__ Summary and Contributions__: This paper proposes to learn features by increasing the inter-class incoherence. A basic assumption is that features for each class lie near a linear subspace. The method is justified by rate distortion argument. Experimental results demonstrate the effectiveness of such a design.
After rebuttal, I'm seeing some of my concerns (e.g., comparison with OLE) being resolved. Some deeper part (e.g., connection with robustness and generalization) are not fully addressed, but I understand it is unfair to ask for one paper solve a whole big research topic. Given these, I keep my current rating.

__ Strengths__: The paper justifies the max-incoherence design from an information theoretic perspective, rather than pure heuristics. Theorem 2.1 shows some nice properties of the optimal solution. The experiments on label corrupted data is interesting.

__ Weaknesses__: Overall, the paper can be considered as applying existing objective/criterion in learning with subspaces into learning of deep features. As already mentioned in the paper, there are existing papers adopting a similar design. To this end, the authors should show more comparison against these relevant methods, e.g., OLE.
It is not clear why a larger intra-class subspace dimension (compared with OLE) is helpful. Perhaps that means more intra-class structure being preserved, and improves robustness. And therefore, gains are observed for classification of label corrupted data. However, these connections are not crystal clear in the paper.
In fact, a core problem in understanding deep learning is robustness and generalization. This paper sort of touches the problem, but does not really advance readers’ understanding on it.

__ Correctness__: I didn't check proof details for Theorem 1. But intuitively they are correct. The empirical methodology is correct.
In addition, I have a concern whether the rate distortion function relies on Gaussian assumption of data. The authors should provide more details on that in order to be self-contained.

__ Clarity__: Yes

__ Relation to Prior Work__: See weakness

__ Reproducibility__: Yes

__ Additional Feedback__: Does the OLE type loss have the same property as theorem 1. What is the connection between rank, nuclear norm and logdet? A theoretical analysis is highly appreciated.

__ Summary and Contributions__: The paper proposes a representation learning method based on the principle of maximum coding rate reduction. The objective has the effect of encouraging the rank of the covariance matrix of the learned representations to be large, while simultaneously encouraging the ranks of the covariances for each individual class to be small. Theoretical properties of minimizers of the proposed objective are derived. Experiments on image datasets demonstrate improved robustness to random label corruptions vs. standard cross-entropy training, and that representations trained in an unsupervised manner possess cluster structure corresponding to ground-truth labels.

__ Strengths__: 1. The paper proposes an intuitively reasonable learning objective for representation learning. This work is relevant to the NeurIPS community given its longstanding interest in supervised and unsupervised representation learning using deep neural networks.
2. Some relevant properties of optimal solutions to the MCR^2 objective are shown theoretically.
3. The empirical results for the method are encouraging. In particular, it is notable that the MCR^2 objective gives additional robustness to random label corruptions without explicitly accounting for label noise in the objective.
4. A range of helpful ablation results are provided in the appendix.

__ Weaknesses__: 1. In Section 2, the authors seek to contrast the MCR^2 objective against mutual information-based approaches like information bottleneck. My impression is that there is a close relationship between these methods that warrants further discussion in the text. After all, the MCR^2 objective admits the following interpretation: first fit Gaussian distributions to the z-vectors with Tikhonov regularization on the covariance matrices to account for possible degeneracy (i.e., the \eps parameter in the text), then compute the mutual information with class labels using the differential entropy of these (regularized) Gaussians.
2. The label noise robustness experiments can be improved by further comparing against other methods like iterative trimmed loss minimization [1].
3. Some of the claims in the text are non-obvious and should be made with appropriate citations. For example, in L48 -- it is stated that the IB objective results in sacrificing "generalizability, robustness, or transferability". Such a statement does not appear to me to be obviously true.
4. The overall clarity of the presentation should be improved (see below).
[1] Shen & Sanghavi. Learning with bad training data via iterative trimmed loss minimization. ICML 2019.

__ Correctness__: The claims and method appear to be correct, although I did not check the proofs in detail.

__ Clarity__: I found the ideas in the paper to be interesting, but the presentation was hindered by the inclusion of vague, hand-wavy prose. For example, the introduction states that the paper aims to learn "explicitly meaningful" representations for the data -- it is not clear what "meaningful" entails, or indeed how the learning objective described in the sequel relates to this stated goal.
Additionally, there is an overabundance of parenthetical information and discussion in the form of footnotes. I found that this detracted from the flow of the narrative.

__ Relation to Prior Work__: I was satisfied with the coverage of prior work in the paper, but the relationship with the information bottleneck method should be expanded on.

__ Reproducibility__: Yes

__ Additional Feedback__: After author response:
Regarding the relation with information bottleneck: I concede that the MCR^2 objective doesn't seek a "minimal" representation in the sense that the mutual information between the input and representation is not penalized; however, it still appears that the method can be viewed as mutual information maximization between the representation and the labels, where the representation takes the form of a regularized Gaussian mixture. I would have preferred to see some more discussion on this point.
Overall, I found that the submission presents an interesting idea with encouraging empirical results. While I would prefer a more focused presentation of the main ideas, I'm willing to increase my score by one.

