Reviewer has chosen to be Anonymous

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.

Nanopublication comments:

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.

By Parikshit Ram https://orcid.org/0000-0002-9456-029X

Review Details

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Reviewer has chosen not to be Anonymous

Summary of paper in a few sentences:

This manuscript is attempting to introduce the ideas of algebraic geometry and differential geometry to a wider audience of machine learning and data science. The manuscript explains these concepts and their applicability to machine learning on a really high level.

Reasons to accept:

The idea of introducing these concepts of algebraic geometry and differential geometry to a wider audience is extremely valuable in my opinion, and I believe that some version of this manuscript should be published and made available to a wider audience. However, I voted as undecided for acceptance because of the reasons below.

Reasons to reject:

I want to start with the caveat that I would love for this manuscript to get published, and the comments below are just some suggestions to make this manuscript more accessible and useful:
- The manuscript is lacking appropriate references to the "plethora" of examples of the application of algebraic geometry or differential geometry to solving ML problems. These references should be such that it will allow the reader to easily recognize the value of these techniques for general ML tasks.
- Moreover, an exposition of (possibly simple) a complete such application in the manuscript would be very useful to the wider audience.
- A lot of examples and technical terms in the manuscript need appropriate references. Or alternately, the manuscript can contain more mathematical details.
- The examples in the (unnumbered) images need context and more elaborate explanation in the text to be useful to the reader.

Nanopublication comments:

Further comments: