIT.

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A Proof of auxiliary lemmas for the DDPM sampler

A.1 Preliminaries

Fix any x t ∈ T t , there exists an index i ( x t ) ∈ I , two points x 0 ( x t ) ∈ B i ( x t ) and ω ∈ G such that

For any r > 0 , define a set

For some sufficiently large constant C 3 > 0 , define

Namely, X t ( x t ) (resp. Y t ( x t ) ) contains the indices of the ε -covering that are close (resp. far) from B i ( x t ) . We require that

which is guaranteed by our assumption that ε = T -c ε for some sufficiently large constant c ε > 0 . Under this condition, for any x, x ′ ∈ X t ( x t ) we have

Hence for any x, x ′ ∈ X t ( x t ) we have

In addition, for any x, x ′ ∈ X , suppose that x ∈ B i and x ′ ∈ B j . For any ω ∈ G , we have

Here step (i) follows from ω ∈ G and the Cauchy-Schwarz inequality; steps (ii) and (iii) follows from ‖ x -x glyph[star] i ‖ 2 ≤ ε and ‖ x ′ -x glyph[star] j ‖ 2 ≤ ε , as well as ‖ ω ‖ 2 ≤ √ d + √ C 1 k log T , which is a property for ω ∈ G .

A.2 Understanding the conditional density p X t | X 0 ( · | x 0 )

Conditional on X t = x t , for any 1 ≤ i ≤ N ε we have

Here the last relation follows from P ( X 0 ∈ B i ( x t ) ) ≥ exp( -C 1 k log T ) due to i ( x t ) ∈ I . We have

For any ˜ x ∈ B i , since X t | X 0 = ˜ x ∼ N ( √ α t ˜ x, (1 -α t ) I d ) , we have

Taking (A.7) and (A.8) collectively to achieve

By similar argument in (A.7), (A.8) and (A.9), we can show that

Combine (A.9) and (A.10) to achieve

Next, we will discuss the implication of the above analysis for any i / ∈ I ( x t ; C 3 ) , i.e., B i ⊆ Y t ( x t ) . For any i / ∈ I ( x t ; C 3 ) , we have

Here step (i) uses the decomposition (A.1); step (ii) follows from ω ∈ G , Cauchy-Schwarz inequality and the fact that ‖ x 0 ( x t ) -x glyph[star] i ( x t ) ‖ 2 ≤ ε ; while the correctness of step (iii) is equivalent to

which follows from i / ∈ I ( x t ; C 3 ) , the assumptions that C 3 glyph[greatermuch] C 1 , and (A.3). In addition, we also have

Here step (i) follows from the decomposition (A.1); step (ii) utilizes the triangle inequality; whereas step (iii) follows from ‖ x 0 ( x t ) -x glyph[star] i ( x t ) ‖ 2 ≤ ε and the condition (A.3). We can substitute the bounds (A.12) and (A.13) into (A.11) to get

provided that (A.3) holds and C 3 is sufficiently large. Since i / ∈ I ( x t ; C 3 ) , we know that α t ‖ x glyph[star] i ( x t ) -x glyph[star] i ‖ 2 2 > C 3 k (1 -α t ) log T , hence when C 3 glyph[greatermuch] C 1 , we learn from (A.6) and (A.14) that

A.3 Proof of Lemma 1

It is straightforward to check that

where step (i) follows from the update rule (2.1), and step (ii) follows from the relation (2.2). Therefore we have

By definition of the set I , we have

as long as C 1 glyph[greatermuch] C cover . In addition, since W t , W t ∼ N (0 , I d ) , by the definition of G we know that

Here step (i) follows from concentration bounds for Gaussian and chi-square variables (see Lemma 9); while step (ii) holds as long as C 1 glyph[greatermuch] C cover . The same bound also holds for P ( W t / ∈ G ) . Taking (A.16), (A.17) and (A.18) collectively yields

as claimed.

A.4 Proof of Lemma 2

For any deterministic pairs ( x t , x t -1 ) , we have

Recall that X t | X t -1 = x t -1 ∼ N ( √ α t x t -1 , (1 -α t ) I d ) , therefore we have

Next, we analyze the density ratio p X t -1 ( x t -1 ) /p X t ( x t ) . It would be easier to do a change of variable

Since √ α t X t -1 | X 0 = x 0 ∼ N ( α t x 0 , ( α t -α t ) I d ) , we can write

We hope to connect the above quantity with the conditional density

Towards this, we can deduce that

where step (i) follows from (A.22) and step (ii) utilizes (A.23). The terms in the exponent can be rearranged into

where we define and

Substituting the above relation into (A.24) yields

Therefore we have

Here step (i) follows from (A.19) and (A.21); step (ii) follows from (A.20) and (A.25); whereas step (iii) follows from the definition of η glyph[star] t and σ glyph[star] t (cf. (2.4)) as well as the fact that

A.5 Proof of Lemma 3

For any ( x t , x t -1 ) ∈ A t , we know that ω ′ := ( x t - √ α t x t -1 ) / √ 1 -α t ∈ G . We will upper bound the integral with two terms

where we recall that

In what follows, we will use ∆( x 0 ) instead of ∆( x t , x t -1 , x 0 ) when there is no confusion. Since ( x t , x t -1 ) ∈ A t , we know that ω ′ := ( x t - √ α t x t -1 ) / √ 1 -α t ∈ G . We decompose ̂ x 0 into

Since X t ( x t ) is a ball in R d centered at x glyph[star] i ( x t ) , it is straightforward to check that x 0 ∈ X t ( x t ) . We also have

Here step (i) holds since for any i / ∈ I ( x t ; C 3 ) and x ′ 0 ∈ B i ,

while step (ii) follows from (A.15). For any i / ∈ I ( x t ; C 3 ) , we know that α t ‖ x glyph[star] i ( x t ) -x glyph[star] i ‖ 2 2 > C 3 k (1 -α t ) log T , hence we can check that

as long as C 3 is sufficiently large and the condition (A.3) holds. Therefore

A.5.1 Step 1: deriving an upper bound for ∆( x 0 )

Suppose that x 0 ∈ B i for some 1 ≤ i ≤ N ε (notice that here we are not requiring that i ∈ I ). We will bound each of | ∆ i ( x 0 ) for i = 1 , 2 , 3 , 4 . We first record two basic facts about the step sizes, which are immediate consequences of Lemma 8:

as long as T is sufficiently large, and

We learn from (A.5) that

We also have

Here step (i) holds since x 0 ∈ B i , hence ‖ x 0 -x glyph[star] i ‖ 2 ≤ ε ; step (ii) follows from (A.4) and the fact that x glyph[star] i ( x t ) , x 0 ∈ X t ( x t ) ; while step (iii) follows from (A.3) and holds provided that C 3 is sufficiently large. Then we have

Here step (a) follows from (A.29) and (A.31); while step (b) utilizes (A.32), (A.3). Similarly we can use (A.30) to show that

Notice that

where step (i) utilizes the Cauchy-Schwarz inequality; step (ii) follows from (A.4), (A.28) and the fact that x 0 ( x t ) , x 0 ∈ X t ( x t ) ; step (iii) holds provided that C 3 is sufficiently large. Therefore we have

where the last relation follows from Lemma 8 and (A.32). Finally we have

Here step (a) follows from (A.32), step (b) follows from (A.4) and the fact that x glyph[star] i ( x t ) , x glyph[star] i ∈ X t ( x t ) Taking the bounds in (A.33) collectively leads to

provided that T is sufficiently large.

A.5.2 Step 2: bounding I 1

For x 0 ∈ X t ( x t ) , we know that x 0 ∈ B i for some i ∈ I ( x t ; C 3 ) , hence α t ‖ x glyph[star] i -x glyph[star] i ( x t ) ‖ 2 2 ≤ C 3 k (1 -α t ) log T . This combined with (A.34) gives

provided that T glyph[greatermuch] k 2 log 3 T . Similarly we can check that for each 1 ≤ i ≤ 4 , | ∆ i ( x 0 ) | ≤ 1 . Then we know that for x 0 ∈ X t ( x t ) , we have exp(∆( x 0 )) ≤ 1 + ∆( x 0 ) + ∆ 2 ( x 0 ) as long as T glyph[greatermuch] k 2 log 3 T . Hence

Here the last step follows from the fact that ∫ p X 0 | X t ( x 0 | x t )∆ 1 ( x 0 )d x 0 = 0 . The integral I 1 , 1 can be upper bounded similar to I 2 , hence we defer its analysis to the next section. For the integral I 1 , 2 , we have

Here step (i) follows from the Cauchy-Schwarz inequality and the facts that | ∆ i ( x 0 ) | ≤ 1 for i = 2 , 3 , 4 , while step (ii) follows from the bounds (A.33) and the fact that α t ‖ x glyph[star] i -x glyph[star] i ( x t ) ‖ 2 2 ≤ C 3 k (1 -α t ) log T .

A.5.3 Step 3: bounding I 2 .

For x 0 ∈ Y t ( x t ) , we know that x 0 ∈ B i for some i / ∈ I ( x t ; C 3 ) , hence α t ‖ x glyph[star] i -x glyph[star] i ( x t ) ‖ 2 2 > C 3 k (1 -α t ) log T . This combined with (A.34) gives

as long as T is sufficiently large. Therefore we have

Here step (i) follows from (A.15) and (A.37); step (ii) holds as long as T is sufficiently large; while step (iii) uses the fact that α t ‖ x glyph[star] i -x glyph[star] i ( x t ) ‖ 2 2 > C 3 k (1 -α t ) log T for i / ∈ I ( x t ; C 3 ) . By similar analysis, we can show that

Here step (a) follows from (A.33a) and the fact that α t ‖ x glyph[star] i -x glyph[star] i ( x t ) ‖ 2 2 > C 3 k (1 -α t ) log T for i / ∈ I ( x t ; C 3 ) ; step (b) follows from (A.15); step (c) holds provided that C 3 is sufficiently large; step (d) follows again from the fact that α t ‖ x glyph[star] i -x glyph[star] i ( x t ) ‖ 2 2 > C 3 k (1 -α t ) log T for i / ∈ I ( x t ; C 3 ) .

A.5.4 Step 4: putting everything together

Taking (A.36), (A.38) and (A.39) collectively, we have

provided that T is sufficiently large. By similar argument, i.e., using the lower bounding exp(∆( x 0 )) ≥ 1 + ∆( x 0 ) -∆ 2 ( x 0 ) in Step 2 and repeat the same analysis, we can show that

This gives the desired result.

A.6 Proof of Lemma 4

Recall that

For any x 0 ∈ X , by the definition of ∆( x t , x t -1 , x 0 ) in (4.4), we have

Here step (i) follows from ̂ x 0 , x 0 ∈ X , hence max {‖ ̂ x 0 ‖ 2 , ‖ x 0 ‖ 2 } ≤ R ; while step (ii) follows from the facts that, for 2 ≤ t ≤ T ,

Hence we have

as long as T is sufficiently large.

A.7 Proof of Lemma 5

Regarding ∆ t, 1 , we first utilize Lemma 3 to show that for any ( x t , x t -1 ) ∈ A t ,

Since log(1 -x ) ≥ -x -x 2 holds for any x ∈ [ -1 / 2 , 1 / 2] , we know that when T glyph[greatermuch] k 2 log 3 T , we have

Hence we have

Here the penultimate step follows from the fact that

∫

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t

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t

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Here we use the fact that Y glyph[star] t -1 | Y t = x t ∼ N ( ( x t + η glyph[star] t s glyph[star] t ( x t )) / α t , ( σ glyph[star] 2 t /α t ) I d ) . Notice that

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Claim 1 . For any x t ∈ T t , we have

and for any 1 ≤ i ≤ j ≤ N ε ,

Since Z ∼ N (0 , I d ) , in view of Lemma 9, with probability exceeding 1 -exp( -( C 1 / 64) k log T ) ,

and for any 1 ≤ i ≤ j ≤ N ε ,

These combined with Claim 1 allow us to show that

Taking the above inequality collectively with Lemma 1 gives

Hence we have

as long as T is sufficiently large.

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A.7.1 Proof of Claim 1

Consider the decomposition x t = √ α t x 0 ( x t ) + √ 1 -α t ω as in Appendix A.1, where x 0 ( x t ) ∈ B i ( x t ) for some i ( x t ) ∈ I and ω ∈ G . Notice that

where ̂ x 0 := E [ X 0 | X t = x t ] is defined in (4.3), whereas x 0 ∈ X t ( x t ) and δ are defined in (A.27). Therefore we can check that

Here step (i) follows from (A.4), the fact that ω ∈ G , and (A.28); step (ii) follows from Lemma 8; while step (iii) holds provided that T is sufficiently large. In addition, for any 1 ≤ i ≤ j ≤ N ε we have

Here step (a) utilizes the Cauchy-Schwarz inequality; step (b) follows from (A.4), the fact that ω ∈ G , and (A.28); step (c) follows from Lemma 8; while step (d) holds when T is sufficiently large.

