Paper: Neural Network Expressive Power Analysis Via Manifold Topology (ICLR-2024)

Reviewer comment (Writing):
A small handful of relatively minor typos (e.g. "quaratically" under main theorem, Definition 2 $\mathcal{M} \subset \mathbb{R}^D$ instead of $\in \mathbb{R}^D$, "fundamental" quote formatting on page 5). While needed to be fixed, these make 0 impact on my overall review.

Author response:
We are grateful to the reviewer for highlighting the typos in our manuscript. We have fixed them in the updated version.

Reviewer comment (Writing):
The term "classification" is overloaded in this paper, and the expected ICLR reader would likely be mislead by some of the writing. Namely, there are times when the author is talking about geometric manifold classification problems, where the goal is determining if one (or a set of) manifold(s) is homeomorphic to another. This is fundamentally different from neural networks classifying two classes. Since explicit clarification and distinction is missing, there are a number of misleading parts of the text. For example, the statement "classifying manifolds beyond three dimensions is known to be equivalent to the word problem on groups [...] and is undecidable" would lead the standard reader to think that this paper's restriction to at most 3-dimensional manifolds actually retains generality for neural network classification theory, but indeed this statement is talking about geometric manifold classification, not neural network classification.

Author response:
We appreciate your attention to the potential ambiguity surrounding the term "classification" as it pertains to both the manifold classification theorem and classification problems in machine learning. In response to your insightful comment, we have carefully revised our manuscript to include clear distinctions and explanations whenever we refer to the manifold classification theorem. This ensures that our usage of the term is precise and unambiguous within the context of our discussion.

Reviewer comment (Novelty):
The authors' claim that their "result reveals for the first time how the topology, as a global structural characterization of data manifold, affects the network expressiveness" is misleading; while this analysis related to specifically Betti numbers and neural network expressiveness, this work is not the first to relate "structural characterization of data manifold" to nextwork expressiveness. While their related works section does cover a good breadth of related work in the intersection of topology and deep learning, one work in particular was missed that is quite close in setting and goal to this paper, namely an ICLR 2021 paper titled ["Deep Networks and the Multiple Manifold Problem"](https://openreview.net/pdf?id=O-6Pm_d_Q-). This referenced paper is of course not a strict superset (e.g. "Deep Networks[...]" only focuses on 1-dimensional manifolds), but Buchanan et al. do lay down a lot of fundamental theory on relating different sizes of a network (e.g. width and depth) to the complexity of the data manifolds and the distributions over them that data is drawn from, much of which this paper misses and could benefit from integrating in terms of practicality.

Author response:
We sincerely thank the reviewer for directing our attention to the paper [3], which was not previously included in our references. While this paper is indeed relevant, it does not diminish the novelty of our claim. Our work uniquely demonstrates for the first time how the topology, as a global structural characterization of data manifolds, impacts neural network expressiveness.

The study in [3] focuses on a specific category of manifolds: smooth, regular simple curves. These manifolds are characterized by being connected and non-self-intersecting, implying a uniform topology with only one connected component. Consequently, their analysis does not necessitate the inclusion of topological descriptors.

In contrast, our research encompasses a broader spectrum of manifolds with diverse topological features. The incorporation of topological complexity into our upper bound represents a pioneering effort in illustrating how varying topological characteristics of data manifolds influence the size and structure of neural networks. This aspect of our work stands as a distinct and significant contribution to the field, highlighting manifold structures and their relationship with network expressiveness.

Reviewer comment (Theory):
I am having a harder time seeing how this theory applies to deep learning, outside of the analysis being over MLP's. If the goal is a new method for constructing MLP's from data, the paper's main neural network construction seems to rely on a decomposition of a data manifolds into a union of fundamental manifolds (e.g. ball, torus), which is hardly known or even computable from fixed, finite samples of practical data manifolds, and likewise does not characterize the type of functions typical MLP's fit. If the goal is to highlight the expected relationship between the sum of Betti numbers and needed network size to classify data, then the experiments shown are not nearly thorough enough to demonstrate this, not only due to unrealistic settings (incredibly high sampled data density, low dimensional, etc.), but because the only trend plotted was over a single family of manifolds (g-genus tori), which alongside its Betti numbers sum has many other geometric properties growing that could explain the expressivity trend. It is thus not clear from these demonstrations why it is meaningful to focus on the Betti numbers.

