Can Diffusion Models Disentangle? A Theoretical Perspective

First of all, I think that the presentation style of the theoretical part of this paper leaves much to be desired: (i) very often notation is used before it is introduced; moreover, in many cases this notation is used without being introduced/defined anywhere in the main part of the paper; (ii) sometimes the notation is inconsistent; (iii) lot's of typos, and I do not mean just textual ones, there are typos in the formulae in the main part of the paper. These issues make it really hard to understand and evaluate theoretical contribution of the paper. Below I list some examples of such issues occurring only in Section 2 and subsection 2.1 (it's just two and a half pages):

1. In lines 068-069 $Y$ stands for classification tasks, in lines 070-071 $Y$ is referred to as labels corresponding to classification tasks.
2. Content random variables $Z$ come from distribution with density $q_{Z}$ in many places in the paper (e.g. line 068), but in line 182 (Assumption 2, item 5) the content density is denoted by $p_{Z}$.
3. Speech features $X\sim q_{\gamma}$ in line 064 and speaker identity $G\sim\gamma$ in line 069. Neither $\gamma$ nor $q_{\gamma}$ are defined. If $\gamma$ is distribution of the random variable $G$ corresponding to speaker identity, then $q_{\gamma}$ should be its density. Why then do speech features $X$ come from the distribution with the density $q_{\gamma}$?
4. Line 077 (Assumption 1, item 3) $d$ is not defined. Functions $I$ and $h$ are not defined either. I understand that they denote mutual information and entropy, but it's better to state that explicitly, especially if you also use the same notation $h$ in line 267 for a multi-class classifier.
5. It's better to be precise and call $B_{t}$ a Wiener process rather than just some Markov process (line 086-087), because later you explicitly use that diffusion process (2) is the one relying on Gaussian noise when you write down Formula (5).
6. In Formula (2) there should be just $X_{t}$ instead of $X_{t}^{\leftarrow}$.
7. Lines 089-090: word "wehe".
8. Formula (3): there should have been $f(X_{t}^{\leftarrow}, A, T-t)$ instead of $f(X_{t}^{\leftarrow}, A, t)$ in the drift coeficient of the reverse diffusion similarly to diffusion coefficient $\nu(T-t)$. The same mistake in the drift coefficient can be found in Formula (7) and the one in lines 147-150, while Formula (10) has the right drift term $f(X_{t}^{2\leftarrow 1}, A, T-t)$.
9. In the same Formula (3) there must be $dB_{t}^{\leftarrow}$ instead of $dB_{t}$ since we switch to the reverse-time dynamics.
10. In lines 097-098 $\chi$ is not defined.
11. In lines 121 we start considering speech features $X^{1}\sim q_{\beta,0}$ while before we had speech features with densities $q_{\gamma}$ and $q_{\alpha}$. None of $\alpha$, $\beta$ or $\gamma$ are defined in the paper up to this moment. Am I correct that $\gamma$ are all possible speech features, and $\alpha$ and $\beta$ correspond to speech features for two particular speakers? Moreover, $\gamma$ is used later in the definition of the implicit bottleneck variable (Formula (11)), and it certainly has another meaning than when it is used in the expressions like $q_{\gamma}$.
12. In Formula (9) the drift term should be $f(X_{t}^{1},A,t)dt$ rather than just $-f(X_{t}^{1},A,t)$ without time differential.
13. In Formula (10) $\hat{\theta}$ is not defined.
14. In Formula (10) you use $B_{t}^{\leftarrow}$ which must be Wiener process in reverse-time, but it is never mentioned in the paper.
15. The acronym AEVC in lines 127-128 is not defined. Does it mean "Autoencoder Voice Conversion"?
16. In Formula (11) in the definition of the implicit bottleneck variable (which is key to the whole theoretical framework you've designed) you use undefined notation $G_{<t}$. This indeed makes it very difficult to understand what is going on after in this section.
17. In the formula in lines 147-150 in the third term you should have $d\tau$ instead of $dt$.
18. In Formula (13) an undefined function $\hat{g}$ is used.
19. In Definition 3 you use $d_{TV}$ distance without mentioning what it is.
20. In Assumption 2, item 1 you define $t^*$, but do not use it anywhere in this Assumption.

