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How does gradient descent work?
This is the companion website for the paper Understanding Optimization in Deep Learning with Central Flows , published at ICLR 2025.
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Less is more: Recursive reasoning with tiny networks
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Why are QR Codes with capital letters smaller than QR codes with lower case?
Take a look at these two QR codes. Scan them if you like, I promise there's nothing dodgy in them. Left is upper-case HTTPS://EDENT.TEL/ and right is lower-case https://edent.tel/ You can clearly see that the one on the left is a "smaller" QR as it has fewer bits of data in it. Both go to the same URl, the only…
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All AI models might be the same
what can language model embeddings tell us about whales speech, and decoding ancient texts? (on The Platonic Representation Hypothesis and the idea of *universality* in AI models)
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The shadows lurking in the equations
If you've ever graphed equations, chances are you have only ever graphed equations in what I refer to as "Binary mode" - which draws a line where the equation is EXACTLY equal, and leaves white everywhere else.
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Important machine learning equations
A reference guide to fundamental machine learning equations with correct implementations and clear explanations of how they connect to each other.
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The cultural divide between mathematics and AI
A reflection on cultural differences observed at the 2025 Joint Mathematics Meeting.
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A Lean companion to Analysis I
Almost 20 years ago, I wrote a textbook in real analysis called “Analysis I”. It was intended to complement the many good available analysis textbooks out there by focusing more on foun…
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Diff Algorithms
How I overcame copying and modifying my own diff library from project to project by diving too deep into diff algorithms and coming out at the other end with new diff library in my hand.
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What Is Entropy?
People say many things about entropy: entropy increases with time, entropy is disorder, entropy increases with energy, entropy determines the arrow of time, etc.. But I have no idea what entropy is, and from what I find, neither do most other people. This is the introduction I wish I had when first told about entropy…
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Derivatives, Gradients, Jacobians and Hessians
This article explains how these four things fit together and shows some examples of what they are used for. Derivatives Derivatives are the most fundamental concept in calculus. If you have a funct…
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Generating an infinite world with the Wave Function Collapse algorithm
Generating an infinite world with the Wave Function Collapse algorithm - Personal blog about game development, programming and making
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Mathematical exploration and discovery at scale
Bogdan Georgiev, Javier Gómez-Serrano, Adam Zsolt Wagner, and I have uploaded to the arXiv our paper “Mathematical exploration and discovery at scale”. This is a longer report on…
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BusyBeaver(6) Is Quite Large
For overdetermined reasons, I’ve lately found the world an increasingly terrifying and depressing place. It’s gotten harder and harder to concentrate on research, or even popular scienc…
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Learning Feynman's Trick for Integrals
Among a few other integral tricks and techniques, Feynman's trick was a strong reason that made me love evaluating integrals, and although the technique itself goes back to Leibniz being commonly known as the Leibniz integral rule, it was Richard Feynman who popularized it, which is why it is also referred to as…
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Why don't you use dependent types?
To be fair, nobody asks me this exact question. But people have regularly asked why Isabelle dispenses with proof objects. The two questions are essentially the same, because proof objects are intrinsic to all the usual type theories. They are also completely unnecessary and a huge waste of space. As described in an…
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Everything is correlated (2014–23)
Anthology of sociology, statistical, or psychological papers discussing the observation that all real-world variables have non-zero correlations and the implications for statistical theory such as ‘null hypothesis testing’.
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A dumb introduction to z3
Exploring the world of constraint solvers with very simple examples.
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Bloom Filters by Example
A Bloom filter is a data structure designed to tell you, rapidly and memory-efficiently, whether an element is present in a set.
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Algebraic Semantics for Machine Knitting
As programming languages researchers, we’re entitled to a certain level of mathematical rigor behind the languages we write and analyze. Programming languages have semantics , which are definitions of what statements in the language mean. We can use those semantics to do all sorts of useful things, like error…
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Compressing Icelandic name declension patterns into a 3.27 kB trie
How I used a compressed trie data structure to encode Icelandic declension data.
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Bloom Filters
The original motivation for the creation of Bloom filters is efficient set membership, using a probabilistic approach to significantly reduce the time and space required to reject items that are not members in a certain set.
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Why Busy Beaver hunters fear the Antihydra
In which I explore the biggest barrier in the busy beaver game. What is Antihydra, what is the Collatz conjecture, how are they connected, and what makes them so daunting?
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Accessible open textbooks in math-heavy disciplines
The challenge The authoring platform of choice in many math-heavy disciplines is LaTeX. It produces typeset documents of excellent quality and handles formulas and mathematical diagrams extremely w…
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A statistical analysis of Rotten Tomatoes
Can Hollywood's stamp of artistic excellence still be trusted?
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Progress toward fusion energy gain as measured against the Lawson criteria
Today we’ve released a pre-publication version of an update to our 2022 paper “Progress toward fusion energy breakeven and gain as measured against the...