__ Summary and Contributions__: - The paper proposes a compression-inspired learning framework that is applicable in supervised and unsupervised settings.
- The authors present a theoretical analysis of the proposed framework showing that, for multi-class data, their loss assigns the embeddings of each class to subspaces that are orthogonal to all other subspaces.
- The proposed framework is empirically evaluated on a variety of unsupervised and supervised tasks, on both synthetic and real data, and compared to prior work.

__ Strengths__: - The paper gives a good overview over different recent representation learning techniques and discusses them under one umbrella.
- The proposed learning framework is applicable in both the supervised setting, where it is shown to promote robustness, and in the unsupervised setting, where it achieves strong clustering results.
- The paper aims to make progress towards learning of more structured, more interpretable representations, which is an important current research direction.

__ Weaknesses__: - My main concern is that, I don’t see the benefits of modeling the data as a union of subspaces, where each subspace corresponds to a class, when the representation space is *learned*. In particular, since these subspaces won’t be orthogonal in practice, on real data. In an unsupervised setting, to recover the subspaces, one needs to perform subspace clustering, which is a hard problem and computationally expensive to perform. In a supervised setting, where estimation of the subspaces is easy, one needs to do nearest-subspace-classification which is more intricate than linear classification. In stark contrast, a linear head trained with a cross-entropy loss learns a representation space with approximately linearly separable regions for each class. As a consequence, classification is simple (linear) and Lp distances in representation space are meaningful (which is not necessarily the case when the classes lie on a union of subspaces).
- I acknowledge the encouraging results regarding robustness of the representations learned with the proposed method. However, there are many other methods which can make neural networks with linear classification head more robust, for example [c]. Therefore I believe a union of subspace structure is not fundamentally required to achieve this.
- While the theoretical analysis reveals interesting properties of the learned representation, it completely ignores the relationship between the individual data points and their representation, defined through the feature extractor. It is well-known that the structure and properties of the extractor crucially impact the learned representation, possibly even more than the loss, see e.g. [ZF14].
[c] Elsayed, Gamaleldin, et al. "Large margin deep networks for classification." Advances in neural information processing systems. 2018.
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Update after rebuttal: Thanks to the authors for their response. I now better see the benefits of encouraging orthogonality between class regions in the feature space, which is why I increased my rating. However, I'm still not sure whether the theoretical result is useful to explain what is going on, as I still believe the network architecture is crucial for the structure in the feature space. Furthermore, as pointed out by the other reviewers, the method seems to have many similarities with previous methods, which should be discussed more precisely.

__ Correctness__: The experiments seem methodologically correct and are clearly described in the main paper and the appendix. The paper is accompanied by well-documented code to reproduce the results.

__ Clarity__: The paper is well written and very polished. It is accompanied by a detailed appendix providing further details and proofs.
Regarding Theorem 2.1. it should clearly be written that it applies to the supervised setting where the class assignments are given (which is stated in the main result in Theorem A.6). This is an important detail that should not be deferred to the appendix.
It might make sense to revisit the footnotes. I feel that moving some of them into the main text might improve clarity.

__ Relation to Prior Work__: While there is no dedicated related work section, related work is sufficiently covered. However, the following references also consider clustering-related representation learning and are evaluated mostly on the same data sets, with the same metrics, and should therefore be included and compared against.
[a] Hu, Weihua, et al. "Learning Discrete Representations via Information Maximizing Self-Augmented Training." International Conference on Machine Learning. 2017.
[b] Ji, Xu, João F. Henriques, and Andrea Vedaldi. "Invariant information clustering for unsupervised image classification and segmentation." Proceedings of the IEEE International Conference on Computer Vision. 2019.

__ Reproducibility__: Yes

__ Additional Feedback__: Minor comments:
Computing the log det has cubic complexity and is known to be prone to instability. Did the authors encounter any problems along these lines?
How is the simplex constraint on \Pi in (9) enforced in practice?
Footnote 3: ...finitely *many* samples...

__ Summary and Contributions__: The authors propose maximal coding rate reduction as a means to learn diverse and discriminative low-dimensional representations from high-dimensional data. The authors empirically show that they establish the new state-of-the-art results in clustering mixed data and theoretically prove the theoretical guarantee for learning diverse features. They also demonstrate the robustness over cross-entropy based classification.
Specifically, the authors argue that a good representation Z for data X is one that maximizes the coding rate reduction, defined by the difference between coding rate (derived from the rate-distortion function given the target distortion \e) for X itself and the coding rate for the mixed distribution, where the subset partition itself is either given or also optimized.

__ Strengths__: The authors nicely bring information theoretic measures, such as the coding rate for the rate-distortion function, to propose the maximal coding rate reduction principle. The idea makes sense and sounds intuitive. They demonstrate pretty impressive performance gain on several datasets over existing baselines.

__ Weaknesses__: Applying the MCR reduction to the large-scale dataset seems computationally very hard.
Why MCR reduction results in robust features isn't unclear (as also noted by the authors). The gap is huge. It'll be interesting to study why this is the case.

__ Correctness__: The claims and method look correct.

__ Clarity__: The paper is well written and easy to follow and understand.

__ Relation to Prior Work__: How this work differs from previous contributions is clearly discussed.

__ Reproducibility__: No

__ Additional Feedback__: