Title: Classification Diffusion Models: Revitalizing Density Ratio Estimation
Venue: NeurIPS 2024
Accepted: yes
Keywords: Density Ratio Estimation
Abstract: A prominent family of methods for learning data distributions relies on density ratio estimation (DRE), where a model is trained to *classify* between data samples and samples from some reference distribution. DRE-based models can directly output the likelihood for any given input, a highly desired property that is lacking in most generative techniques. Nevertheless, to date, DRE methods have failed in accurately capturing the distributions of complex high-dimensional data, like images, and have thus been drawing reduced research attention in recent years.  
In this work we present *classification diffusion models* (CDMs), a DRE-based generative method that adopts the formalism of denoising diffusion models (DDMs) while making use of a classifier that predicts the level of noise added to a clean signal. Our method is based on an analytical connection that we derive between the MSE-optimal denoiser for removing white Gaussian noise and the cross-entropy-optimal classifier for predicting the noise level. Our method is the first DRE-based technique that can successfully generate images beyond the MNIST dataset. Furthermore, it can output the likelihood of any input in a single forward pass, achieving state-of-the-art negative log likelihood (NLL) among methods with this property.

Classification Diffusion Models: Revitalizing Density Ratio Estimation

Abstract

Aprominent family of methods for learning data distributions relies on density ratio estimation (DRE), where a model is trained to classify between data samples and samples from some reference distribution. DRE-based models can directly output the likelihood for any given input, a highly desired property that is lacking in most generative techniques. Nevertheless, to date, DRE methods have failed in accurately capturing the distributions of complex high-dimensional data, like images, and have thus been drawing reduced research attention in recent years. In this work we present classification diffusion models (CDMs), a DRE-based generative method that adopts the formalism of denoising diffusion models (DDMs) while making use of a classifier that predicts the level of noise added to a clean signal. Our method is based on an analytical connection that we derive between the MSE-optimal denoiser for removing white Gaussian noise and the cross-entropy-optimal classifier for predicting the noise level. Our method is the first DRE-based technique that can successfully generate images beyond the MNIST dataset. Furthermore, it can output the likelihood of any input in a single forward pass, achieving state-ofthe-art negative log likelihood (NLL) among methods with this property. Code is available on the project's webpage.

1 Introduction

A classical family of methods for learning data distributions relies on the concept of density-ratio estimation (DRE) [46]. DRE techniques transform the unsupervised task of learning the distribution of data into the supervised task of learning to classify between data samples and samples from some reference distribution [15, 4, 35, 7]. These methods have attracted significant research efforts over the years [27, 14, 35, 47], particularly for their inherent capability to directly output the likelihood for any given input. However, to date, they have not succeeded in capturing the distribution of complex high-dimensional data, like natural images. Instead, their generative performance was demonstrated only on low-dimensional toy examples and on the simple MNIST handwritten digits dataset [28, 35, 7]. As illustrated in Fig. 1, while the state-of-the-art DRE method, telescoping density-ratio estimation (TRE) [35], succeeds in capturing the distribution of the MNIST dataset [28], it fails on the slightly more complex CIFAR-10 dataset [26].

As opposed to DRE methods, denoising diffusion models (DDMs) [38, 19] have had unprecedented success in generative modeling of complex high-dimensional data, including images [9, 36], audio [25, 5], and video [12, 1]. This has made them perhaps the most prominent technique for learning data distributions in recent years, with applications in solving inverse problems [21, 37], image editing [33, 16, 3, 20] and medical data enhancement [44, 8], to name just a few. However, assessing the likelihood of data samples is a challenging task with DDMs; it requires many neural function

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evaluations (NFEs) to calculate the likelihood-ELBO [19], or to approximate the exact likelihood using an ODE solver [43].

DDMs are based on minimum-MSE (MMSE) denoising , while DRE methods hinge on optimal classification . In this work, we develop a connection between the optimal classifier for predicting the level of white Gaussian noise added to a data sample, and the MMSE denoiser for cleaning such noise. Specifically, we show that the latter can be obtained from the gradient of the former. Utilizing this connection, we propose classification diffusion model (CDM), a generative method that combines the formalism of DDMs, but instead of a denoiser, employs a noise-level classifier. CDM is the first instance of a DRE-based method that can successfully generate images beyond MNIST (Fig. 1). In addition, as a DRE method, CDM is inherently capable of outputting the exact log-likelihood in a single NFE. In fact, it achieves state-of-the-art negative-log-likelihood (NLL) results among methods that use a single NFE, and comparable results to computationally-expensive ODE-based methods.

Our experiments shed light on the reasons why DRE methods have failed on complex highdimensional data to date, and why CDM inherently avoids these challenges. Furthermore, we show that CDM can serve as a more accurate denoiser, in terms of MSE, than a DDM with a similar architecture. This typically translates into better FID scores. Representative generated samples are shown in Fig. 1. We hope that our approach will spark new interest in DRE methods and ultimately unlock their full potential.

2 Background

2.1 Density Ratio Estimation

Learning data distributions via DRE was first proposed by Gutmann and Hyvärinen [15]. Their noise contrastive estimation (NCE) method uses the fact that the ratio between an unknown distribution p d ( x ) and a known reference distribution p n ( x ) can be extracted from the optimal binary classifier for discriminating samples from p d ( x ) and p n ( x ) . Once this ratio is extracted from the classifier, it can be multiplied by the known p n ( x ) to obtain p d ( x ) . Specifically, let C denote the class of a sample x , with C = 1 , 0 corresponding to the event that x is a sample from p d ( x ) , p n ( x ) , respectively. The optimal classifier for predicting C from x outputs both P ( C = 1 | x ) and P ( C = 0 | x ) . Using Bayes' rule, we can use these values to compute the density ratio

where we assumed that the classes are balanced, so that P ( C = 1) = P ( C = 0) = 1 2 .

Unfortunately, this method fails in practice when p d ( x ) and p n ( x ) differ significantly from one another, as is the case when p d ( x ) is the distribution of images and p n ( x ) corresponds to white Gaussian noise. This is because when training a classifier to discriminate between images and noise, the classifier can achieve very high accuracy even without learning meaningful information about images. When this point is reached, the weights of the classifier practically stop updating. Rhodes et al. [35] referred to this issue as the density-chasm problem , and suggested to overcome it by making the classification problem more difficult. To do so, their TRE method uses a sequence of distributions p x 0 ( x ) , p x 1 ( x ) , . . . , p x m ( x ) , which are closer to one another, such that p x m ( x ) is the reference distribution and p x 0 ( x ) is the target distribution. The intermediate distributions { p x i ( x ) } m -1 i =1 do not have to be known; the only requirement is that it would be possible to sample from them. For example, x i can be defined as x i = √ ¯ α i x 0 + √ 1 -¯ α i x m , where x 0 ∼ p x 0 , x m ∼ p x m , and ¯ α i is a sequence that decreases from 1 to 0. Then, using (1), each ratio p x i ( x ) /p x i +1 ( x ) can be extracted by training a binary classifier to distinguish between samples from p x i ( x ) and p x i +1 ( x ) , and the ratio between the target and the reference distributions can be calculated as

While this method overcomes the density-chasm-problem for each pair of consecutive distributions, it still fails in learning the distribution of datasets that are more complicated than MNIST, as illustrated in Fig. 1. This is because each ratio p x i ( x ) /p x i +1 ( x ) is extracted from a binary classifier trained only on inputs x from the distributions p x i and p x i +1 . For instance, the classifier producing the ratio p x 0 ( x ) /p x 1 ( x ) is trained on inputs close to the real data, while the one producing p x m -1 ( x ) /p x m ( x ) is trained on inputs close to the reference distribution. This can lead to a mismatch between training and test time, since at inference, all the ratios are evaluated at the same input x . Moreover, even if each individual ratio is nearly accurate, the accumulation of small errors can result in a significant overall error. Our method is also based on a classification objective, however it avoids these problems by employing an additional loss, which is based on our main result (Theorem 3.1).

2.2 Denoising Diffusion Models

DDMs [38, 19], are a class of generative models that sample from a learned target distribution by gradually denoising white Gaussian noise. More formally, DDMs generate samples by attempting to reverse a forward diffusion process with T steps that starts from a data point x 0 and evolves as x t = √ α t x t -1 + √ 1 -α t ˜ ε t , t = 1 , . . . , T , where { ˜ ε t } are iid standard Gaussian vectors. Samples along this forward diffusion process can be equivalently expressed as

where ¯ α t = ∏ t s =1 α s . The coefficients { α t } are taken to be such that { ¯ α t } is a monotonic sequence with ¯ α T ≈ 1 . This enforces the density p x T to be close to the normal distribution N (0 , I ) .

The reverse diffusion process is learned by modeling the distributions of x t -1 given x t as a Gaussian with mean

and covariance σ t I , where ε θ ( · , · ) is a neural network and { σ t } are fixed hyperparameters. Training is done by minimizing the ELBO loss, which reduces to a series of MSE terms,

At convergence to the optimal solution, the neural network approximates the timestep-dependent posterior mean

To generate samples, DDMs sample x T ∼ N (0 , I ) and then iteratively follow the learned reverse probabilities, terminating with a sample of x 0 . In more detail, at each timestep t , the model accepts x t and outputs a prediction of the noise ε t (equivalently, a prediction of the clean signal x 0 ), from which x t -1 is obtained by sampling from the reverse distribution. The process described above is that underlying the DDPM method [19]. Here we also experiment with DDIM [39] and DPM-Solver [32], which follow a similar structure.

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3 Method

We start by deriving a relation between classification and denoising, and then use it as the basis for our CDM method. A summary of the notations we use can be found in App. A.

Let the random vector x t be defined as in (3) for timesteps t ∈ { 1 , . . . , T } and set two additional timesteps, 0 and T +1 , corresponding to clean images and pure Gaussian noise, respectively. Namely, we define ¯ α 0 = 1 and ¯ α T +1 = 0 . We denote the density of each x t by p x t ( x ) . Our approach is based on training a classifier that takes as input a noisy sample x t and predicts its timestep t . Formally, let t be a discrete random variable taking values in { 0 , 1 , . . . , T +1 } , with probability mass function p t ( t ) = P ( t = t ) , and let the random vector ˜ x be the diffusion signal at a random timestep t, namely 1 ˜ x = x t. Note that the density of each x t can be written as p x t ( x ) = p ˜ x | t ( x | t ) and by the law of total probability, the density of ˜ x is equal to

We are interested in a classifier for predicting t from ˜ x . It is well known that given any sample x drawn from (7), the optimal such classifier (in terms of the cross-entropy loss) outputs the probability vector ( p t | ˜ x (0 | x ) , p t | ˜ x (1 | x ) , . . . , p t | ˜ x ( T + 1 | x )) , where p t | ˜ x ( t | x ) = P ( t = t | ˜ x = x ) . As we now show, the denoiser in (6) corresponds to the gradient of this classifier.

Theorem 3.1. Let F ( x , t ) = log( p t | ˜ x ( T +1 | x )) -log( p t | ˜ x ( t | x )) with t , ˜ x and x t as defined above. Then √

regardless of the choice of the probability mass function p t , provided that p t ( t ) > 0 for all t .

The proof, provided in App. B.1, consists of three key steps:

Using Bayes rule, we write p x t ( x ) as a function of p x T +1 ( x ) and the optimal classifier.

Then, we take the derivative of the log of both sides and use the fact that ∇ x log p x T +1 ( x ) has a closed form solution.

Finally, we use Tweedie's formula [34, 45, 11] to connect between ∇ x log p x t ( x ) and E [ ε t | x t = x ] .

Theorem 3.1 suggests that we may train a classifier and use its gradient as a denoiser according to relation (8). This paradigm is illustrated in Fig. 2. Once we have constructed a denoiser, we can apply any desired sampling method (e.g. DDPM, DDIM, etc.) to generate images from the learned distribution. However, as we show in Sec. 4.1, naively training such a classifier with the standard

Algorithm 1 CDMTraining

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Algorithm 2 DDPM Sampling Using CDM

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cross-entropy (CE) loss leads to poor results. This is because a classifier may reach a low CE loss even without learning the correct probability p t | ˜ x ( t | x ) for any t . This phenomenon can be observed in Fig. 3, which illustrates the reason that existing DRE methods fail to capture the distribution of high-dimensional complex data like images. We discuss this in more detail in Sec. 4.1.

