Machine Learning in a Non-Euclidean Space
1. There are different examples of non-Euclidean geometry, among them spherical geometry and hyperbolic geometry.
2. A hyperbolic space is a space of negative constant curvature.
3. There are different models of hyperbolic geometry, the most famous being the Poincaré ball model.
4. For datasets with a hierarchical structure, it is better to represent it in a hyperbolic space, because both a hyperbolic space and a hierarchical dataset have an inherent exponential growth.
Our conversation.
M: Aniss, could you give us some intuition behind hyperbolic geometry and allow us to understand for what kind of data it is relevant? I know hyperbolic geometry is one sort of non-Euclidean geometry, the one which has a negative curvature.
A: Let’s start from the beginning! First, what is non-Euclidean geometry? There is something called the 5th postulate of Euclide (also called the parallel postulate), which is equivalent to the following Playfair’s axiom: “In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.”.
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