Paper: Continual Nonlinear ICA-Based Representation Learning (ICLR-2024)

Reviewer comment (Writing):
What is the actual form for the domain variable u?

Author response:
$u$ is an indicator of domain which can be represented with a vector or a scalar.

Reviewer comment (Experiments):
This paper only considers a general continual learning setting, which relies on the task information. However, the proposed approach can not be used in task-free continual learning.

Author response:
Thank you for your question and we would like to re-emphasize the difference between ours with traditional continual classification tasks. In the traditional continual classification task, each task is independent. The purpose of continual learning is to make the model remember each task as much as possible. In this situation, not knowing the identities of tasks will indeed affect testing. However, for our identifiable nonlinear ICA learned in sequentially arrived domains, we need multiple domains to make the model identifiable. Once we have enough domains, knowing the identity of the domain (task) or not is not important at all for the test phase. Regardless of which domain the tested data comes from, even a completely unseen domain, we can still recover meaningful latent variables (component-wise identifiable). Figure 5 (a) also validates this point as the model is always tested on all 15 domains while the training data increases gradually from 1 domain to 15 domains.

For the training phase, it should be noted that our observations are generated from changing distribution through **fixed** transformations. Thus, it would be easy to capture the boundary of each domain for model training.

Reviewer comment (Novelty):
The theoretical framework is based on the existing work (Kong et al., 2022).

Author response:
Thanks for your comment. Compared with the framework from [1], we introduced the subspace identifiability for changing variables, which is essential to investigate the case where few domains are observed. Beyond that, we have demonstrated that, in the context of continual learning, the identifiability of certain variables becomes compromised with the introduction of new domains, as outlined in Proposition 2 and empirically shown in Figure 6.

Reviewer comment (Experiments):
The number of baselines in the experiment is small and more continual learning experiments should be performed.

Author response:
The reason we have not conducted comparisons of our method with other nonlinear ICA approaches is due to the distinctive structure of our model. Specifically, our approach involves partitioning latent variables into two distinct categories: changing and invariant components. Thus, other nonlinear ICA frameworks cannot be directly compared as a baseline.

Reviewer comment (Experiments):
Since this paper employs the generative model. The lifelong generative modelling experiments should be provided.

Author response:
We appreciate your comment, but we would like to clarify our understanding of your question to provide a more accurate response. Section 4 of our paper is dedicated entirely to the lifelong generative modeling experiment. If you could provide more specific details or context regarding your query, it would greatly assist us in addressing your concerns or questions more effectively.

Reviewer comment (Writing):
Between Eqs (21) and (22), is it necessary to keep the \( \mathbf{0} \) argument? This appears more confusing/unnecessary than helpful, but there is the possibility that something is being overlooked.

Author response:
> between Eqs (21), (22), is it necessary to keep the argument $0$ ?

Thanks for your suggestion which helps improve the presentation and lighten the notations. We have updated the manuscript to remove the argument $0$.

Reviewer comment (Experiments):
The method is purely validated on synthetic toy datasets. There are many available datasets (e.g., image datasets like dSprites) to validate the approach on more complex distributions without drastically increasing the number of latent factors. Such an exploration would improve the paper significantly. This could be particularly interesting to "convert" an existing benchmark into the continual ICA setting.

Author response:
> The method is purely validated on synthetic toy datasets.

Thanks for your suggestion. In light of your comment, we are currently conducting the experiment on image datasets and will report the results once we get them.

Reviewer comment (Novelty):
Contribution is not clear since relevant previous work is ignored. The claim that "[Nonlinear ICA] still relies on observing sufficient domains simultaneously..." is not necessarily true: for example [1] shows the identifiability of hidden Markov nonlinear ICA. As is well known, HMMs can be learned sequentially / in online fashion with the latent states analogous to different domains. Authors have not covered relation to this work.