To obtain the correct probability p t | ˜ x ( t | x ) for any t , we suggest training the classifier with a combination of a CE loss on its outputs, and an MSE loss on its gradient, according to relation (8). The network's gradient can be efficiently computed using automatic-differentiation. Following Ho et al. [19], we use the same weight for all timesteps in the MSE loss. Our full training scheme is described in Algorithm 1. Here, f θ ( x )[ t ] denotes the model's t -th logit, which serves as an approximation for log p t | ˜ x ( t | x ) (up to an additive constant that cancels out in the SoftMax operation), and F θ ( x , t ) = f θ ( x )[ T +1] -f θ ( x )[ t ] . The added timesteps, corresponding to entries 0 and T + 1 of the classifier, are trained only using the CE loss. This is because the prediction of the noise is trivial when t = T + 1 and meaningless when t = 0 (since there is no noise). Importantly, this behavior is automatically achieved without any modification to the algorithm. Specifically, in line 4 of the algorithm, F θ ( x T +1 , T +1) = 0 , and in line 5, ε θ ( x 0 , 0) = 0 , preventing the MSE loss from updating the weights for these timesteps.

Algorithm 2 shows how to generate samples with CDM using the DDPM sampler (a similar approach can be used with other samplers). Note that each step t in DDPM sampling using CDM is given by

where z ∼ N (0 , I ) . Therefore, each step steers the process in the direction that maximizes the probability of noise level t , while minimizing the probability of noise level T +1 . This can be thought of as taking a gradient step with size (1 -α t ) / √ α t , followed by a step in a random exploration direction with magnitude σ t , similarly to Langevin dynamics.

3.1 Exact Likelihood Calculation in a Single Step

To compute the likelihood of a given sample, DDMs are required to perform many NFEs in order to compute a lower bound on the log likelihood using the ELBO [19], or can approximate the exact likelihood [43] using an ODE solver based on repeated evaluations of the network. In contrast, as a DRE-based method, a CDM is able to calculate the exact likelihood in a single NFE. In fact, a CDM can compute the likelihood w.r.t. the distribution p x t of noisy images, for any desired timestep t . Specifically, we have the following (see proof in App. B.3)

Theorem 3.2. For any t ∈ { 0 , 1 , . . . , T +1 } ,

where N ( · ; 0 , I ) is the probability density function of a standard multivariate Gaussian distribution.

Note that the first term in (10) only depends on the pre-selected probability mass function p t (which we choose to be uniform in our experiments), and the second term can be obtained from the t -th and ( T +1) -th entries of the vector at the output of the classifier (after applying SoftMax). This implies

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4 Experiments

We train several CDMs on two common datasets. For CIFAR-10 [26] we train both a class conditional model and an unconditional model. We also train a similar model for CelebA [31], using face images of size 64 × 64 . In Sec. 4.1, we demonstrate why existing DRE methods fail on complex high-dimensional data like images, and show how the incorporation of the MSE loss in our method, according to Theorem 3.1, overcomes these challenges. In Sec. 4.2, we compare our method with pre-trained DDMs with similar architectures, to disentangle the benefits of our method from other variables. We evaluate the performance of CDM as a denoiser, assess its generation quality using FID [17], and measure its likelihood modeling capabilities using NLL. Finally, in Sec. 4.3, we demonstrate the use of different noise schedulers, one specifically tuned for likelihood estimation, and one corresponding to the flow matching optimal-transport scheme [29, 30]. We show that incorporating these schedulers into our method leads to state-of-the-art NLL results among methods capable of outputting the likelihood in a single forward pass.

4.1 The Importance of Using Both Losses for Achieving an Optimal Classifier

In Theorem 3.1, we established that the MMSE denoiser corresponds to the gradient of the optimal noise-level classifier. A natural question is whether we can train our model only with the MSE loss. Unfortunately, the answer is negative. This is because the MSE achieved by the model does not change if we add a function of t to its output, as such an additive term vanishes when taking the gradient with respect to x . The CE loss is important for removing this degree of freedom. Namely, without the CE loss, the model can function as a denoiser but is useless for the purpose of outputing the likelihood in a single step.

Can we train the model only with the CE loss, then? In theory, training the model only with the CE loss should be sufficient. However, as we will demonstrate next, incorporating MSE is crucial in practice for achieving an optimal classifier.

Table 1 reports the MSE, CE and classification accuracy achieved by models trained with different losses. We emphasize that the model trained using only MSE in this comparison, is a CDM model trained using Algorithm 1, and is not equivalent to a DDM model trained using (5). As evident from

the table, when using only the CE loss, the MSE is high, and when using only the MSE loss the CE and classification accuracy are poor. An important point to notice is that even when training with the CE loss, the classifier's accuracy is rather low (though greater than the 0% achieved when training only with the MSE loss). This is a key prerequisite for making DRE methods work. Specifically, as shown in [35], the classification problem should be sufficiently hard in order to avoid the densitychasm problem , otherwise the classifier can easily discriminate between the classes even without having learned the correct density ratio. Yet, as we illustrate next, only making the classification problem harder is still insufficient for learning the probability p t | ˜ x ( t | x ) with only the CE loss.

Figure 3 shows the logits f θ ( x t ) for noisy images with different noise levels, comparing a model trained using CE to a model trained with both CE and MSE. As can be seen, in both scenarios the prediction near the true label is the same, namely the CE works well in the vicinity of the correct noise level. However, the model trained without the MSE loss exhibits significantly higher predicted logits for more distant noise levels compared to the model trained using both CE and MSE. Moreover, the logits of the model trained without MSE do not decrease monotonically as the distance from the actual noise level increases, which is in contrast with the expected behavior. This demonstrates the importance of the MSE loss for obtaining good prediction globally. As can be seen in Theorem 3.1, the denoiser at timestep t depends on the predictions of the classifier in both the t -th and the ( T +1) -th entries. Therefore, the addition of the MSE loss enforces the classifier to achieve accurate predictions in both entries, thereby ensuring accurate predictions globally.

4.2 Denoising Results, Image Quality and Negative Log Likelihood

We compare our method with pre-trained DDMs of similar architectures. Since CDM is a classifier and DDM is a timestep-conditional denoiser, we take the architecture of our CDM to be identical to the DDM, except for altering the last two layers to output a vector of logits, and removing all timestep conditioning layers. These changes have a negligible effect on the number of parameters in the model. For more details please refer to App. C.

