# Name of academic field studying geometric structure of data sets [closed]

I have questions about the geometric structure of data sets, esp. as it relates to the relationships between predictors. Is there a name for this field?

• Topological data analysis?
– Dave
Commented Dec 14, 2023 at 17:47
• @Dave TDA is an interesting suggestion. In persistent homology the persistence of topological invariants like Betti numbers acriss a sequence of simplicial complexes is dependent on the geometry of the sampled manifold. This differs from a theoretical discussion of homology and homotopy classes not depending strongly on geometry. Commented Dec 14, 2023 at 17:58
• Topology is not geometry. From a broad perspective of geometry, I see almost no difference between "geometric structure of datasets" and "data analysis:" very little of the latter is non-geometric and any "geometric structure of data" is perforce a form of data analysis.
– whuber
Commented Dec 14, 2023 at 19:00
• @whuber To add to that, I am unaware of any geometry-free data analysis techniques that are actually used. Although theoretically possible by starting with a ASC, in practice geometric simplicial complexes are obtained first which have a place in the theory around ASCs. Commented Dec 14, 2023 at 19:17
• Note that pretty much all of machine learning can be described as finding transformations that simplify the geometry of datasets. For instance, if you have a datasets of images to classify, maybe each image can be seen as a vector in widthxheightx3-dimensional space; but the classes are completely messy in that space. Now you train a neural network on this dataset, and its output is a vector in k-dimensional space where k is the number of classes; and now the k classes look like nice geometric clusters in this space.
– Stef
Commented Dec 15, 2023 at 13:45

Wikipedia uses the term "geometric data analysis" but I find that the page hardly scratches the surface, and the examples given seem hodge-podge to me. Let me provide a template for how you might progress into this subject.

# Basic

An excellent starting point is Euclidean geoemtry. A really basic description is a Euclidean distance matrix and an angle matrix, and plot them as heatmaps.

You can start with plotting the univariate and bivariate plots that seem appropriate to the domain.

Parallel axis plots can also be your friend if you want to visually learn about the distances and angles in relation to the original variables.

# Intermediate

Learn what a metric space is so that you understand the general notion of distance.

You can readily generalize your notion of "angle" using inner product spaces .See dot product and generalizations. Hilbert spaces are a good case study of this abstraction which has applications in physics.

Some of the classic correlation coefficients can be interpreted as such angles. It can be helpful to think of correlation as an angle which is the approach Aguste Bravais took before Galton or Pearson. Correlation is usually thought of as a relationship between two variables, however Seilis 2022 discusses "correlations" as angles between more general subspaces and cites some related literature.

With that generalization of distance and angle, you can repeat the basics I suggested above.

Geometry in a modern mathematical sense geometry is generalized to transformational invariance. There is a lot of history there even though the subject is only a few hundred years old (breaking news compared to Euclidean geometry). An interesting inflection point is the Erlangen program. I recommend Bronstein et al 2021 for a nice introduction to some of the history and basic version of the math, and further discussion of geometry in relation to innovations in deep learning.

The abstract task becomes uncovering the transformational invariants (AKA symmetries) of a latent manifold for which your data is assumed to be a noisy sampling from. This requires mathematical modelling and is sometimes enhanced by machine learning in which you wish to approximate an unknown shape.

A special case that I recommend is discrete differential geometry. I have been working my way through Discrete Differential Geometry - CMU 15-458/858 . I have benefited from combining what I have learned about curvature in that course with statistics.

# Warning

Some data sets have interesting geometry, but many don't. Many data sets can be seen as samples from roughly-elliptical or a union of roughly-elliptical shapes.

Learn this stuff either because you enjoy it or because you have a data set that seems to have some interesting property that seems related to geometry. But don't expect it to revolutionize how you conduct data analysis in most cases.

• Thanks Galen, for the in depth answer! One question I have is whether TDA concerns itself with the predictor (X's) or response (Y's) information? I am specifically interested in the structure of the predictor information, using it to specify design matrices. For example, if Predictor A is nested within Predictor B, and B is nested within C, then can it be formally determined if A is also nested within C? More generally, can model design matrices be specified using only pairwise relationships between predictors, or is higher dimensional analysis of the predictors required? Commented Dec 14, 2023 at 23:44
• @ChrisScience TDA is an entire field of techniques, so I am not sure I can unilaterally answer. But in the basic approaches I have seen such as persistent homology, there is no notion of predictor and predicted variables. Sometimes topological features are extract and then used in machine learning problems, but that is not TDA per se. Commented Dec 15, 2023 at 2:36
• The "nesting" relationship would have to be more clearly defined for me to definitively answer, but if you are referring to a subset relation then for w/e sets you are referring to the answer would be "yes": subset relations are partial orders and therefore are transitive. That is more a point about order theory than what I would immediately recognize as a question of topology. Commented Dec 15, 2023 at 2:37
• "More generally, can model design matrices be specified using only pairwise relationships between predictors, or is higher dimensional analysis of the predictors required?" -- I am not sure what you mean by this. You'll have to be more precise. But comments are not for discussion. I suggest you formulate any additional questions as using the site's question feature rather than in the comments. Commented Dec 15, 2023 at 2:41