__ Summary and Contributions__: This work proposes a practical method for disentangling, i.e., Geometric Manifold Component Estimator (GEOMANCER). GEOMANCER does not learn a global nonlinear embedding, instead, it learns a set of subspaces to assign to each point, where each subspace is the tangent space of one disentangled submanifold. Thus, GEOMANCER can be used to disentangle manifolds for which there may not be a global axis-aligned coordinate system. Experimental results on both synthetic data and Stanford 3D data are included in this paper.

__ Strengths__: The intuition here is interesting, i.e., "restrict to Lie groups – groups that are also manifolds, we could use the properties of infinitesimal transformations as a learning signal", and in practice, it seems first perform manifold clustering, then learning the local structure for each sub-manifold, can achieve similar results as GEOMANCER. This work may also bring some new interesting theoretical thinking in this direction. Theorem 2 (section 2), the analysis of the 2nd order Laplacian matrix, is quite informative & as the zero eigenvalue associated with factors of a product manifold, and the eigenfunction for subspace tangent.
Experimental results on synthetic data are provided in section 4, and show that GEOMANCER worked well, in particular for orthogonal group SO(n), the proposed algorithms still learns the correct local tangent space.

__ Weaknesses__: The first step in GEOMANER (see algorithm 1) is to construct NN-graph and then use local PCA to estimate tangent space. If understood correctly, there is no outlier filtering or robust related consideration added here, and raise questions for the robustness of the proposed algo given step 1 is like a foundation for following algos.
Section 3. page 6, "The appropriate number of submanifolds m can be inferred by looking for a gap in the spectrum .... ", estimation of number of submanifolds is interesting and seems one way to evaluate the proposed GEOMANCER, but seems only results in synthetic data is included (e.g., Fig. 5(a)).
Experimental results on Stanford 3D object data (Bunny and Dragon) can be viewed as the key part in this paper to support the claimed contribution. However, it seems GeoManCer only works well when applied to the true latent space vectors, and it fails when applied to pixels. Section 4 also claim the existing algorithms to disentangle directly from pixels fail as well [19], and applying GeoManCer to the latent vectors learned by the β-VAE is not improved either.
Also, it would be better to see the downstream application based on the Disentangling learning results, e.g., classification or cluster (and classification has well defined metric for evaluation), then it can be a better judgement for the proposed contributions.

__ Correctness__: Yes

__ Clarity__: Yes

__ Relation to Prior Work__: This work did a nice job to cite 60+ related work from disentangling, VAE, CNN, manifold learning, etc. Still, it seems some manifold learning papers are missing here, for example LTSA, Z. Zhang, H. Zha, Principal manifolds and nonlinear dimensionality reduction via tangent space alignment in 2004, this is one of the earliest papers in the machine learning community to directly mention the local tangent space learning. There are also following works after 2004, and some of them are more similar to the proposed GEOMANCER, for example D. Gong, X. Zhao, G. Medioni, Robust Multiple Manifolds Structure Learning, ICML 2012.

__ Reproducibility__: Yes

__ Additional Feedback__: Update --- After discussions with other reviewers & meta-reviewer, we decided not take into account author's response given it is not follow the NeurIPS author response template. I keep my original score for this submission as "6 - marginally above the acceptance threshold".

__ Summary and Contributions__: This work presents a new manifold learning method called GeoManCEr that decomposes data into multiple disentangled manifolds. This method is derived similar to Laplacian-based manifold learning methods such as Laplacian eigenmaps and diffusion maps, but here the approximation of the Laplace-Beltrami operator is replaced with an approximation of the second order connection Laplacian. Similar to the mentioned spectral methods, the embedding is performed here via spectral decomposition of the Laplacian. However, in this case the eigenvectors are organized into projection matrices on disentangled tangent spaces, requiring some additional steps to clean spurious eigenvectors (e.g., coming from the "classic" Laplacian operating on functions), cluster together the components of each disentangled manifold, and to organize the resulting coordinates so that each submanifold can select its appropriate tangent spaces and represent data points (and tangent vectors) in them. The method is demonstrated on simple synthetically generated entangled manifolds, where the intrinsic manifold metric is known in advance, as well as a toy example of rendered 3D objects with different rotations and light source positions.

__ Strengths__: This paper provides new insights into the problem of disentangling independent latent factors, viewed here through the lens of factorizing groups of transformations on a data manifold. The authors base their construction on the de Rham decomposition, which itself is based on the holonomy group that considers parallel transport over loops on a manifold. Essentially, the authors seek to extract multiple representations of input data, such as each of them encodes a submanifold with holonomy group independent from all other submanifolds. This provides an important formalism to an important problem that is often ill defined, with mostly heuristic qualitative goals that depend on specific applications rather than studied with rigor.
The construction itself here is based on extending the work of Singer and Wu on vector diffusion maps, which enriches more traditional manifold learning by encoding information about tangent spaces and the operation of the connection Laplacian on tangent vector fields. Through careful spectral consideration, the authors identify here the eigenfunctions (or "eigentensors") of the second order connection Laplacian that correspond to projections on tangent spaces of individual disentangled manifolds, and provide a constructive method to extract and cluster them, thus assigning each point multiple representations corresponding to these tangent spaces.
An important result established, and verified empirically, is the ability to identify in an unsupervised way the number of disentangled components that should be considered. This is given here by an elegant analogue to the identification of connected components in spectral graph theory. There, the multiplicity of the zero eigenvalue of a graph Laplacian gives the number of connected components, while the corresponding eigenvectors (up to demixing them) identify the association of nodes to components. Similarly, the authors show here that the multiplicity of the zero eigenvalue (or "sufficiently small" eigenvalues, w.r.t. a spectral gap, in practice) provides a reliable indication of the number of submanifolds to consider, while joint decomposition of the corresponding eigentensors together with cosine-similarity clustering could yield reasonable disentangled projections, although it should be noted that this is not shown to work in realistic applications (see weaknesses below).

__ Weaknesses__: The main weakness if the proposed approach here is that it is unclear whether it provides a realistic direction in practical applications. Indeed, while the results shown here for rather simple artificial data seem nice, the application to 3D renderings, which are also rather simple and synthetic, already struggles. The authors do address this point partially by identifying the lack of accurate metric information as a crucial missing ingredient. However, it is not clear how realistic would it be to expect such metric information to be provided, or how sensitive the proposed is to the various artifacts and approximation errors that would clearly be expected in real data. Indeed, data "manifolds" rarely actually correspond to clean manifold models, as they have density variations, dimensionality variations, noise, etc., and much work has been invested in coping with such artifacts. There is also the question of scalability of the proposed method, as it is only demonstrated for very simple examples, but many applications that require disentangling in fact involve much more high dimensional data and complex structure. Realistic and challenging data analysis settings have already been studied extensively within the field of diffusion based manifold learning considered here (e.g., works by R Coifman et al., B Nadler et al., T Berry et al., and A Singer et al., come to mind), and it should be noted in this context that even the VDM approach extended here was developed and successfully applied to challenging tasks in organizing CryoEM data with extremely low SNR. Therefore, the underlying foundations of the proposed approach should provide a sufficient starting point to expect some more promising application. Simply put, if the proposed approach struggles with very simple 3D images with uniformly (densely) sampled variations along two clearly independent manifolds, how would it be realistic to apply in practice?

__ Correctness__: The claims and methodology seem well established, and the authors clearly state some deficiencies and identify one of the main gaps or challenges remaining for practical uses of their proposed approach.

__ Clarity__: The paper is well written. It does require some manifold (non-Euclidean) geometry background to fully understand it at times, but is sufficiently clear given such background.

__ Relation to Prior Work__: The paper provides sufficient background to understand the presented ideas, although it does rely on some prior understanding of nontrivial ideas from differential geometry. The introduction provides reasonable coverage of prior and related work on disentangling.
One aspect that could be improved is to provide some preliminaries on diffusion maps and its extensions, which either serve as the foundation for the discussed prior work by Singer and Wu or as related extensions. The authors only mention briefly they extend Laplacian Eigenmaps and Vector Diffusion Maps (VDM), but it would be good to provide further discussion of the works of Coifman and Lafon (ACHA 2016), Nadler et al. (ACHA 2016; NeurIPS 2016), Salhov et al. (ACHA 2012; Machine Learning 2016), Wolf and Averbuch (ACHA 2013), Fan and Zhao (ICML 2019), etc. to present a more complete overview of this well studied field using diffusion in manifold learning, both for scalar functions and for vector fields on Riemannian manifolds. This can also help address the question of how to extract or approximate intrinsic metric information, along the lines of the LEM column shown in Table 1.

__ Reproducibility__: Yes

__ Additional Feedback__: *** Updates following author response ***
Unfortunately, the author response cannot be taken into consideration as it does not follow the NeurIPS author response template and does not meet the one-page limit.
=== Original review ===
While the focus of the paper seems to be on theoretical aspects, and the motivation for disentangling data manifolds is naturally well understood, it would be good to provide some demonstration of the insights or data organization provided by the application of the proposed approach, rather than just quantify the angle between (tangent) subspaces compared to baselines.

__ Summary and Contributions__: This paper proposes an algorithm to construct a space to enable the parallelogram commutation. The commutation may not work in some sub-space, and the algorithm will reorganize the sub-space through the learning component.

__ Strengths__: This work is well-motivated by the fundamental theory of differential geometry, i.e. de Rham decomposition, which motivates the local disentanglement of commutative global manifold. The local disentanglement comes from Theorem 2, which is enabling mechanism in the paper, by finding the objective of Laplacian \triangle^2 to be near zero eigenvalues. Therefore, authors find the disentangled submanifolds, or local coordinates, by the matrix decomposition, i.e. SVD, on data instances, on the second-order connection Laplacian.

__ Weaknesses__: My expertise lies in the disentanglement of the latent spaces with regularization, discriminators, and prior distributions, i.e. \beta-VAE. Therefore, I had to review this paper from the user's perspective.
- The proposed method relies on the matrix decomposition on the local PCA estimation, and this happens for all instances. This seems a very arduous computation even with the parallel computing by GPU. Any opinion on this complexity? Simiarily, there are series of matrix multiplications to be reviewed from the computational complexity perspective.
- The result is not much supportive considering the result in Table 1 without the true latent information, which limits the practical application of the proposed algorithm.

__ Correctness__: Seems to be correct up to my understanding, but this should be discussed by other reviewers.

__ Clarity__: Yes. The paper itself is clearly written for a researcher with knowledge on linear algebra, abstract vector space, and differential geometry.

__ Relation to Prior Work__: The prior work is not much covered in this line of differential geometry and latent disentanglement. I was not able to find some, either.

__ Reproducibility__: Yes

__ Additional Feedback__: Please focus on the computational complexity question, and I suggest that you create a subsection to discuss the complexity.
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Read the rebuttal. The rebuttal format was not following the Neurips guideline, so the panel decided to ignore the rebuttal.

__ Summary and Contributions__: The authors introduce the GeometricManifold Component Estimator (GEOMANCER) which is a really cool name. The provide a partial answer to is it possible to learn how to factorize a Lie group solely from observations of the orbit of an object it acts on? The develop a geometric theory based on holonomy and provide an algorithm based on a discrete Hodge Laplacian which is an approximation of the giving an approximation to the de Rham decomposition from differential geometry. The paper
reduces the question of whether unsupervised disentangling is possible to the question of whether unsupervised metric learning is possible, providing a unifying insight into the geometric nature of representation learning.

__ Strengths__: The paper is rigorous, the mathematical ideas are potentially powerful and the exposition outside of the intro and the parallelogram are clear and motivating. The use of de Rham decomposition is nice and appealing. The relations to classic Laplcains is also interesting.

__ Weaknesses__: The paper does not make clear the relation to other works on learning group structure such as The Geometry of Synchronization Problems and Learning Group Actions Tingran Gao, Jacek Brodzki & Sayan Mukherjee
Discrete & Computational Geometry (2019) or The Diffusion Geometry of Fibre Bundles: Horizontal Diffusion Maps
Tingran Gao. Also the need for the implicit fiber construction over standard diffusion maps was not made clear.

__ Correctness__: Yes,

__ Clarity__: Overall yes. The intro I felt was a bit watered down.

__ Relation to Prior Work__: No as stated above.

__ Reproducibility__: Yes

__ Additional Feedback__: