Machine Learning

The Peter-Weyl theorem is one of the most beautiful results in Harmonic Analysis, and it has found surprisingly wide applications in machine learning (drug discovery, molecular dynamics, robotics, particle physics, graphs, etc). In many real-world problems, data is scarce, complexity is high, or we simply want our models to generalize better and produce more robust solutions. In these cases, it makes sense to leverage any prior knowledge we have about the problem structure. For example, in meteorology the data lives on a sphere (the Earth), or we might know that the underlying dynamics roughly satisfy the Navier-Stokes equations. The desire to incorporate this kind of domain-specific knowledge into machine learning models gave rise to what we now call Physics-Informed ML and Geometry-Informed ML. The former led to Neural ODEs and Physics-Informed Neural Networks (PINNs), but in this post we focus on the latter. ...

What do books, art, music, video games, and mathematics have in common? I remember the last time I went to the cinema and was genuinely impressed by a movie’s visual effects: Avatar (2009). Since then, my impression is that we’ve been living through diminishing returns: more polish, higher resolution, better lighting, but fewer moments that feel like a new era. The same seems true in video games: Red Dead Redemption 2 (2018) felt like a leap, and many people are hoping GTA VI will be the next one. But few expect another order-of-magnitude jump anytime soon. ...

How do we know if a machine learning model will perform well on unseen data? What happens if you continue to add samples to the dataset? Is it better to have a more complex model or a simpler one? These questions have been around for many years and are central to the field of statistical learning theory. Generalization theory provides mathematical guarantees and bounds on the generalization capability of families of functions. I have prepared a few notes on the basics of generalization theory. The only prerequisite is probability theory, and it is intended to be self-contained. It includes the most important results, such as Dudley’s Theorem and McDiarmid’s Theorem. ...

We all know the slogan “knowledge is power”. In this essay, I want to argue for the opposite: too much knowledge can hinder creativity Not because learning is bad, but because what you already know quietly narrows what you consider plausible. In the age of AI, where solutions are increasingly cheap, this matters: the bottleneck shifts from “finding an answer” to “finding an original direction”. So… should we stay ignorant? Creativity needs ingredients: skills, taste, vocabulary, technique, references. But there’s a trade-off between exploration (searching broadly) and exploitation (reusing what already works) [2]. Expertise pulls you toward exploitation and AI makes exploitation nearly frictionless. ...

I think there are 3 major milestones remaining for humanity, and one of them is clean and abundant energy. Fusion energy has the potential to provide a nearly limitless source of clean energy by replicating the processes that power the sun. However, achieving controlled fusion reactions on Earth has proven to be a formidable and very challenging task. The most well-known fusion prototype is the tokamak, which uses a magnetic field to confine the plasma within a toroidal chamber (doughnut-shaped). ...

For some time, I’ve been thinking about how to generalize self-attention mechanisms. Most existing attention mechanisms rely on pairwise similarities (dot products) between query and key vectors. However, higher-order relationships (involving triples or tuples of elements) could capture richer interactions. I then found that several people are already exploring this idea under the name “higher-order attention” [5]. However, this approach comes with a performance cost. Traditional self-attention has a complexity of O(n^2), while higher-order attention is even more computationally expensive. In this post, I’d like to share my perspective on this topic, connecting it with kernel methods and generalized metric spaces. ...

Initially, I wanted this post to focus solely on metrics in machine learning. However, the concept of metrics is far more universal, and it doesn’t make sense to treat it as an isolated problem. This is more of a philosophical post, the ultimate goal is to make you think and reflect. We live surrounded by metrics: grades from teachers, performance reviews from employers, publication counts in academia, FLOPs in computing, ELO in chess, salary, IQ for intelligence, movie ratings on IMDB, book ratings on Goodreads, stars/reviews on Amazon, election results in democracies, F1 score in machine learning, GDP for countries, EBITDA in finance, likes/followers on Instagram, time spent on TikTok for content recommendation algorithms, etc. ...

Gaussian Process Regression is one of the most elegant and theoretically rich algorithms in machine learning. With this post, I want to celebrate the mathematical beauty underlying Gaussian Processes. I will divide this post into two sections: theory and practice, accompanied by code examples. One of the key advantages of Gaussian Processes compared to Deep Learning methods is that they inherently provide interpretability (through confidence intervals and uncertainty estimation). They also offer excellent extrapolation properties, as we will see, and a way to incorporate knowledge about the structure of the data into the model. However, these benefits come at a cost. The algorithm has a wide variety of hyperparameters that are difficult to configure; for instance, kernel selection alone is challenging. Understanding and having a good intuition for the inner workings of this algorithm (and the data) is key to making the most of it. ...

A few days ago, this paper was published in Nature [1] claiming a huge improvement in LLM inference using analog computing: “Our architecture reduces attention latency and energy consumption by up to two and four orders of magnitude” This could mean we may soon run LLMs on devices no bigger than smartphones while consuming less power than a light bulb. After taking a deep dive, I believe this might turn out to be one of the most influential results of 2025. ...

This post is an extract of my Master’s Thesis in Mathematics, it can be found in this GitHub repository for a complete and autocomplete document: repo. In our continuous pursuit of enlightenment, knowledge, and understanding of the world around us, we turn to the tool of mathematics. For thousands of years, mathematicians have sought to translate our sensory experiences into a language that captures their essence in the most abstract, universal, and general form. ...