Author response:
Thank you for your insightful observation. It is indeed correct that Hidden Markov Models (HMMs) can be learned sequentially. However, as delineated in reference [1], the identifiability of the model is contingent upon the concurrent observation of a sufficient number of latent states or domains. On one hand, the validity of Equation (22) is compromised in scenarios where the model experiences 'forgetting', particularly when only a single latent state is observed at any given time. This scenario presents a stark contrast to the framework and contributions of our study, which focuses on a genuinely continual learning approach. On the other hand, to preserve the identifiability of the model in a sequential learning context (as proposed in [1]), where only one latent state is observed at a time, the algorithm we have used in our paper could be adapted as it only focuses the optimization procedure.

Reviewer comment (Novelty):
Related to the above, [1] and [2] give a broad identifiability theorem that applies to a situation with a changing latent variable that can be interpreted as the changing domain that alters the probability distributions. It seems to already cover mostly what is considered in the identifiability theorems here, and provides stronger results in a more general setting. It is therefore important to contrast to this work and explain what is novel here. While sequential learning is different, this comment refers only to the identifiability theorem. Other important and similar work is also ignored and needs to be discussed such as [3]-[6]. Most of these works do not assume auxiliary variables.

Author response:
Thank you for your feedback. We agree that [1] and [2] are highly relevant, which we will discuss further in the revised manuscript. At the same time, we would also like to point out the differences with our setting (and therefore one is not "stronger" than another): 

(1) While [2] is predicated on exploiting the temporal structure of latent variables, our paper delves into a scenario where the latent variables are independently and identically distributed (i.i.d) within each domain. This distinction is crucial as the reliance on the use of temporal structures in [1] and [2] can indeed facilitate stronger identifiability results.

(2) Our papers separate the latent variables into invariant and changing part, which is practical in real-world scenarios. We thoroughly investigated the conditions under which these two groups can be segregated (requiring at least $n_s+1$ domains) and also provided the sufficient condition that each changing variable is identified up to a nonlinear transformation ($2n_s+1$ domains). This exploration into the separability and identification of latent variables underscores the novelty and depth of our study. 

(3) In contrast to [1], which presupposes the conditional distribution of latent variables within the exponential family framework, our work does not make such an assumption.

Reviewer comment (Novelty):
The paper frames itself as "Causal Representation Learning" but it seems like it's much more related to nonlinear ICA and doesn't learn latent causal relationships — see [6] and [7]. Authors are recommended to reconsider the use of this term.

Author response:
Thanks for your suggestion. We have reconsidered the use of the term and will use "continual identifiable representation learning" instead.

Reviewer comment (Experiments):
Experimental evaluation is lacking. The authors only evaluate their model on synthetic data, with no real data used, and the importance of this method to practicable applications is not clear.

Author response:
Thanks for your suggestion. We are currently conducting the experiment on image datasets and will report the results once we get them.

Reviewer comment (Experiments):
Experimental baselines do not include any relevant previous works; for example, [1] could be reasonably adapted to this data.

Author response:
Thanks for pointing this out. The setting we considered is slightly different from the regular nonlinear ICA setting; that is, our setting divides the latent variables into changing and invariant parts, which may be considerably more general. (In this case, regular nonlinear ICA considers all latent variables to be "changing".) For this setting involving changing and invariant parts, the proposed method by [8] is the state-of-the-art method which we adopt and compare to, and we are not aware of any other methods that consider the same setting. If there is any other specific method considering this setting that the reviewer thinks should be compared, we would greatly appreciate if you could kindly let us know.

Reviewer comment (Theory):
In theory and estimation, the identifiability theorem appears not to consider observation noise, yet the estimation method VAE clearly includes that. This mismatch and its impact on identifiability can be significant (Theorems 1, 2 in [2]) but is ignored here.

Author response:
Thank you for your valuable feedback. We acknowledge that our current identifiability theory does not incorporate observation noise and does not facilitate dimension reduction. However, it is important to emphasize that the crux of our research is the identification of nonlinear ICA in a continual learning context. This aspect, to the best of our knowledge, has not been well explored in existing literature.