As shown in Fig. 4, the denoising performance of our CDM surpasses that of pre-trained DDMs at high noise levels, and is comparable to them at lower noise levels. These quantitative results are corroborated by the qualitative examples in Fig. 5, which showcase image denoising results across various noise levels.

The good denoising performance of CDM translates into high quality image generation. This is illustrated qualitatively in Fig. 1, which shows samples from models trained on CelebA and on CIFAR-10 (unconditional). To quantitatively compare the generation quality of CDM to that of pre-trained DDMs, we use 50k FID [17] against the train-set. For both CDMs and DDMs, we compare images sampled using the DDPM [19], DDIM [39], and DPM-Solver (DPMS) [32] samplers, using 1000, 50, and 25 timesteps, respectively. The results, shown in Table 2, demonstrate that CDM is at least comparable to pre-trained DDMs in image quality, outperforming them in most cases.

Additionally, we evaluate CDM's effectiveness in applying classifier-free guidance (CFG) [18] for conditional sampling tasks. As expected, incorporating CFG improves image quality beyond unconditional generation, as reflected in the conditional CIFAR-10 FID results of Table 2. More details and qualitative results are provided in Appendix C.5. These results showcase the effectiveness of CDM for image generation, showing it to be equal or better than a similar DDM.

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Finally, we calculate log-likelihoods and compare our NLL results to recent leading methods in Table 3. CDM demonstrates comparable performance on NLL estimation for CIFAR-10 compared to DDMs. Notably, CDM stands out as a more efficient method than existing ones, requiring only a single forward pass for NLL computation. Table 3 also includes CDM(unif.), and CDM(OT) on which we elaborate in Sec. 4.3 below. These variants of our model, improve the NLL predictions and achieve state-of-the-art results among methods requiring a single step.

4.3 Different Noise Scheduling for Better Likelihood Estimation

To see the effect of using a timestep scheduler tuned for DRE tasks, we repeat the CIFAR-10 unconditional experiment, with a different noise scheduler. Following Rhodes et al. [35] we use the scheduler defined as √ 1 -¯ α t = t T +1 and choose T = 1000 similarly to our previous experiments. Utilizing this scheduler, we achieve a better NLL of 2.98 at the expense of a higher FID of 10.28, when using the DDIM sampler with 50 steps. This trade-off highlights that the scheduler optimal for learning the data distribution may not be ideal for sampling.

To further explore the influence of the noise scheduler, we train and evaluate a CDM with the flowmatching optimal-transport (OT) scheduler [29, 30] on unconditional CIFAR-10. In this scheduler, x t = T -t T x 0 + t T ε t , where ε t ∼ N (0 , I ) and t ∈ { 0 , . . . , T } . This scheduler leads to a state-of-theart single-step NLL of 2.89 and to an FID of 7.07 with 1000 sampling steps. Please see App. B.4 for more details.

Future research could explore schedulers aimed at further enhancing the NLL. Further analysis of the difference between the schedulers from a classification perspective can be found in App. E.2

5 Related Work

Using the concept of DRE for learning data distributions was initially studied by Gutmann and Hyvärinen [15]. Their noise contrastive estimation (NCE) method approximates the ratio between the density of the data distribution and that of white Gaussian noise. However, it struggles in practical scenarios where the gap between these distributions is large, as is the case for natural images [35]. Conditional noise contrastive estimation (CNCE) [4] is a slightly improved version of NCE, in which

the classification problem is designed to be harder. Specifically, CNCE is based on training a classifier to predict the order of a pair of samples with closer densities, e.g. achieved by pairing a data sample with its noisy version.

Telescoping density-ratio estimation (TRE), proposed by Rhodes et al. [35], avoids direct classification between data and noise. Instead, it uses a gradual transition between those two distributions, and trains a classifier to distinguish between samples from every pair of adjacent densities. Such a classifier learns the ratio between adjacent distributions, and the overall ratio between the data and noise distributions is computed by multiplying all intermediate ratios.

Choi et al. [7] extended this idea from a finite set of intermediate densities to an infinite continuum. This was accomplished by deriving a link between the density ratios for infinitesimally close distributions and the principles of score matching [23, 40, 42], motivating the training of a model to predict the time score ∂ ∂t log p x t . In contrast, we draw a different connection which shows that an MMSE denoiser can be obtained as the gradient of an optimal noise level classifier. Also, to obtain the log ratio between the target and reference distributions, Choi et al. [7] need to solve an integral over the time-score using an ODE solver, while in our method this ratio can be calculated in a single NFE.

Yair and Michaeli [47] extended the concept of TRE, proposing the training of a single noise level classifier instead of training a binary classifier for each pair of neighboring densities. While this method is conceptually similar to ours, our approach distinguishes itself by incorporating the MSE loss as outlined in Theorem 3.1. As demonstrated in our experiments, this proves to be crucial for achieving an optimal classifier and high-quality image generation.

6 Discussion and Conclusion

We developed an analytical connection between an MSE-optimal denoiser for removing white Gaussian noise and a cross-entropy-optimal classifier for predicting the noise level. We used this connection to propose CDM - a DRE based generative technique that is based on a noise-level classifier. Importantly, our classifier is trained using both a classification loss (CE) and a regression loss (MSE). We showed that this key component is what sets CDM apart from existing DRE based methods, and makes it the first instance of a DRE-based technique that can successfully generate images beyond MNIST.

Our approach is not free of limitations. A key challenge is that CDMs can be more computationally expensive than DDMs. Indeed, while DDMs require a single forward pass for each denoising step, CDMs require both a forward pass and a backward pass. Nevertheless, the computational cost of performing a forward and a backward pass through a network depends on its architecture. In this work, we chose to use the same architecture as that used by DDPM [19], in order to isolate the impact of our algorithmic approach from the choice of the model architecture when comparing to DDMs. However, an important future direction would be to explore architectures that are particularly optimized for CDMs and that alleviate the gap in computational complexity. Such architectures should have the property that performing a forward pass and a backward pass through them is computationally similar to performing only a forward pass in a regular DDM. This could potentially be achieved e.g. , by relying only on the encoder part of the U-Net. However, we leave this exploration for future work.