There is also a mistake in Formula (5): it suggests that the score function is $-(X_{t}-X_{0})/\sigma^{2}(t)$ for all $t$, but this is not true. E.g. in case of Variance Preserving diffusions [1] the score function is $-(X_{t}-a(t)X_{0})/\sigma^{2}(t)$ where $X_{t}|X_{0}\sim a(t)X_{0}+\sigma(t)\mathcal{N}(0,1)$ and $a(t)$ varies in $t$.

The next weakness regards the notion of implicit bottleneck variable. First of all, I feel that the definition itself is ambiguous in the sense that the functions $\eta$, $\zeta$ and $\gamma$, if they exist, are not unique, and they are not even unique up to some linear transform that could preserve "information" contained in, for example, $\zeta(X_{t}^{\leftarrow})$. A simple example is as follows: suppose $X_{t}^{\leftarrow}=|X_{0}^{\leftarrow}|$, then we can have either $\zeta(X_{t}^{\leftarrow})=X_{0}^{\leftarrow}$ and $\eta(z,g)=|z|$, or $\zeta(X_{t}^{\leftarrow})=|X_{0}^{\leftarrow}|$ and $\eta(z,g)=z$. And further assumptions (Assumption 2) rely heavily on properties of $\zeta^{\star}=\zeta(\hat{X}_{<t^{*}}^{\leftarrow})$, e.g. on values like $I(\zeta^{\star};X)$ that are different in the two cases.

The former issue leads to the next concern: we really do not know much about this implicit bottleneck variable $\zeta^{\star}$. What are the sufficient conditions for it to exist? Are items 1 and 4 in the Assumption 2 feasible? This issue is very important, because you build your theory in Section 2 based on the random variable $\zeta^{\star}$ which is not even guaranteed to exist (it corresponds to the case $t^{*}=0$ if I'm not mistaken). Apart from this, I also have some less important considerations which I'll list below as questions.

As for experimental part, I also feel it has serious drawbacks. As far as I understand, the whole theoretical framework derived in Section 2 is tested only on synthetic datasets while the authors claim that they describe the mechanism behind disentanglement in general diffusion models. Moreover, after this claim they start describing their setup in "intuitive" voice conversion terms which may be a bit confusing. If they do so, it would be good to demonstrate some of their findings on at least one example not related to voice conversion.

In the experiments on synthetic datasets, the authors consider linear subspace diffusion models which decompose speech features following Equation (18). The papers you refer to before this equation to justify this decomposition deal with discriminative tasks, but you consider voice conversion, a generative task. I don't know papers that successfully use this type of decomposition in generative tasks, and it seems very unlikely that such a simplified point of view on speech features decomposition can lead to a good quality of generated speech. So, I do not think that the results of these experiments on synthetic datasets can be extrapolated to real voice conversion.

The experiments on realistic datasets correspond to theoretical findings in Section 3 which are actually more related to ensembling theory than to disentanglement in diffusion models. Altough I do not have much experience in ensembling methods, I think that the first conclusion the authors make ("Adding target speakers reduces speaker distortion") is not surprising and can be explained not only by Theorem 4, but by the reduced variance as well: when the number of target speakers for voice conversion increases, the variance introduced by different performance of classifiers on different (imperfectly generated) voices gets lower resulting in improved performance in classification task when some kind of aggregation mechanism is used. Two other conclusions ("Different VC excels at different tasks" and "Tradeoff between classifier accuracy and diversity") are quite interesting, but they can not be explained by the theoretical framework the authors have developed.

Overall, I think that the theory trying to explain disentanglement in diffusion models presented in this paper lacks convincing experimental material, has some issues regarding feasibility of the assumptions crucial for this theory, and the text itself has to be made more readable by getting rid of numerous typos/mistakes/inconsistencies in notation.

