Sklearn's documentation refers to "flat" or "non-flat" geometry of clusters to describe the use-cases of their implemented clustering algorithms. Those terms are not directly defined. However, the following quote states:
Non-flat geometry clustering is useful when the clusters have a specific shape, i.e. a non-flat manifold, and the standard euclidean distance is not the right metric.
So far, I do understand that a manifold type of shape is poorly represented by a cluster centroïd and that clustering such dataset should rather rely on local density, nearest neighbors, or connectivity constraint. However, I do not understand how it relates to the concept of flatness according to Wikipedia's definition:
In geometry, a flat is a subset of a Euclidean space that is congruent to a Euclidean space of lower dimension.
It seems contradictory to me. For instance, a hypersphere is not geometrically flat but would be flat using sklearn's terminology.
My questions are then:
- Are flat and non-flat geometry a legit terminology in machine learning and statistics?
- If yes, what is the mathematical definition?
- If no, what is the most appropriate alternative? (e.g. manifold vs convex ???)