Broader Impact CDM is a generative model and thus may potentially suffer from the same limitations as other generative techniques. These include biases in the generated images, as well as malicious and offensive use, such as creating Deepfakes for disinformation. However, CDMs may also impact domains that rely on generative models in a positive way. This is because, different from most generative models, CDMs are able to compute the likelihood for any input in a single step. This may be used e.g. , for out-of-distribution detection or for ranking the likelihoods of different restored images in image restoration tasks. Such capabilities may be crucial in fields like medical imaging.

Acknowledgments

This research was partially supported by the Israel Science Foundation (ISF) under Grant 2318/22.

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A Notation

B Proofs

B.1 Theorem 3.1 Proof

Using Bayes rule,

Combining (11) and (12) yields

Taking the logarithm of both sides, we have

Taking the gradient of both sides w.r.t x , and noting that the first term on the right hand side does not depend on x , we get that

Since p x T +1 ( x ) = N ( x ; 0 , I ) , we have that ∇ x log( p x T +1 ( x )) = -x . Therefore, overall we have

As for the left hand side of (15), since x t = √ ¯ α t x 0 + √ 1 -¯ α t ε t and ε t ∼ N (0 , I ) , using Tweedie's formula [34, 45, 11] it can be shown that

For completeness we provide the full proof for (17) in App. B.2. Substituting (16) into (17) and multiplying both sides by √ 1 -¯ α t gives

In particular, for t = T +1 , this relation reads

which completes the proof.

Note that the proof is correct for any choice of p t (provided that p t ( t ) > 0 for all t ∈ { 1 , .., T +1 } ), since the term that depends on p t does not depend on x and thus drops when taking the gradient. In practice, as mentioned in the main text, we chose to use t ∼ U ( { 0 , 1 , ..., T +1 } ) .

B.2 Proof of Tweedie's Formula

Let us show that if y = µ x + σ z with z ∼ N (0 , I ) statistically independent of x , then ∇ y log( p y ( y )) = -1 σ E [ z | y = y ] .

From the law of total probability,

Taking the gradient w.r.t. y gives

Dividing both sides by p y ( y ) , we get

Now, substituting µ = √ ¯ α t , σ = √ 1 -¯ α t , y = x t , y = x t , and z = ε t , leads to

demonstrating (17).

B.3 Proof of Theorem 3.2

Substituting p x T +1 ( x ) = N ( x ; 0 , I ) into (13) leads to

which proves Theorem 3.2.

B.4 Proof of CDM for the Flow Matching Optimal Transport Scheduler

In the case of the flow matching optimal-transport scheduler [29, 30], x t is defined as

where ε t ∼ N (0 , I ) and t ∈ { 0 , . . . , T } . In this case, the objective of conditional flow matching is [29, 30]

and the global minimum of this objective is achieved by

We will start by expressing this solution in terms of the optimal denoiser E [ ε t | x t ] . Substituting (24) into (26) and using the fact that E [ x t | x t = x ] = x gives

Next, following exactly the same logic as in App. B.1, we can write

Finally, by substituting (28) into (27) we get

C Implementation details

C.1 Architectures

For a fair comparison, for each dataset we used the same architecture for our method and for DDMs. As baslines we took the pre-trained model for CelebA 64 × 64 from the DDIM Official Github [39] and the EMA pre-trained model for CIFAR-10 from the pytorch diffusion repository, who converted the pre-trained model from the Official DDPM implementation from tensorflow to pytorch.

For conditional CIFAR-10 we trained the DDM model by ourselves because there exist no pre-trained models for CIFAR-10 capable of handling CFG. We used the same architecture from [19] for both models and trained them for the same number of iterations (more details are in App. C.2.2). To condition the model on class labels, we learned an embedding for each class using nn.Embedding and injected it at all points where the time embedding was originally applied. In the case of DDM, we added the class embedding to the time embedding, while for CDM, we replaced the time embedding with the class embedding.

In contrast to DDM architectures, our model does not need the layers that process the timestep input so we removed them. In addition, our model outputs a probability vector in contrast to DDMs which output an image, therefore, we replaced the last convolution layer that reduces the number of channels to 3 in the original architecture by a convolution layers outputs 1024 and 512 channels for CelebA 64 × 64 and CIFAR-10, respectively. Following this layer, we performed global average pooling and added a linear layer with an output dimension of T +2 . The resulting change in the number of parameters is negligible.

̸

Inspired by Yair and Michaeli [47], we added a non-learned linear transformation at the output of the network, which performs cumulative-summation (cumsum). This enforces (for the optimal classifier) the t -th output of the model before this layer to be log r t ( x ) = log p t | ˜ x ( t | x ) p t | ˜ x ( t -1 | x ) = log p t | ˜ x ( t | x ) -log p t | ˜ x ( t -1 | x ) for t = 0 and r 0 ( x ) = log p t | ˜ x (0 | x ) , so that after the cumsum layer, the t -th output is ∑ t i =0 log r i ( x ) = log p t | ˜ x ( t | x ) .

C.2 Hyperparamters

C.2.1 CelebA 64 × 64

We trained the model for 500k iterations with a learning rate of 1 · 10 -4 . We started with a linear warmup of 5k iterations and reduced the learning rate by a factor of 10 after every 200k iterations. The typical value of the CE loss after convergence was ∼ 3 . 8 while the MSE loss was ∼ 0 . 0134 so we chose to give the CE loss a weight of 0 . 001 to ensure the values of both losses have the same order of magnitude. In addition We used EMA with a factor of 0.9999, as done in the baseline model.

C.2.2 Unconditional and Conditional CIFAR-10

We trained the model for 500k iterations with a learning rate of 2 · 10 -4 . We started with a linear warmup of 5k iterations and reduced the learning rate by a factor of 10 after every 200k iterations. The typical value of the CE loss after convergence was ∼ 4 . 4 while the MSE loss was ∼ 0 . 03 so we chose to give the CE loss a weight of 0 . 001 to maintain values at the same order of magnitude. In addition, we used EMA with a factor of 0.9999, as done in the baseline model.

For the CDM(unif.) model we used the same hyperparametes except for learning rate, which we set to 1 · 10 -4 .

The CDM(OT) model was trained with the same hyperparameters, except for the learning rate, which was set to 1 · 10 -4 . Additionally, we trained the model without a learning rate schedule for 1M iterations, as the NLL continued to decrease beyond 500k iterations.

