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A Princeton probabilist explains why enormous random matrices stop behaving randomly and start behaving like a single fixed object. Almost nobody watches it. This is Ramon van Handel at Harvard's Science Center, April 2025, on the strong convergence phenomenon. Every weight matrix in every model starts as random numbers.... The claim here is that as dimensions grow, the extreme behavior of such matrices, the largest eigenvalues, the operator norm, stops fluctuating and locks onto a deterministic limit. That is why initialization works at all. Why spectral norms are predictable. Why large networks behave more consistently than small ones instead of less. Watch how he sets up what strong means. The distinction between convergence of averages and convergence of the extremes carries the entire result. A machine learning engineer I know rewatched the setup twice and said scaling laws stopped feeling like empirical luck. Free on YouTube from the Harvard Mathematics Department, subtitles on. Random at small scale. Deterministic at large. Nobody told the engineers.show more

Poiler

1,474 subscribers

104,017 Aufrufe • vor 3 Monaten •via X (Twitter)

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12 Kommentare

Profilbild von Rob Sneiderman

Rob Sneidermanvor 3 Monaten

A visual companion for the first step in this story: the thin-shell phenomenon. As dimension grows, a Gaussian vector still has random coordinates, but its normalized length becomes sharply concentrated near 1. Randomness remains locally; globally, the geometry becomes predictable. That concentration-of-measure viewpoint is one of the cleanest entry points into high-dimensional probability.

Profilbild von Sonny from Laguna

Sonny from Lagunavor 3 Monaten

It’s almost like there’s some kind of law of large numbers at play…

Profilbild von Yagami

Yagamivor 3 Monaten

@grok Give me the gist of this article

Profilbild von JÜRGEN

JÜRGENvor 3 Monaten

Great you said almost nobody watches it. which means you've employed reverse psychology which means I definitely can't watch it now. thanks.

Profilbild von Michael

Michaelvor 3 Monaten

wait this is literally just how statistics works, same thing with atoms determinism is non-determism at scale is this something thats not taught to ml students?

Profilbild von ᛕᎥᕼᗷᗴᖇᑎᗴ丅Ꭵᑕᔕ

ᛕᎥᕼᗷᗴᖇᑎᗴ丅Ꭵᑕᔕvor 3 Monaten

Gerald Weinberg did say something similar in 1975 in his "An Introduction to General #Systems_Thinking." Many have a problem in realizing this simple truth:

Profilbild von Huysolo

Huysolovor 3 Monaten

Random at small scale. Deterministic at large. Nobody told the engineers.

Profilbild von michael

michaelvor 3 Monaten

Is this why metronomes fall into synchronicity after a period?

Profilbild von PeteSK

PeteSKvor 3 Monaten

Thanks for posting these I have them filed for watchin!

Profilbild von AFX LAB

AFX LABvor 3 Monaten

Traditionally, eigenvalues are diagnostic. We compute them to understand a system after the fact. But if they instead become active state variables that exert forces, then the optimizer isn't just following the landscape, its reshaping the landscape while it moves through it.

Profilbild von ራስ ባሪያው Rass Bariaw

ራስ ባሪያው Rass Bariawvor 3 Monaten

🗝️

Profilbild von DanNoaHided

DanNoaHidedvor 3 Monaten

I spoke to a PHD electrical engineer and when he told me I could count to more than 10 on both hands I was shocked!

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Pearl Research Labs

23,995 Aufrufe • vor 3 Monaten

MIT filmed the single hour that makes all of calculus one subject instead of two. Free, uncut, chalk only. Almost nobody watches it. This is 18.01, Single Variable Calculus, lecture 19, taught by David Jerison. The First Fundamental Theorem of Calculus. Slopes and areas were separate problems for centuries. Different tools, different people, different questions. This lecture shows they were the same thing pointed in opposite directions. Every gradient computed in every training run rides on that connection. Most engineers use it daily and have never seen the proof that makes it legal. Watch the board work in the middle. He builds the link line by line, real students in the room, no software, no animation. A machine learning engineer I know rewatched it before an interview and said the chain from loss to update finally read as 1 object instead of 3 tricks. Full course free on YouTube under Creative Commons. You've been using the theorem. You've never seen it earned.