Author response:
Thank you for your detailed feedback and for highlighting key concerns about the practical applicability of our theoretical findings. We value your insights and would like to address these concerns in the context of our study’s objectives and contributions.

Our primary goal is not to devise a method for constructing MLPs from data. Rather, our research aims to deepen the theoretical understanding of how the topology and geometry of data manifolds influence neural network size. We accomplish this by establishing an upper bound for network size required for classifying data from different manifolds. This approach is consistent with numerous theoretical studies in the field that focus on the expressive power of neural networks. Similar works like [1, 2] have provided various upper bounds for neural network sizes in approximating distinct function families, and while these results might not directly apply to datasets, they are invaluable for enhancing our fundamental comprehension of neural networks.

The insights from our theoretical exploration shed light on the intricate relationship between the complexity of data manifolds and the requisite size of neural networks for effective classification. This enhanced understanding has significant potential to guide future research in deep learning, especially regarding the optimization of network architectures based on the characteristics of the underlying data.

Regarding the necessity and relevance of topological complexity and Betti numbers in our study, we invite you to refer to Q1 and Q3 in our summary. 

In essence, while our results are rooted in theory, they form an integral part of the evolving understanding of neural networks, laying a foundation for further empirical exploration and practical application in the field of deep learning.

Reviewer comment (Novelty):
In the study of data manifolds and MLP classifiability, why should the research community focus analysis on Betti numbers for data manifolds instead of other previously studied measures of extrinsic geometry?

Author response:
RfL: Thank you for your thoughtful feedback and for recognizing the significance of starting our exploration from a low intrinsic dimension. We deeply appreciate your insights and the increased score you have provided.

In response to your concerns about the justification of our theory regarding the influence of Betti numbers on neural network size, we acknowledge the need for more empirical evidence, as you have rightly pointed out. We'd like to address this in two key areas:

1. **Interplay of Intrinsic and Extrinsic Properties:** As highlighted in our summary response Q1, both intrinsic and extrinsic properties play crucial roles in the analysis of classification neural networks. This is because, during classification, these properties are compressed together. In our summary response Q1, we demonstrate that curvature alone is insufficient to determine the necessary size of a neural network. We provided an example where two datasets, despite having identical curvature measures, require neural networks of different sizes for effective classification. This underscores the need to consider a more comprehensive range of properties.

2. **Empirical Evidence from Related Studies:** We base our theory on existing empirical studies that demonstrate a correlation between complex Betti numbers and the size of networks required for classification. For instance, the study referenced as [1] conducts an extensive examination of synthetic data. Their findings, particularly in Figure 5, reveal that datasets with larger Betti numbers necessitate larger neural networks for comparable levels of convergence compared to those with smaller Betti numbers. Another relevant study, [2], investigates changes in Betti numbers across latent representations in neural network layers. Their experiments, which include both synthetic and real-world datasets, although focusing on a different aspect, also highlight the relationship between Betti numbers and the size of neural networks in a classification context.

We believe these points reinforce the basis for our theoretical approach, and we are committed to further empirical investigations to strengthen our theory. Your feedback is invaluable to our ongoing research, and we thank you again for your constructive critique.

[1] William H Guss and Ruslan Salakhutdinov. On characterizing the capacity of neural networks using algebraic topology. arXiv, 2018.

[2] Gregory Naitzat, Andrey Zhitnikov, and Lek-Heng Lim. Topology of deep neural networks. JMLR, 2020.

Reviewer comment (Experiments):
The paper contains only one experiment on a synthetic dataset, which is a union of solid tori. A typical paper from ICLR is expected to have a larger experimental section. Why not to try more complex datasets: entangled tori, nested spheres, etc. (see Naitzat, 2020).

Author response:
We are grateful for your insightful suggestions regarding the experimental aspects of our work. Your feedback highlights important areas for potential expansion and improvement.

We would like to clarify that the core contribution of our paper is the establishment of an upper bound, as detailed in our main theorem. This result stands on its own merit, grounded in rigorous mathematical proof. Therefore, the necessity of experimental validation for this specific aspect of our work is not as critical, since the theorem is proven through theoretical methods.

That said, the inclusion of the experimental section was motivated by an intriguing observation: in certain settings, the impact of topological complexity, as predicted by our theorem, becomes quite apparent. We acknowledge that these findings may not be universally applicable across all settings. However, the mere existence of settings where the topological complexity bound is observed to be tight serves as a compelling illustration of our theoretical results.

We understand your perspective on the breadth of experimental validation and appreciate the opportunity to consider how our work might be further enriched. We believe this discussion enhances the understanding of our work's scope and its application in various contexts.

Reviewer comment (Writing):
In Section 3, you write "To explore the influence of manifold topology on network size, our analysis framework primarily leverages manifold classification theorems". There seems to be an ambiguity of the term "classification". Here, you, probably mention **classification** of manifolds by their structure. While in ML, by classification we mean a separation of two manifolds in the shared ambient space, via a hyperplane of a non-linear separating surface.

Author response:
We appreciate your attention to the potential ambiguity surrounding the term "classification" as it pertains to both the manifold classification theorem and classification problems in machine learning. In response to your insightful comment, we have carefully revised our manuscript to include clear distinctions and explanations whenever we refer to the manifold classification theorem. This ensures that our usage of the term is precise and unambiguous within the context of our discussion.

Reviewer comment (Theory):
Naitzat, et al. 2020 showed that the topological complexity of data manifold (and its representations in NN) decreases with depth, while in your work you find that width is more important (beta^2 vs. log(beta)). Can you explain it?

Author response:
Thank you for highlighting the comparison with the findings in Naitzat, et al. 2020. We would like to clarify that our results do not contradict their experimental outcomes. Firstly, our analysis indicates that the size of the neural network, in terms of the total number of neurons, is proportional to \(\beta^2\), focusing on the overall network size rather than just its width. It's a common understanding that the size of a neural network can exceed its depth, which aligns with our findings.

Secondly, while Naitzat et al. 2020 [1] observed that the topological complexity of internal representations decreases with network depth, this aligns with the expectation in classification tasks. Typically, the final outputs are binary (0/1), leading to a collapse of topological complexity at the middle layers of the network. In contrast, our work specifically examines how the initial topological complexity of input manifolds influences the overall size and depth of a neural network. Our focus is on the input side of the network, rather than the evolution of topological features within the network layers.

We believe these clarifications underscore the distinct yet complementary nature of our work with respect to the study by Naitzat et al. 2020.

[1] Gregory Naitzat, Andrey Zhitnikov, and Lek-Heng Lim. Topology of deep neural networks.
JMLR, 2020.

Reviewer comment (Theory):
**Assumptions on manifolds:** The types of manifolds that are considered are relatively limited. Some of the assumptions (such as compactness) are reasonable when thinking about data manifolds. Other assumptions, such as $d$-dimensionality, where $d \leq 3$, seems somewhat limiting (for instance, [1] provided evidence that ImageNet has an intrinsic dimension between 26 and 43). That being said, other works in this research area also put constraints on some part of the set-up (or have correspondingly weaker conclusions) so this may be the price of progress.

Author response:
**Q1. Assumptions on manifolds’ dimensions.**

The constraint to manifolds with dimension \(d \leq 3\) may appear overly restrictive at first glance. However, it's important to emphasize that the analysis of 3-dimensional manifolds is far from trivial. These manifolds present unique complexities and require intricate theoretical approaches. For a more detailed discussion on this, please refer to Q1 in our summary response.

Reviewer comment (Theory):
**Hints at the proof technique:** For papers such as this where the main contribution is a theorem, it is useful to give a high-level overview of the proof, even if the proof itself is relegated to the Appendix. This was done to some extent in this paper, but some additional details could probably be included for clarity. For instance, it seemed that some classification theorem was being used (and this presumably was what determined the choice to constrain to *solid manifolds*), but it was not clear what this was from the body of the work. Further spelling out some of these dependencies might illuminate why various assumptions were made.

Author response:
**Q2. Suggestions on the manuscript.**

We are grateful for your insightful suggestions regarding the proof sketch, typos, and references. In response, we have thoroughly revised and updated our manuscript to reflect these valuable inputs.

Reviewer comment (Writing):
“indication function” $\mapsto$ “indicator function”.

Author response:
**Q2. Suggestions on the manuscript.**

We are grateful for your insightful suggestions regarding the proof sketch, typos, and references. In response, we have thoroughly revised and updated our manuscript to reflect these valuable inputs.

Reviewer comment (Novelty):
There are a number of works that look at topological properties of ReLU networks that were not cited in the paper that may be worth considering (e.g., [2]).

Author response:
**Q2. Suggestions on the manuscript.**

We are grateful for your insightful suggestions regarding the proof sketch, typos, and references. In response, we have thoroughly revised and updated our manuscript to reflect these valuable inputs.

