Machine Learning in a Non-Euclidean space

Chapter 0. A Microsoft DS and an ML Doctoral Researcher thinking together

3 min readJun 26, 2023

How it all started.

Recently I launched a series on the mathematics needed for any Machine Learning practitioner. I quickly came across Euclidean spaces and Euclidean geometry. As a teacher at EPITA Paris and a DS at Microsoft, I have built a lot of intuition based on Euclidean geometry.

Typically, in many cases, I understand we want two things similar in our human reasoning to be also similar from a Euclidean perspective, i.e. close together in say a 2-dimensional space. For example, in NLP, we typically want two words that are semantically close to each other to also be close to each other in their n-dimensional representations (embeddings).

Though, as I progressed in my personal reading, I came across non-Euclidean geometry. The authors seemed to argue that it was important to learn more about this space. I then remembered a couple of models leveraging non-Euclidean geometry, e.g. Poincaré Embeddings for Learning Hierarchical Representations. I also remembered discussing non-Euclidean Variational Autoencoders (VAE) with my good friend, Aniss Medbouhi.

**Resource**: Adapted from Wikipedia. The typical objective of an autoencoder is to reproduce the input x after compressing it in a latent space. Such latent space is typically Euclidean. What if we use a hyperbolic space for certain datasets to better reproduce them in the output space?

Aniss is doing a PhD at KTH Royal Institute of Technology. He is working on geometric and topological methods for representation learning, and applications to Brain-Computer Interfaces. I quickly remembered him presenting his master’s thesis focusing on computational topology including non-Euclidean geometry applied to Machine Learning.

For those of you that don’t know KTH, it is a fantastic university, the best university for engineering in Scandinavia. I have done my masters in Machine Learning there myself, as a double degree student from France. That is where I met Aniss, where he was also doing a double degree from another engineering school in France.

Back then, I was amazed by the variety of high-quality courses in Machine Learning and Deep Learning. Landing at Microsoft right after my masters, I was armed with some solid understanding of the field thanks to KTH.

Why is Aniss a great partner for this series?

He has done great work in topology and has a heavy background in maths.
He is doing research in hyperbolic geometry applied to Machine Learning in a top lab in Sweden: the Robotics Perception and Learning Division, in Prof. Danica Kragic’s team.
He is passionate about what he does and loves using simple tools to explain complex math.
He is a really cool guy!

Like many of you out there, I have a limited amount of time for learning, but I understand that, in this fast-moving space, it is key to stay sharp. So, I have chosen not to go all-in looking for multiple resources and reading them in isolation. Rather, I will approach this complex space with my friend Aniss. He is going to help me, and by the same token, help all of us, get intuition around ML in non-Euclidean spaces. I’ll do my homework, and so should you. In this series, I’ll share my learnings and my discussions with Aniss.