Originality: although it is not groundbreaking, the paper is interesting to read. From the methodological point of view the originality comes from being the first paper showing how to perform regression with covariance matrices where the target is defined across persons. Information is missing on the difference from other techniques working with matrices for within-subject target and also comparison to Spoc is missing. Quality: the paper is quite easy to follow. I miss the following: 1) information on regression strategies within person and point out the differences to the method presented here. 2) a block diagram with the steps to be performed in each of the two cases they describe (for the covariance based approaches). 3) in case of rank deficient matrices: missing comparison between the method for rank deficient matrices versus methods for full rank matrices but using regularised covariance matrices. 4) significance results of the comparisons between methods. 5) comparison to SPoC. The paper states: "based on spoc method" but it is not clear if spoc could have been useful in this case nor how the methods would compare. 6) there is not clear claim about weaknesses of their approach. Clarity: the paper is clear, but some information is missing (see previous point). Significance: the paper is interesting for experts. It would fit the conference. It provides a new experimental approach. Clarifying its relation to Spoc might increase the significance of the paper.

The theoretical sections of the paper appear sound, with the Riemannian approaches and their respective invariance properties being properly established. The authors also discuss multiple possible functions that could be applied on the signal powers to obtain the target variable, and prove how using a linear regression model with the Riemannian feature vectors would be optimal for the identity, log and square roots of the signal power. However, they fail to discuss how often these types of scenarios occur in actual MEG/EEG dataset, and also how the performance would deteriorate in case where a different function of the source signals powers is used. The construction of the toy dataset is well thought out to exploit the invariances provided by the Riemannian metrics and demonstrate their performance in the ideal scenario. But as mentioned previously, some additional toy examples that examine the performance of the different models in sub-optimal conditions would also be useful. In addition, it would be interesting to see how the performance of the log-diag model on the toy dataset is affected by the use of supervised spacial filters, or how the geometric distance changes when supervised or unsupervised spacial filters are used. This comparison of the effects of supervised and unsupervised spacial filters with the geometric distance should also be performed on the Cam-CAN dataset evaluations, or the authors should at least address why this scenario was left out of the evaluation. Finally, the inclusion of the model inspection results seems unnecessary, since it does not relate to any of the methods proposed in this paper. Even though the Riemannian measures used in this paper are already existed, here they are applied in a new scenario, and good performance methods further demonstrates the usefulness of Riemannian techniques for MEG/EEG data analysis. The application of the geometric distance in conjunction with spatial filtering to deal with rank-reduced covariance matrices is an interesting approach, but it would be more interesting to see a bigger focus and a more detailed analysis on the interplay of the Riemannian metrics with the different spacial filters discussed in this paper. The claim that this paper presents the “first study to apply a covariance-based approach to regression problem in which the target is defined across persons and not within persons” is too broad and not entirely correct, as some other works can also be placed into this category (e.g. Dmochowski et al. 2012, Parra et al. 2019). However, the fact that this paper introduces the first benchmark on MEG-based age prediction for the Cam-CAN dataset could prove to be a significant contribution if this problem is seen as interesting by the community and is taken up as a standard benchmark for novel methods. The paper is generally clear and not too difficult to read, with the exception of sections 2.1 and 2.2 which are somewhat dense and more difficult to get through. Additionally, there are several small errors or typos throughout the text. No code was provided for the methods or evaluations, and while most of the details on the software used and preprocessing steps taken were discussed, but some specifics, such as if/how subjects were rejected and how the time-samples were drawn from the Cam-CAN dataset. A link to the toy dataset or corresponding code was also not provided.

The paper studies regression with rank-reduced covariance matrices from MEG data. Rank-reduced covariance matrices are an issue because they are the product of modern EEG/EMG artefact and noise reduction methods. The authors study two Riemannian approaches by projecting MEG covariance matrices into a tangent space: Firstly, the Wasserstein metric and secondly, a combination of linear projection and geometric distance. To evaluate the performance of the method the algorithm is applied to artificial data set and to the Cam-CAN dataset. Results of the proposed method are also compared to results of other state of the art methods. The results suggest that the proposed method is only outperformed by the complex biophysics driven source modelling. The proposed framework is novel and based on the Riemannian Geometry, which is becoming a more and more important approach used for the MEG/EEG analysis. Related work is properly cited. The paper is well structured and clear. The authors reasoning and rationale is sound. The discussion of pros and cons of the method could be increased. Particularly, because the used cross-validation approach may lead to an optimistic assessment of the regression. The reviewer understands that the whole Cam-CAN dataset was randomly sampled and that authors used a 10-fold cross-validation to test the regression. For example, a leave one-subject out cross-validation would have made a more realistic assessment of the performance. Nevertheless, the idea and the results are significant and other researchers will build on this idea. ======== Response to authors’ feedback: Thank you for the clarifications.