[1] Score-Based Generative Modeling through Stochastic Differential Equations, Song et al.

Personally I experienced some difficulties in following the math. Especially the derivations of the different bounds are rather complex, and I am not sure I fully understand the process. One might also comment that - because the basic theory around diffusion models is quite general - perhaps the focus of this paper specifically on speech tasks is rather limiting. It would be interesting to see if these conclusions hold also across modalities, for instance around image based tasks such as detection / segmentation / recognition from disentangled features.

• In general the paper lacks focus, and the interleaving of information-theoretic results regarding disentanglement and details specific to the voice conversion application leads to both heavy notation, and a confusing narrative. The paper's contributions are not clear.
• The experiments are meant to provide validation for the presented theorems, but strong links are not established. Any novel theoretical results may be better served by a separate publication focusing on them. Similarly, to the extent that the experimental methods are able to leverage the theoretical results to make progress, the practical contributions could be made more clear.
• The significance of the experimental results is not established. What is baseline and SOTA performance on these tasks? Furthermore, the methods employed are not described at all in the main body of the paper, and completely relegated to the appendix.
• The paper requires major revision to make both the exposition and the contributions relative to existing work clear.

Unfortunately, I found the paper quite difficult to parse - either due to my own failings or because the paper is indeed hard to parse. The presentation seems more like a sequence of mathematical observations / proofs strung together into a complicated story, rather than a clear theme. I am happy to change my mind if the authors can help clarify things in the rebuttal process, and I am leaving a large number of questions below to help me understand things better -- with probably more follow up questions if the authors engage.

I think the paper is very poorly motivated in setting up what the authors want to solve, and examples of the exact specifics. It is not clear what the paper is trying to accomplish. Is it a proof that under certain assumptions, a conditional diffusion model can learn somewhat entangled distributions and adapt-and-vote can help in training classifiers on out of distribution data ? In spite of all the proofs, I am left wondering what the main contribution of the paper is ? Is it adapt-and-vote ? If so, how does it compare to other methods ? It seems that Table 2 and other tables only show comparisons between different types of voice cloning models, using adapt and vote. Is it possible to compare to some other techniques out of distribution ? What about showing disentangling without voice cloning ?

On the other end of spectrum, related to details in the paper. I found it hard to read because the authors make a lot of implicit assumptions about the terminology and probably wanted to keep the presentation more abstract. However this leads to a paper that's really difficult to parse. To give some examples - in section 2, is Z a variable that is computed from the data X, or an independent variable ? Is G computed from X, or just a one-hot label ? These things get resolved reading the paper in more detail, but it takes an inordinate amount of effort to parse, and it leads me to believe there are probably important technical details in the paper that I missed out on, in spite of my best (under a finite budge of time) effort. Please see my questions in the section below.

Finally. About disentangling representations. Generative models using independent variables as inputs has been a guiding principle in representation learning for a long time. It helps make the claim, at test time that new combinations of data can be generated because the variables input into the generative model are independent. However, it seems to me that this is an artificial setup - for example in the case of style and content, it's really hard to define what is considered content and what is considered style. After learning such models, we often find (and this paper also finds in some cases) (and this is certain true for variational auto encoder like models) that arbitrary combinations of variables that are treated as independent by models have strong dependence on each other. This can result in holes where generating from the prior leads to weird results. The paper proposes to use mutual information over the population as a proxy for disentanglement, in Definition 2, which I appreciate as a good attempt, but I wonder if such a thing is possible when it come to variables like speakers and content. It is possible that certain speakers have certain topics / content they are more likely to generate. In these cases such a disentanglement may not be possible, or, in some cases even desirable, now that we have really powerful models. Given these issues, how should I think of $\epsilon$ in $\epsilon$-disentanglement ? i.e. what does this way of thinking about abstraction of disentanglement lead to ?