A.8 Proof of Lemma 6

We can upper bound | ∆ t, 2 | by

Here step (i) follows from Lemma 4; step (ii) follows from the update rule (2.1); step (iii) utilizes the Cauchy-Schwarz inequality. In view of (2.2), we have

where we use the fact that E [ ‖ W t ‖ 2 2 ] = d . Then we have

Here step (a) utilizes (A.41) and the fact that E [ ‖ W t ‖ 2 2 ] = d ; step (b) follows from Lemma 1; while step (c) makes use of the assumption that k ≥ log d and holds provided that C 1 glyph[greatermuch] c 0 + c R .

A.9 Proof of Lemma 7

We first decompose K t into

Here step (i) follows from the fact that ∫ p Y glyph[star] t -1 | Y t ( x t -1 | x t ) ( x t -1 -µ glyph[star] t ( x t ))d x t -1 = 0 for any x t ∈ R d , and K 1 and K 2 are defined to be the two integrals over A t and A c t in step (ii). The following two claims provide bounds for the two integrals K t, 1 and K t, 2 respectively.

Claim 2 . Suppose that T glyph[greatermuch] k 2 log 3 T . Then for each 2 ≤ t ≤ T , we have

Claim 3 . Suppose that T glyph[greatermuch] 1 . Then for each 2 ≤ t ≤ T , we have

Proof. See Appendix A.9.2.

Then we conclude that

as claimed. Here step (a) follows from Claim 2 and Claim 3; while step (b) utilizes Jensen's inequality, and holds provided that T is sufficiently large.

A.9.1 Proof of Claim 2

The term K t, 1 can be upper bounded by

Here step (i) follows from Jensen's inequality; step (ii) utilizes Lemma 3; step (iii) follows from the definition of Y glyph[star] t in (4.1) and of µ glyph[star] t in (4.7), and Z t ∼ N (0 , I d ) is independent of X t ; step (iv) follows from Lemma 8 and the fact that Z glyph[latticetop] t ε t ( X t ) | X t ∼ N (0 , ‖ ε t ( X t ) ‖ 2 2 ) and hence

A.9.2 Proof of Claim 3

The term K t, 1 can be upper bounded by

The second term γ 2 can be easily bounded by

In what follows, we will bound the first term γ 1 . Note that

We have

Here step (i) follows from Cauchy-Schwarz inequality; step (ii) follows from Lemma 1 and the definition of µ glyph[star] t in (4.7); while step (iii) uses the fact that α t ≥ 1 / 2 (see Lemma 8). Recall the definition of s glyph[star] t ( · )

which leads to the following upper bound

Hence we have

Here step (i) follows from the elementary inequality 8( x 4 + y 4 ) ≥ ( x + y ) 2 ; step (ii) follows from Lemma 8 and (A.42); step (iii) follows from W t , W t ∼ N (0 , I d ) and the proviso that T being sufficiently large. Hence we have

as long as C 0 glyph[greatermuch] c R and k ≥ log d . Regarding γ 1 , 2 , we have

Here step (i) follows from the definition of Y glyph[star] t in (4.1) and of µ glyph[star] t in (4.7); step (ii) follows from the Cauchy-Schwarz inequality; step (iii) utilizes Lemma 8 and Lemma 1; while step (iv) follows from Z t ∼ N (0 , I d ) and holds provided that T is sufficiently large and k ≥ log d . Taking the above bounds collectively yields

B Proof of Theorem 2

In view of the update rule (2.1), the variables X 0 , X 1 , . . . , X T are jointly Gaussian, and we can check from (2.2) that

hence the score functions

We first derive the density of X t -1 conditional on X t = x t . Since the joint distribution of ( X t -1 , X t ) is

we can derive that

In addition, with perfect score estimation, we can use (2.3) and (B.2) to achieve

which indicates that

Then we can check that for any x t ∈ R d ,

One can check that

We combine the above two relations as well as the assumption that k ≤ d/ 2 to achieve

By taking expectation w.r.t. x t , we have

where we use the fact that

C Technical lemmas

This section collects a few useful technical tools that are useful in the analysis.