C.3 Compute Resources

C.3.1 CelebA 64 × 64

Training the model on CelebA 64 × 64 takes 108 hours on a server of 4 NVIDIA RTX A6000 48GB GPUs. Sampling 50k images for FID calculation takes 16 hours on the same hardware.

C.3.2 Unconditional and Conditional CIFAR-10

Training the model on CIFAR-10 takes 35 hours on a server of 4 NVIDIA RTX A6000 48GB GPUs. Sampling 50k images for FID calculation takes 9 hours with classifier free guidance and 5 hours without classifier free guidance on the same hardware.

C.4 Data Augmentation

Following the models to which we compared, for all the datasets we normalized the images to the range [ -1 , 1] and applied random horizontal flip at training.

C.5 Classifier Free Guidance

We used CFG to sample conditioned examples in Sec. 4.2 following [18]. We trained our own conditional models on the CIFAR-10 dataset, both for DDM and for CDM, and used label dropout of

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C.6 The Addition of The Timesteps 0 and T +1

As outlined in Sec. 3, we introduce additional timesteps, corresponding to x 0 , x T +1 , which are not present in DDMs. This inclusion is fundamental to our method formulation. Importantly, this addition does not alter the number of sampling steps, as we continue initiating the reverse process from t = T and finish it with the denoised result of t = 1 , following the approach in DDM.

D Validating the Log Likelihood Computation on Toy Examples

We validate our efficient likelihood computation by applying it to toy examples with known densities, allowing us to compute the analytical likelihood and verify that our computations align with theoretical expectations.

D.1 Images With Independent Pixels

We start by experimenting with the uniform distributions p γ = U [ -γ 2 , γ 2 ] 3 × 32 × 32 with γ ∈ { 0 . 25 , 0 . 5 , 1 , 2 , 3 } . We trained a CDM separately for each of these densities following the procedure outlined in Algorithm 1. These distributions correspond to 32 × 32 color images of iid uniform noise. The probability density function (pdf) of the uniform distribution U [ -γ 2 , γ 2 ] is given by

The pdf p γ ( x ) in the d -dimensional space is the product of the individual pdfs for each dimension. Since the dimensions are independent, the joint pdf is given by

The logarithm of the pdf p γ ( x ) is given by -d ln( γ ) . The log-likelihood is the expectation of ln p γ ( x ) . Since ln p γ ( x ) is constant, its expectation is that constant. Therefore, the log-likelihood normalized by d is -ln( γ ) . In the left subplot of Fig. 7 we compare the analytical log likelihood computed using (32) with the estimated log likelihood by our model.

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̸

To further validate our likelihood calculation, we tested it on toy examples of images with correlated pixels. Specifically, we used images of size 3 × 32 × 32 sampled from multivariate Gaussian distributions with µ = 0 and Σ = I . The normalized log likelihood per dimension for a multivariate Gaussian is given by

We defined a sequence of distributions with Σ = √ 1 -γ Σ CIFAR-10 + √ γ I , where Σ CIFAR-10 is the empirical covariance matrix of the CIFAR-10 dataset, and γ ∈ { 0 , 10 -5 , 10 -3 , 10 -1 } . In the right subplot of Fig. 7, we compare the analytical log likelihood computed using (32) with the estimated log likelihood by our model.

E More Experiments

E.1 Analysis of the Effect of Training with Different Losses

Figure 8 provides further qualitative analysis of the effect of training with different losses on the quality of the model's image denoising capabilities. The results illustrate that the model trained using only CE loss achieves poor denoising quality compared to the CDM model trained with both CE and MSE.

E.2 The Influence of Different Schedulers on the Classifier Performances

First, to assess the classifier's performance, we present the confusion matrices of models trained with and without MSE loss in Fig. 9a. Notably, at lower noise levels, the classifier exhibits high confidence, while at higher noise levels, confidence diminishes. This finding corresponds to the DDPM scheduler [19], which partitions the noise levels to be more concentrated for high timesteps and less concentrated for low timesteps. The similarity between the confusion matrices underlines that CE loss alone is adequate for accurate predictions around the real noise level. However, as demonstrated in the main text, this does not imply that the classifier is optimal in terms of learning the correct logits for any given t .

