Machine Learning in a Non-Euclidean Space

<h2>What you will learn in this article.</h2> 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. <h2>Our conversation.</h2> 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.”. This might seem obvious, but actually, it is not always true. Indeed, it is true only in Euclidean spaces, and when this axiom does not hold, it means that we are actually considering non-Euclidean geometry. This axiom is also equivalent to the Triangle postulate, which says, “The sum of the angles in every triangle is equal to 180°”. <a href="https://pub.towardsai.net/machine-learning-in-a-non-euclidean-space-8f3d13f0a317">Learn More</a>