Our work revolves around the concept that the identifiability of nonlinear ICA, reliant on multiple domains. This principle holds true even in scenarios where auxiliary variables may not be directly observable; the essence of making the model identifiable remains rooted in utilizing the distribution change of latent variables.

The primary objective of this paper is to elucidate a methodology for addressing scenarios where these distributional changes in latent variables cannot be observed concurrently. This approach mirrors the human cognitive process of sequentially observing and refining understanding to discern true latent factors.

We also recognize the significance of considerations regarding noise and dimension reduction. These factors are indeed crucial in the broader context of model development and application. There are multiple possible solutions to solve this problem. For example, we can use PCA to first reduce the dimension of the observation and then apply our nonlinear ICA framework. We can also follow the work of [7] to propose an explicit noise model. Therefore, we plan to extend our research to include these aspects in our future work.

Reviewer comment (Theory):
The identifiability theorem does not appear to consider dimension reduction into the latent space, which is in practice very important. Without that you are usually stuck with high-dimensional latent space, and the theorems presented here are unlikely to hold with high-dimensional latent variables, making it unclear how useful the work is in practice.

Author response:
Thank you for your valuable feedback. We acknowledge that our current identifiability theory does not incorporate observation noise and does not facilitate dimension reduction. However, it is important to emphasize that the crux of our research is the identification of nonlinear ICA in a continual learning context. This aspect, to the best of our knowledge, has not been well explored in existing literature.

Our work revolves around the concept that the identifiability of nonlinear ICA, reliant on multiple domains. This principle holds true even in scenarios where auxiliary variables may not be directly observable; the essence of making the model identifiable remains rooted in utilizing the distribution change of latent variables.

The primary objective of this paper is to elucidate a methodology for addressing scenarios where these distributional changes in latent variables cannot be observed concurrently. This approach mirrors the human cognitive process of sequentially observing and refining understanding to discern true latent factors.

We also recognize the significance of considerations regarding noise and dimension reduction. These factors are indeed crucial in the broader context of model development and application. There are multiple possible solutions to solve this problem. For example, we can use PCA to first reduce the dimension of the observation and then apply our nonlinear ICA framework. We can also follow the work of [7] to propose an explicit noise model. Therefore, we plan to extend our research to include these aspects in our future work.

Reviewer comment (Theory):
The term "component-wise identifiability" is used inaccurately; the elements are not identified component-wise, rather they are identified up to component-wise nonlinearity. This should be fixed to avoid misleading the reader into thinking exact identification of each component is possible.

Author response:
Thanks for your suggestion. We will add a footnote to avoid such kind of confusion.

Reviewer comment (Writing):
The statement "Importantly, the guarantee of identifiability persists even when incoming domains do not introduce substantial changes for partial variables" is unclear because the term "partial variables" has not been defined by that point and needs clarification.

Author response:
Sorry for the confusion. What we want to express here is even when a new domain comes in, there is a part of changing variables that don't undergo any distribution change. Kindly refer to Section 3.2.1 for the details.

Reviewer comment (Theory):
It is not explained clearly enough that the conditions in Theorems 1 and 2 are sufficient conditions, not necessary. There is confusing writing related to this. For example, the statement "However, when considering domains u0, u1, u2, the component-wise identifiability for z1 disappears, and instead, we can only achieve subspace identifiability for both z1 and z2" is not necessarily true. According to the theorems, you can only guarantee subspace identifiability (sufficiency), but since Theorem 1 does not give necessary conditions, one cannot say that the component-wise identifiability for z1 disappears; it could still be component-wise identified.

Author response:
Sorry for the confusion. We will refine our paper and change the expression to "the component-wise identifiability cannot be guaranteed anymore".

Reviewer comment (Presentation):
Figure 5 coming before Figure 4 is extremely confusing.

Author response:
For Figure 5(a), the model is trained on increasing domains and tested on fixed all 15 domains. For the top-right one in Figure 4, the number of training and the number of testing domains are equated.