Lemma 8. When T is sufficiently large, for 1 ≤ t ≤ T , we have

In addition, for 2 ≤ t ≤ T , we have

Proof. See Appendix A.2 in [14].

Lemma 9. For Z ∼ N (0 , 1) and any t ≥ 1 , we know thatProposition 2.1.2

In addition, for a chi-square random variable Y ∼ χ 2 ( d ) , we have

Proof. See Proposition 2.1.2 in [27] and Section 4.1 in [12].

Lemma 10. Suppose that T is sufficiently large. Then we have

Proof. See Lemma 3 in [14].

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers the problem of convergence of probability distribution learned by diffusion models to target data distribution when the data distribution lies has some low-dimensional structure. Previous work either provides a convergence rate bound in Wasserstein distance for low-dimensional distribution or provides a bound in total variation distance with polynomial dependence on the ambient dimension. This paper improves the convergence rate to logarithmically depend on the ambient dimension and polynomially on the intrinsic dimension.\n\nSTRENGTHS:\n- It is interesting that the authors do not only consider the case of data on a k-dimensional manifold but consider a general version of a low-dimensional structure provided by the covering number of the support.   \n- The paper is overall well-written and easy to follow.\n\nWEAKNESSES:\n- The result in the paper is proved under the assumption of the bounded support of the data distribution (i.e. $|| x || \\leq R$ where $R = \\textrm{poly}(T)$). The results in the previous works are under the assumption that the data has a bounded second moment therefore, the bounded support assumption is more restrictive than the assumptions considered in the previous work. \n- The argument about the uniqueness of coefficients that leads to dimension-independent discretization error in Section 3.2 is not convincing because it proves a lower bound on the error incurred in one denoising step which is an upper bound on the total variation distance (the authors also mention it).\n\nQUESTIONS:\n- I understand that showing the lower bound on the total variation error in section 3.2 might be difficult but can you experimentally show that the claimed coefficient in eq.(2.4) leads to a better convergence compared to other coefficients (e.g., coefficients used in practice) on some synthetic low-dimensional distribution? \n- The final bound on T in theorem 1 seems to depend on polylogarithmically on the dimension of the distribution (instead of polynomially in previous works). Is the polylogarithmical dependence of dimension necessary? \n\nMinor comment: it might be worth including a discussion in the paper on why earlier approaches get the dimension-dependent error.\n\nLIMITATIONS:\nYes, the authors discuss the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-07"}

comments: SUMMARY:
In DDPM-style diffusion models, we need to discretize a continuous
process.  Benton et al [3] showed that ~ d/eps^2 steps suffice over d
dimensions.  But what if (as is typical) the data lie in a space of
lower intrinsic dimension (e.g. k-sparse or a k-dimensional manifold)?
This paper shows how to replace the d in the iteration complexity by
k^4.

STRENGTHS:
This paper shows that DDPM can be nearly dimension independent, up to
log factors only depending on the intrinsic dimension.  This holds for
a very general definition of intrinsic dimension---just the log
covering number.  I was pretty surprised to learn that such a result
is possible.

Doing so requires particular choices of step size, which they show is
necessary, at least for the typical Girsanov-style proof approach.

WEAKNESSES:
Obviously it would be nice to see k rather than k^4.  We don't
typically expect intrinsic dimension << d^{1/4}, so this isn't giving
a real quantitative improvement.

There's very little intuition for what's going on.  Why are these the
right step sizes?

QUESTIONS:
How does the schedule (2.4) compare to the suggestion from [3]?

What happens if X isn't exactly low dimensional, but is close?  Say,
there is eps TV mass outside the cover.