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a class of generative models based on neural networks which learn the conditional probability distribution of the noise level given a noisy image. It is trained by combining a maximum-likelihood (cross-entropy) objective with a denoising score matching objective. The resulting generative model can be sampled from by iterative denoising like in diffusion models, and also allows for direct likelihood evaluation (in principle).\n\nSTRENGTHS:\n- The proposed model is original and very interesting. Such an approach has the potential to significantly improve upon diffusion models.\n\n- The experimental results are very convincing (except for NLLs, see below) and show great promise.\n\n- The analysis of the shortcomings of density ratio estimation is also on point and very clear. Although these issues were already evidenced in the previous work of Choi et al., their results were limited to MNIST while CDMs achieve satisfying results on CIFAR10 and CelebA 64x64, and have the added potential benefit of direct likelihood evaluation.\n\nWEAKNESSES:\nThe biggest weakness of the paper to me is the method used for likelihood evaluation, which has several issues:\n- An implicit assumption in the paper is that the marginal distribution of the noise level learned by the model matches the correct one used during training, that is, $p_\\theta(t) = p(t)$. Although reasonable, it is very difficult to verify, as it requires marginalizing over the high-dimensional variable $x$.\n- There is no guarantee that the resulting density function (eq. (10)) is normalized (i.e., integrates to 1). This can lead to arbitrarily small _reported_ NLLs (potentially even smaller than the entropy of the data). Although the soundness of the approach is verified in a synthetic but high-dimensional example, I think it would be necessary to verify at least this property on a natural image dataset, though this is also very hard to do (but see below).\n- There is also no guarantee that the density function in eq. (10) is the density of samples generated by the reverse diffusion (Algorithm 2). CDMs thus contain a priori two models of $p(x)$, one for sampling and one for likelihood evaluation, which could be different. I would also like to see in the final version of the paper (not for the rebuttal) a comparison between the NLLs of these two models on natural images (the latter can be evaluated with an integral along the sampling trajectory using the change of variable formula, as pointed out in the text). If they match, this also verifies that the reported densities are correctly normalized (see previous point).\n- Finally, restricting NLL comparisons to \"models where likelihood can be evaluated in single shot\" seems slightly arbitrary, especially since here sampling is not in a single shot. It would make sense to allow a similar computational complexity for sampling from the model and evaluating the log-density at a data point.\n\nI might be missing subtle details in my assessment though, and I would be happy to update my score if the authors point out misunderstandings.\n\nQUESTIONS:\n- In the proposed approach, $t = T+1$ plays a special role, which seems slightly unsatisfying to me. Having correct log-density estimates using eq. (10) requires an accurate estimation of $\\log p(T+1|x)$ for $x$ a clean image, which is very challenging as natural images are very far from typical realizations of Gaussian noise (unless the explicit form of the Gaussian density is enforced here?). However, the MSE loss (which indirectly involves $\\nabla_x \\log p_\\theta(T+1|x)$) seems to fix this problem. Still, is there a way to avoid it altogether with a more symmetric formulation? I liked how the approach of Choi et al. elegantly solved it by considering a path $(x_t)$ of typical realizations of each $p(x|t)$, though I understand that in their setting log-density estimation requires computing a (one-dimensional) integral.\n\n- Out of curiosity, what does Figure 3 look like when the model is trained with MSE loss only? A first observation is that one can add an arbitrary function of $t$ only to the model outputs without changing the MSE loss, so the predicted conditional distributions cannot be right. However, I wonder if this is the only degree of freedom: does the difference between the \"true\" and predicted $p(t|x)$ depend on $x$? If true, it would show that the CE objective just fixes this constant and is thus not so important.\n\n- I am very puzzled by the DDM denoising result at high noise in Figure 4. Why would they be so bad? Predicting the mean of the dataset is really easy, so this is not a capacity issue but rather a training issue? And why CDM would be specifically better at large noise levels? Is this because $T+1$ plays a special role?\n\nMinor suggestion:\nI understand that the \"classification\" term comes from historical considerations in the density ratio estimation problem. However, it seems ill-adapted here: $t$ is really a continuous variable (which has been discretized, but this does not seem necessary), and therefore $p(t|x)$ is simply a probabilistic model of a conditional distribution that is trained by maximum-likelihood. The title of the paper also sets the incorrect expectation that it is about unifying diffusion models with image classifiers.\n\nLIMITATIONS:\nAs mentioned in the discussion, an important aspect that is not studied in this paper is the architecture of the network to approximate $\\log p(t|x)$. The community has extensively explored architecture to model scores $\\nabla_x \\log p(x|t)$, but architectures for energies have been much less studied. As rightfully pointed out, here $f$ is a UNet, so that $\\nabla f$ is the gradient of a UNet and thus significantly differs from a UNet (which would be the goal to emulate score-based approaches). Let me suggest a few references that consider this problem:\n- Tim Salimans and Jonathan Ho. Should EBMs model the energy or the score? In Energy Based Models Workshop-ICLR, 2021.\n- Samuel Hurault, Arthur Leclaire, and Nicolas Papadakis. Gradient Step Denoiser for convergent Plug-and-Play. In International Conference on Learning Representations, 2021.\n- Regev Cohen, Yochai Blau, Daniel Freedman, and Ehud Rivlin. It has potential: Gradient-driven denoisers for convergent solutions to inverse problems. In Advances in Neural Information Processing Systems, 2021.\n\nAll in all, I very much like this paper, and strongly recommend acceptance. It proposes novel ideas of interest to the generative modeling community, and the strong numerical results demonstrate that this approach has promising potential.", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nA common approach for likelihood estimation is the density ratio estimation (DRE). DRE is a method of modeling the density of a target distribution (or data distribution) in the form of a density ratio with a known reference distribution. The standard normal distribution is often chosen as the reference. To train the density ratio models, a classification-based method called noise contrastive estimation (NCE) is generally used.\n\nHowever, in high dimensions, the difference between the reference and the data distribution is large, making the classification problem very easy. Consequently, the model does not learn accurate density ratios. To overcome this saturation problem, some have proposed to create intermediate distributions between the target and reference, similar to diffusion models. For example, the transporting ratio estimation (TRE) first creates a bridge between the target and the reference distribution through their linear combinations. The density ratio between two adjacent timesteps—$t$ and $t+1$-th timestep—is trained using the NCE method. However, despite the diffusion-like modifications, TRE has failed to resolve the saturation problem in high-dimensional data.\n\nTherefore, to overcome the limitation of the previous approaches and estimate the log-likelihood efficiently for the high-dimensional datasets, the paper proposes a novel DRE-based diffusion-based generative model. Most importantly, the paper proposes to predict the timestep $t$ for a given observation $x$—$p(t | x)$—when $x$ is sampled uniformly from all intermediate distributions in the bridge. It means that the model predicts how noisy the given input is, unlike the classification problems proposed in TRE. The paper highlights two important benefits of the proposed method.\n\nFirst, the authors draw the analytical connection between the noise-predicting classifiers and the Bayes-optimal denoisers at each time step. This connection means conventional parameterizations in diffusion models, such as epsilon/score/denoiser, can also be written by the noise-predicting classifiers. Thus, the noise-predicting classifiers can be trained via denoising score matching (DSM) as well as the cross-entropy loss. This training overcomes the saturation problems that originated from classification-based training.\n\nSecond, even compared to the TRE, regardless of its scalability issue, the noise-predicting classifiers provide efficiency at the log-likelihood evaluation. The proposed method can estimate the log-likelihood of the (clean) data in one-g, while the TRE requires $T$-number of model evaluations.\n\nIn addition, the paper mentions a concurrent work that proposes a similar noise-predicting classifier, but the authors emphasize that the concurrent work didn't incorporate the DSM loss, which is the key ingredient of the proposed approach's scalability. \n\nFinally, the authors demonstrate the proposed method's effectiveness via various experiments, including generative modeling of image generation benchmark datasets.\n\n\n------------------------\nUpdate the rating from 7 to 8 after the authors' rebuttal\n\nSTRENGTHS:\nOne of the paper's key contributions is the introduction of a novel solution for likelihood estimation. The proposed method addresses significant challenges in the density ratio estimation and offers a fresh perspective on the parameterizations in diffusion-based models.\n\nMoreover, the authors effectively motivate the use of DSM by establishing the connection between the time-prediction classifiers and the conventional parameterizations. This training overcomes the limitations of previous DRE-based approaches.\n\nFurthermore, the proposed method exhibits comparable performance and scalability compared to popular diffusion-based generative models, further underscoring the practical effectiveness of the proposed DRE-based method.\n\nWEAKNESSES:\nWhile the proposed method's originality and novelty are clear, its significance is still doubtful. While achieving efficient likelihood estimation is the primary motivation, the paper doesn’t provide sufficient discussion about it—instead, it focuses on demonstrating the improved generation performance of the proposed DRE-based method. Thus, the proposed method would be much more impactful if the authors could demonstrate what becomes available when the likelihood estimation for high-dimensional data works.\n\nAdditional discussion about the importance of the classification loss may seem required. It is clear that the time-prediction classifier links to the conventional denoiser parameterization, and thus, this connection equips the use of DSM to train the classifiers, which enables the proposed model's applicability to high-dimensional data. However, this also implies the possibility of training the proposed models without any classification objective.\n\nQUESTIONS:\nN/A\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a clever connection between Density Ratio Estimation (DRE) and Diffusion Models (DM), showing that optimal denoisers are also optimal noise classifiers. Doing so allows them to construct a new type of loss based on noise classification. This allows DRE methods to inherit the benefits of diffusion models, like strong sampling, while adding new capabilities like single step likelihood estimation.\n\nSTRENGTHS:\n- The mathematical connection is clever and elegant.  \n- I liked the framing / connections to DRE literature.  \n- The single step likelihood calculation gives a distinguishing feature that sets the method apart from diffusion models. \n- Reasonable ablations were considered, and clear and readable notation was used.  \n- The method requires an additional backward pass for each denoising/sampling step. In the spirit of the NeurIPS guidelines, I count this clearly labeled limitation as a strength. It's intellectually interesting and also clearly distinguishes that the computation done by their model is different from regular diffusion models in a non-trivial way.\n\nWEAKNESSES:\nThe proposed approach is functionally equivalent to a diffusion model (using Eq. 8 and with necessary incorporation of MSE loss that matches typical diffusion models), so you might hope to see that you can substantially improve on diffusion models by adding the CE loss as a kind of regularizer. But there are several reasons why it isn't clear that this gives a susbtantial improvement. \n- The architecture has to be changed. There's no obvious way to get around that, but it does marginally affect the ease of use for these ideas, and the directness of comparisons (though on this point I agree with the authors' statement that they chose a minimal intervention).  \n- MSE improvements (Fig. 4) are only marginally different in absolute terms, which is what matters for log likelihood. See some questions about this result. \n- The authors only used / compared with relatively old diffusion models, so it's not clear if modest benefits to MSE or FID in some cases would still be seen in more SOTA models. I would expect much more comprehensive experiments to support the idea that using CE loss can improve sampling (Table 2). \n\n\nThe strongest claim of improvement is the single step likelihood calculation. However, there are three weaknesses associated with this result. \nTable 3 is a bit of a dubious comparison for two reasons. \n- Unlike the ELBO, it's not clear if the proposed estimator gives an upper bound on NLL. I suspect it doesn't. If there are errors in p(t|x), then because it shows up with both signs in Eq. 10, it could contribute to the error in either direction. So we are comparing bounds on NLL to an estimate of NLL which could look lower due only to error.  \n- The NFE comparisons are also a bit dubious. Function evaluation complexity varies wildly between methods. We should have a fair comparison to the associated diffusion model with the same architecture, but it wasn't clear which one that is in the table. Though it's a bit moot as I wouldn't trust the result anyway, as we don't have a bound here. \n- A compelling application of the single step log likelihood evaluation was not discussed. If the log likelihood had been used and evaluated for some downstream application, it might have assuaged worries about the accuracy of the estimate. \n\n\nMinor comments:\n- I didn't like the abstract sentence \"directly output likelihood...lacking in most generative techniques\", since most common generative techniques today do output likelihood. You clarify later what you mean by \"directly\" so it is Ok. I just wanted to say that it was off-putting on first read.\n\nQUESTIONS:\nI was really surprised by the fact that your architecture has no timestep conditioning, but matches or outperforms DDMs that do have timestep conditioning in denoising MSE. Have I understood that correctly (i.e. that the DDM in Fig. 4 does have timestep conditioning?) If so, that seems like a really interesting conclusion for diffusion research, as the final appearance of the timestep only shows up in a simple way in Eq. 8, potentially leading to a simplification of other architectures. It would be interesting to see if this effect persists for stronger architectures. On the other hand, maybe the right way to interpret this result is that your architecture essentially outputs the result for all time-steps at once. Eq. 8 selects the correct one gets a score through backprop. Still, that may be an interesting possibility to try to explore for standard diffusion models. \n\nIn Fig. 3, I can understand why the classifier is limited, but it wasn't totally clear to me why MSE helps. But there may not be an easy answer to that (besides the superficial one that it is a regularizer of some sort).\n\nLIMITATIONS:\nLimitations were well addressed.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-07"}

