Machine Learning meets Riemannian Geometry

This is the second article I am writing about Machine Learning in Riemannian Manifold. This time it is the extension of Principal Component Analysis(PCA) to Riemannian space. Readers are expected to have an understanding of PCA. To recall PCA once again, the following article can be helpful Understanding the Mathematics behind Principal Component Analysis. I also recommend readers to go through my previous article on Geodesic Regression where necessary concepts on Riemannian geometry are discussed.

Figure 1: Principal Geodesic Analysis (Image by author)

A glimpse at PCA

PCA is a well-known technique for data analysis by representing the data in terms of its principal constituents. Two of the well-known applications of PCA…


Space-time and Gravity

Source: Wikipedia

We are pretty much obsessed with seeing things from a Newtonian point of view that it has become a natural intuition. Newtonian view of the world seems obvious and satisfying. However, reality can be different than what it seems. For example, a body need not necessarily require a force to change its state from rest to motion. “The Theory of General Relativity” (GR) is a clearer viewpoint to seeing things around us and this overthrows even the most fundamental principles of Newtonian physics.

Albert Einstein is a celebrity physicist and a very well-known name. He started his work on GR…


Machine Learning meets Riemannian Geometry

Riemannian Geometry can be safely tagged as a “revolutionary” theory in mathematics. Firstly, the theory put forward a radical view of space and geometry by generalizing the “flat” Euclidean space to curved manifolds. Later, it was the basis for a major Physics revolution when Albert Einstein made use of the theory to explain space and gravity which we know as the “Theory of General Relativity”.

There have been uses of Riemannian geometry in Machine Learning as well. In this article, we will learn about Geodesic Regression which is an extension of Linear Regression to Riemannian space. It is assumed that…

Paribesh Regmi

into Machine Learning and Mathematics, and physics !

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