LIMITATIONS:
None.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-13

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates score-based diffusion models when the underlying distribution is near low-dimensional manifolds in a higher-dimensional space. It addresses the gap in theoretical understanding of diffusion models, which are suboptimal in the presence of low-dimensional structures. For the DDPM, the error dependency on the ambient dimension $d$ is generally unavoidable during each denoising step in the existing literature. However, the authors identify a unique set of coefficients that yields a convergence rate of $O(k^2/\\sqrt{T})$, where $k$ is the intrinsic dimension and $T$ is the number of steps. The analysis employs novel tools that characterize the algorithmic dynamics in a deterministic manner. Additionally, the paper establishes that the DDPM sampler's error, influenced by time discretization and score estimation, is nearly dimension-free, with the ambient dimension $d$ appearing only in logarithmic terms.\n\nSTRENGTHS:\n**1. Theoretical Advancement in Understanding Diffusion Models.** The paper makes a good contribution to the theoretical understanding of score-based diffusion models in the context of low-dimensional manifolds. By identifying a unique set of coefficients that yield a convergence rate dependent on the intrinsic dimension $k$ rather than the ambient dimension $d$, the authors provide a refined theoretical framework that addresses previously suboptimal theoretical support.  \n\n**2. New Analytical Tools and Methodology.** The paper introduces a new set of analytical tools to characterize the algorithmic dynamics of the DDPM sampler in a deterministic manner. This innovative approach allows for a more precise analysis of the error sources—time discretization and score estimation errors.\n\n**3.** In general, the paper is well-written and joyful to read.\n\nWEAKNESSES:\n**1. Strict Assumption on the Data Distribution** In Line 126, the authors assume that the support set of the data distribution is bounded. This assumption is overly restrictive and may not hold for many real-world data distributions, such as Gaussian distributions, which have unbounded support. The authors should consider relaxing this assumption or providing justification for its necessity, as well as discussing the implications of this restriction on the generalizability and robustness of their findings. \n\n**2.  Lack of Empirical Demonstration.** Although this paper presents a solid and sharp theoretical analysis of the convergence rate of DDPM, the authors don't provide any experimental results to support their theory. Without experimental evidence, it is difficult to assess the real-world performance and robustness of the proposed coefficient design and convergence rate improvements.\n\nQUESTIONS:\n**Q1.** Is it widely used in the literature to employ $\\epsilon$-net and cover number to measure the intrinsic dimension of a data distribution? The authors should provide a more thorough discussion on this point. \n\n**Q2.** In Line 126, the authors assume that the support of the data distribution is bounded. However, the support of the Gaussian distribution in Theorem 2 is not bounded. It is believed that this is not a good example to demonstrate the uniqueness of coefficient design.\n\nLIMITATIONS:\nNot applicable", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper discusses the DDPM sampler's capability to adapt to unknown low-dimensional structures in the target distribution, and studies how this informs the coefficient design.\n\nSTRENGTHS:\nThe paper is written clearly and has a nice structure in general. It contributes to an important topic in diffusion models about adapting to lower dimensional structure. In the first part of the paper, the authors show that, with a particular coefficient design (2.4), the TV error of the DDPM sampler has an upper bound that depends on the intrinsic dimension. In the second part of the paper, the authors exemplify the unique choice of the coefficients.\n\nWEAKNESSES:\nThe main weakness of the paper is the generality. The adaptivity to the lower dimensional structure is based on a particular coefficient design, however, this design is not shown to be unique for the TV error, or for general target data. It is this reviewer's opinion that it is better to write the contribution section more precisely in terms of the setup and the scope of the results. The concerns are specified in the questions section.\n\nQUESTIONS:\n1. How does this paper's result compare with [1], which does not require knowing/estimating the manifold? \nCan the result in the paper lead to a tighter bound when a subspace is given a priori? For example, how does it compare to [2]?\n\n[1] Rong Tang and Yun Yang. Adaptivity of diffusion models to manifold structures. In International Conference on Artificial Intelligence and Statistics, 2024. \n[2] Kazusato Oko, Shunta Akiyama, and Taiji Suzuki. Diffusion models are minimax optimal distribution estimators. arXiv preprint arXiv:2303.01861, 2023. \n\n2. Section 3.2 about unique coefficient design is very interesting, however, the results lack generality. \n    - Theorem 2's result is derived in the case that the target data distribution is a standard Gaussian distribution. What can you get when considering a general data distribution?\n    - As the authors noted at the end of the section, the uniqueness shown is with respect to the upper bound of the TV error. How does one make use of the uniqueness with respect to the upper bound in practice? What are the possible ways to address the coefficient design for the actual TV error, or tighten the gap?\n\nLIMITATIONS:\nThe authors have disccussed the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-26"}