RMLR: Extending Multinomial Logistic Regression into General Geometries

Summary

- Instead of adopting complex approaches for extending MLR to Riemannian manifolds via general geometry extensions such as gyro structures and generalized SINE rules, this study generalizes to Riemannian manifolds using a simple approach based on the logarithm map. - The authors show the several experimental results with various types of datasets.

Strengths

1. This study generalizes to Riemannian manifolds using a simple approach based on the logarithm map, avoiding complex approaches like gyro structures and generalized row of sines. 2. The geometric aspects of this paper are well-founded. Conducting geometrically valid computations using the logarithm map and parallel transport is a standard method for tangent space analysis. 3. Evaluation is conducted on various types of datasets.

Weaknesses

1. I find this paper somewhat confusing to read. What does the claim "our framework only requires the explicit expression of the Riemannian logarithm" in the introduction mean? For example, parallel transport requires explicit expressions for each of types of Riemannian manifold or metrics (Table 12). Is there a contradiction with the authors' claim?

Questions

1. Does Eq. 8 satisfy the axioms of distance? 2. The authors adopt parallel transport to determine A~_k in Eq. 11 but projecting a point in Euclidean space to the tangent space might be simpler. Parallel transport can be computationally intensive and needs to be defined according to the type of manifold. Why did the authors choose parallel transport? Also, regarding Weakness 1, if parallel transport needs to be determined individually, is there a contradiction with the authors' claim that only the logarithm map is needed?

Rating

7

Confidence

4

Soundness

4

Presentation

4

Contribution

4

Limitations

- Discussed in Appendix A.

Summary

The authors extend multinomial logistic regression into spaces where they only require a logarithmic map. They do so in order to accomplish tasks such as classification

Strengths

The paper is well organized and written. There is a good balance of theoretical results and practical applications. It is nice to see a thorough exposition of the SPD manifold with all the commonly used metrics.

Weaknesses

I don't understand why lines 29-31 seem to look down on the use of things such as tangent spaces and coordinate systems because the proposed method relies on the log map which itself effectively requires tangent spaces and coordinate systems. The experiment section could use more thorough explanations which are found in the appendix.

Questions

Can the authors elaborate on how their work is a distinct contribution compared to the SOTA methods which are mentioned? I understand this is more general, as is shown in Table 1 but I fail to see the entire picture.

Rating

7

Confidence

3

Soundness

3

Presentation

3

Contribution

2

Limitations

NA

Summary

This paper extends the multiclass logistic regression into general Riemannian spaces, contributing to the field of Riemannian deep learning. Starting from the concept of Riemannian hyperplanes, the present work constructs the distance from Riemannian points to Riemannian hyperplanes and derives Riemannian Multinomial Logistic Regression (RMLR) on Riemannian manifolds. The RMLR framework is then showcased under 5 geometries on the SPD manifold, and SO(n). Extensive experiments on different Riemannian backbone networks, including Riemannian feedforward, Riemannian residual, and Riemannian graph neural networks, validate the effectiveness of the proposed RMLR. Especially, the results in Tab. 9 on direct classification (LogEig v.s. SPD MLR) show a clear advantage of the proposed classifiers (up to 18.34 improvement).

Strengths

1. The proposed RMLR framework (Thm. 3.3) can be easily implemented in different geometries. For a specific geometry, one only needs to put the involved operators into Eq. 11. 2. The proposed RMLR not only generalizes the Euclidean MLR, but also incorporates several previous MLRs, such as gyro SPD MLR, gyro SPSD MLR, and flat SPD MLR (Tab. 1). Besides, it further can deal with the geometry which is non-flat or agnostic to gyro structure. 3. A complete study of 5 families of deformed SPD metrics is presented in Tab. 2 and Fig. 1. 4. 5 SPD MLRs and one Lie MLR are specifically implemented. The experiments on different network backbones including Riemannian feedforward, Riemannian residual, and Riemannian graph neural networks, validate the effectiveness of the proposed RMLR. 5. The presentation is clear, such as SPD and Lie MLRs in Thms. 4.2 and 5.2.

Weaknesses

1. More details on the optimization for learning the parameters of the MLR should be presented. 1. For the SPD manifold, there are at most three hyperparameters: $\theta, \alpha, \beta$. Although these indicate the generality of the proposed framework, how to select the parameter should also be discussed from a practical view.

Questions

1. What are the complexities (memory and time) of different metrics, theoretically and experimentally? 2. For the SPD metrics, how can the researcher select the involved hyper-parameters in practice?

Rating

8

Confidence

4

Soundness

4

Presentation

3

Contribution

4

Limitations

N/A

Official Comment by Reviewer dBzm

Thanks for the reply. 1 Interesting. I hope the SO(n) computation package will be released. This will facilitate building networks in the Lie group. 2-3 Interesting and worth reading, thanks! Generally, Riemannian deep learning can benefit from the proposed RMLR framework. Apart from the current geometries discussed in this paper, the proposed RMLR has the potential to be implemented into other geometries, facilitating Riemannian neural networks. I have no further concerns and have raised my score to 8. Good luck.