Reviewer comment (Theory):
The work references the use of manifold classification theorems, what manifold classification theorem was actually used?

Author response:
**Q4. What manifold classification theorem was actually used?**

In the proof of Lemma 1, we use the classification theorem for closed 1-manifolds and the classification theorem for closed surfaces.

Reviewer comment (Experiments):
The settings discussed in this paper seem unrealistic from a realistic machine learning perspective. It is generally recognized that data manifolds are embedded in low-dimensional manifold, but anything below 3 dimensions seems too low-dimensional. In three dimensions or less, the target is likely to be relatively simple and easy to analyze by machine learning. Therefore, even if correctly shown in these settings, many theories do not hold true in higher dimensions. In other words, this result is not recognized as a result that can be passed on a general level. On the other hand, if it does not hold in these settings, it will not hold in general. So I recognize that the discussion in this paper is positioned as a pilot study that suggests that it may be important to consider topological complexity.

Author response:
**Q1. Manifolds in three dimensions or less are too easy.**

The complexity of low-dimensional manifolds (3 or lower) should not be underestimated. It is not true that if a theory does not hold in low-dimensional cases, then it won’t hold in general. They differ from high-dimensional manifolds (5 or higher) in the theoretical framework. Low-dimensional manifolds often present unique complexities and challenges in analysis. Please refer to Q1 in our summary response.

Reviewer comment (Theory):
The discussion in this paper uses Betti numbers for data manifolds. However, since the data is typically given as a point cloud, the definition of the Betti number is not unique. This is the same as the discussion of filtration in persistent homology. In other words, it is not appropriate as a theory to evaluate a classification network if the results differ depending on the method used to calculate the Betti number, even for the same data.

Author response:
**Q2. Betti numbers are not accessible from point clouds.**

Contrary to the belief that Betti numbers are inaccessible from point clouds, they can indeed be accurately computed from sufficiently dense samples. For more information, please see our detailed response to Q3 in the summary.

Reviewer comment (Theory):
The assumption is that the phase of the data manifold is known, but it seems difficult to know that in advance. Are we not specifically discussing building a network, but merely discussing relationships? In other words, is the goal to suggest the value of considering topology?

Author response:
**Q3. Is the goal of this paper to suggest the value of considering topology?**

Our paper emphasizes the intrinsic role of topology in classification problems. We discuss this further in response to Q1 in the summary, addressing the reviewer's concerns about the prior knowledge of the data manifold's phase.

Reviewer comment (Theory):
The network discussed in this paper introduces \( N_p \) and \( N_{\phi} \) first. However, a general classification network does not include \( N_p, N_{\phi} \), nor is there a need to introduce it. Also, from the proof, the result of the main theorem seems to be strongly influenced by \( N_p, N_{\phi} \). In other words, the authors have failed to show that topology considerations are important for general classification networks; the only thing that can be shown is the obvious fact that networks with the additional topology analysis functionality introduced by the author himself are affected by topology complexity.

Author response:
**Q4. Real neural networks do not work as the construction in the paper.**

We aim to establish an upper bound on network complexity in classification scenarios. And the methodology is to prove by construction. While classification neural networks may not function as a straightforward homeomorphism followed by a classifier, the derived upper bound remains valid, suggesting that real networks could have fewer parameters. Building an upper bound on the size of neural networks to analyze its expressive power is a common approach in many theoretical papers, like [1, 2].

Reviewer comment (Theory):
The assumption in this paper seems to be that the classes of classification are disjointed. Is that a realistic setting?

Author response:
**Q5. The assumption in this paper seems to be that the classes of classification are disjointed. Is that a realistic setting?**

The assumption that classes in classification are disjoint is both intuitive and realistic, as overlapping manifolds would imply ambiguously labeled points, which is uncommon. This theoretical setting is also employed in other studies, such as [3].

Reviewer comment (Experiments):
This discussion seems to be overly argumentative for the purpose of simply binary classification. Perhaps it would be more appropriate to discuss this in a setting such as expressive learning.

Author response:
**Q6. It would be more appropriate to discuss this in a setting such as expressive learning.**

We appreciate the suggestion to frame our discussion in the context of expressive learning. Typically, neural network expressiveness is analyzed through function approximation under assumptions of continuity or smoothness, as in [2]. However, classification involves discontinuous real functions, presenting greater analytical challenges due to their less favorable properties.