Review Details

Reviewer has chosen to be *Anonymous*

**Overall Impression:** Weak

**Suggested Decision: ** Reject

**Technical Quality of the paper:** Weak

**Presentation:** Weak

**Reviewer`s confidence:** High

**Significance:** Moderate significance

**Background:** Incomplete or inappropriate

**Novelty:** Limited novelty

**Data availability:** Not all used and produced data are FAIR and openly available in established data repositories; authors need to fix this

**Length of the manuscript:** The authors need to elaborate more on certain aspects and the manuscript should therefore be extended (if the general length limit is already reached, I urge the editor to allow for an exception)

**Summary of paper in a few sentences: **

The paper is essentially a survey of Geometrically motivated methods and their applications to machine learning. The author provides a set of methods and mentions their applications under two categories of algebraic geometry and differential geometry.

**Reasons to accept: **

I vote for rejection

**Reasons to reject: **

The paper attempts to provide a comprehensive survey of Geometrically motivated methods for learning. First and foremost, there is a large abundance and depth of such method and the survey doesn’t provide any of those. The survey is more of a short list of methods and their applications which have been selected in what seems to be some random or author dependent priority without any clear motivation for the selection. The survey is lacking proper depth, and professionalism in describing the basic mathematics or even intuition behind these methods. The reader is left with some list of method names but without any clear understanding of their mathematical basis. It is even hard to understand the advantage and disadvantages that each method has in various applications under such circumstances. The level of scientific writing, I’m afraid, is lacking and below standard.

In some places there are attempts to explain some intuition about existing methodology and its mathematical concepts however it is done in a rather vague manner, for example, “Forman-Ricci curvature (Weber, Jost, & Saucan, 2016), which measures the amount of “stuff” weighing down and spreading out each edge in the network, signaling network growth/shrinkage and importance of that edge to the overall shape of the network (with highly-curved edges serving as part of the network’s underlying skeleton).” How could anybody not familiar with the concept understand that clearly? Some basic formalism could help here.

The illustrations are simplistic and often lacking clear purpose: for example, the tangent space drawing is rather trivial. The illustration have no numbering and captions. In page 6 there is some illustration of 2D points with some coloring related to labels – what is this related to? What point is the author trying to make via this illustration. I am afraid this is below any publication standard.

What does the illustration in page 4 trying to convey? Some of the concepts there are not even discussed in the survey.

The Language is vague, using undescribed nouns and concepts as if the authors expects the reader to be completely on same page with him\her: Last paragraph on page 3 is unclear and missing relevant references to allow the reader to understand better: what are paired-ranking problems, what ‘items’ are you referring too?

Another example, “The image below shows a 3-D brain image with 4 landmarks and the disc created by the Ricci-flow-based map between spaces.” What spaces are you referring to? Where are the 4 landmarks in the example? They are not marked no the image, how could anyone understand the illustration or the point you are trying to convey.

The author describes many mathematical terms and algorithms but doesn’t provide adequate reference. references to Bertinini algorithm or to a book in algebraic geometry, what about references to the MDS algorithm, PCA, etc. I am afraid this is again below any publication standard.

There are many more incidents as the one I describe above which renders this manuscript unpublishable in my view. I hope the author can use my comments to improve the writing and presentation.

**Further comments: **

The paper attempts to provide a comprehensive survey of Geometrically motivated methods for learning. First and foremost, there is a large abundance and depth of such method and the survey doesn’t provide any of those. The survey is more of a short list of methods and their applications which have been selected in what seems to be some random or author dependent priority without any clear motivation for the selection. The survey is lacking proper depth, and professionalism in describing the basic mathematics or even intuition behind these methods. The reader is left with some list of method names but without any clear understanding of their mathematical basis. It is even hard to understand the advantage and disadvantages that each method has in various applications under such circumstances. The level of scientific writing, I’m afraid, is lacking and below standard.

In some places there are attempts to explain some intuition about existing methodology and its mathematical concepts however it is done in a rather vague manner, for example, “Forman-Ricci curvature (Weber, Jost, & Saucan, 2016), which measures the amount of “stuff” weighing down and spreading out each edge in the network, signaling network growth/shrinkage and importance of that edge to the overall shape of the network (with highly-curved edges serving as part of the network’s underlying skeleton).” How could anybody not familiar with the concept understand that clearly? Some basic formalism could help here.

The illustrations are simplistic and often lacking clear purpose: for example, the tangent space drawing is rather trivial. The illustration have no numbering and captions. In page 6 there is some illustration of 2D points with some coloring related to labels – what is this related to? What point is the author trying to make via this illustration. I am afraid this is below any publication standard.

What does the illustration in page 4 trying to convey? Some of the concepts there are not even discussed in the survey.

The Language is vague, using undescribed nouns and concepts as if the authors expects the reader to be completely on same page with him\her: Last paragraph on page 3 is unclear and missing relevant references to allow the reader to understand better: what are paired-ranking problems, what ‘items’ are you referring too?

Another example, “The image below shows a 3-D brain image with 4 landmarks and the disc created by the Ricci-flow-based map between spaces.” What spaces are you referring to? Where are the 4 landmarks in the example? They are not marked no the image, how could anyone understand the illustration or the point you are trying to convey.

The author describes many mathematical terms and algorithms but doesn’t provide adequate reference. references to Bertinini algorithm or to a book in algebraic geometry, what about references to the MDS algorithm, PCA, etc. I am afraid this is again below any publication standard.

There are many more incidents as the one I describe above which renders this manuscript unpublishable in my view. I hope the author can use my comments to improve the writing and presentation.

## 2 Comments

## Unable to handle

Submitted by Matjaz Perc on

Sorry, I am unable to handle this because of traveling.

## Meta-Review by Editor

Submitted by Tobias Kuhn on

The reviewers have acknowledged that topic of the paper is relevant, and a paper which provides a comprehensive survey of the relationship between algebraic geometry and machine learning would have significant value. However, both reviewers observe that the survey is currently too shallow, and does not adequately reflect the current state of the art in the field. It is a short manuscript which does not introduce the core concepts in sufficient depth or with sufficient mathematical rigor. There are inadequate references to papers in the field, and as such cannot really be considered a "survey". The author is advised to reconsider the scope, audience, and objectives of the manuscript.

Karin Verspoor (http://orcid.org/0000-0002-8661-1544)