comments: SUMMARY:
This work develops a new generative framework called the classification diffusion models (CDMs) based on the density ratio estimation (DRE), by establishing an interesting connection between the DDPM's denoiser and noise-predictive classifier, which also helps the exact likelihood computation in a single pass. As is claimed by authors, the proposed method is the first DRE-based technique to successfully generate images of the CIFAR-10 dataset.

STRENGTHS:
1. This work is well-written and clearly formulated. 
2. The proposed algorithm is theoretically grounded. 
3. The experiments are convincing, which verify the effectiveness of proposed methods.

WEAKNESSES:
1. The main concern is the capability or potential of DRE-based methods to successfully learn data distributions, particularly when applied to datasets with large scales in practice. As is claimed by authors, the current development of DRE-based modeling is up to the CIFAR-10 dataset (MNIST before), which is too small to indicate the validity compared to more standard methods (e.g. DDPM). 
2. Another major concern is that CDMs can be more computationally challenging than DDPMs in sampling. According to Algorithm 2, CDMs require an extra backward propagation (BP) than DDPMs for each time step when denoising, which is quite expensive.

QUESTIONS:
1. Please provide more details of questions raised in the weaknesses section above.
2. Check minor grammar typos, e.g. in Line 258, " trains a classifiers...".

LIMITATIONS:
As is stated by authors, it is worthy to explore BP-friendly architectures to alleviate the computation inefficiency of CDMs.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-13