MIT filmed the single hour that makes all of calculus one subject instead of two. Free, uncut, chalk only. Almost nobody watches it. This is 18.01, Single Variable Calculus, lecture 19, taught by David Jerison. The First Fundamental Theorem of Calculus. Slopes and areas were separate problems for centuries. Different tools, different people, different questions. This lecture shows they were the same thing pointed in opposite directions. Every gradient computed in every training run rides on that connection. Most engineers use it daily and have never seen the proof that makes it legal. Watch the board work in the middle. He builds the link line by line, real students in the room, no software, no animation. A machine learning engineer I know rewatched it before an interview and said the chain from loss to update finally read as 1 object instead of 3 tricks. Full course free on YouTube under Creative Commons. You've been using the theorem. You've never seen it earned.

obssnnn

125,618 Aufrufe • vor 3 Monaten

Gilbert Strang, the legendary mathematician who taught linear algebra for 61 years and became the most watched math professor in history: "I used to think a matrix was just a grid of numbers, until I proved that its rows and columns always agree on one number no matter how you look at them. That fact still feels like magic to me after sixty years." this is the exact proof sitting quietly underneath every factor model a risk desk trusts with real capital, and almost nobody outside a math department has ever seen it. strip away the notation and the idea is almost absurdly simple. take any matrix, any grid of numbers, and count how many of its rows are truly independent, meaning none of them can be built out of the others. now count the independent columns instead, a completely different question on the surface. those two numbers, row independence and column independence, always turn out exactly equal, no matter how large or lopsided the matrix is. nobody presenting a clean risk model out loud credits a decades old proof for the reason the math even holds together. zoom out to what this means for anything built on a grid of numbers today. a portfolio, a covariance matrix, a neural network's weights, all of them hide a true dimension smaller than their size suggests, and that hidden number is exactly what this proof pins down. the industry sells complexity as scale, more assets, more parameters, more rows and columns. but the real question was never how big the matrix is. it's how many independent directions are actually hiding inside it. the size of the grid was never the real story. it was the one number both sides of it were quietly agreeing on the whole time.

Gilbert Strang, the legendary mathematician who taught linear algebra for 61 years and became the most watched math professor in history: "I used to think a matrix was just a grid of numbers, until I proved that its rows and columns always agree on one number no matter how you look at them. That fact still feels like magic to me after sixty years." this is the exact proof sitting quietly underneath every factor model a risk desk trusts with real capital, and almost nobody outside a math department has ever seen it. strip away the notation and the idea is almost absurdly simple. take any matrix, any grid of numbers, and count how many of its rows are truly independent, meaning none of them can be built out of the others. now count the independent columns instead, a completely different question on the surface. those two numbers, row independence and column independence, always turn out exactly equal, no matter how large or lopsided the matrix is. nobody presenting a clean risk model out loud credits a decades old proof for the reason the math even holds together. zoom out to what this means for anything built on a grid of numbers today. a portfolio, a covariance matrix, a neural network's weights, all of them hide a true dimension smaller than their size suggests, and that hidden number is exactly what this proof pins down. the industry sells complexity as scale, more assets, more parameters, more rows and columns. but the real question was never how big the matrix is. it's how many independent directions are actually hiding inside it. the size of the grid was never the real story. it was the one number both sides of it were quietly agreeing on the whole time.

MindArch

18,494 Aufrufe • vor 3 Monaten

Richard Feynman took live questions from a random Auckland audience in 1979 and answered the ones physicists still argue about in 2026. The camera kept rolling. Almost nobody watches it. This is the final lecture of the series, the one where the script ends. A Nobel laureate standing at a microphone, fielding whatever the room throws at him about the deepest open problems in physics. He won the 1965 Nobel Prize for the theory behind every answer he gives here. This is him off-book, no prepared blackboard, thinking out loud in front of ordinary people. Watch how he handles the questions he can't answer. He says so, flat out, then explains exactly where knowledge stops and guessing begins. No hedging. No performance. A quant researcher I know rewatched it twice and sent 1 timestamp to his entire team with the message: this is how you say I don't know. 47 years on YouTube. Fewer views than a mousepad review. Admitting ignorance was his edge. It still works.

Richard Feynman took live questions from a random Auckland audience in 1979 and answered the ones physicists still argue about in 2026. The camera kept rolling. Almost nobody watches it. This is the final lecture of the series, the one where the script ends. A Nobel laureate standing at a microphone, fielding whatever the room throws at him about the deepest open problems in physics. He won the 1965 Nobel Prize for the theory behind every answer he gives here. This is him off-book, no prepared blackboard, thinking out loud in front of ordinary people. Watch how he handles the questions he can't answer. He says so, flat out, then explains exactly where knowledge stops and guessing begins. No hedging. No performance. A quant researcher I know rewatched it twice and sent 1 timestamp to his entire team with the message: this is how you say I don't know. 47 years on YouTube. Fewer views than a mousepad review. Admitting ignorance was his edge. It still works.

xanax

60,676 Aufrufe • vor 3 Monaten

. Qutaiba Idlbi قتيبة ادلبي Qutaiba Idlibi, Director of the U.S. Affairs Department at the Syrian Ministry of Foreign Affairs: “There is a convergence between the Syrian state and the United States on the necessity of imposing stability in Syria and unifying the country under the authority of a central government that asserts its authority and sovereignty.”

. Qutaiba Idlbi قتيبة ادلبي Qutaiba Idlibi, Director of the U.S. Affairs Department at the Syrian Ministry of Foreign Affairs: “There is a convergence between the Syrian state and the United States on the necessity of imposing stability in Syria and unifying the country under the authority of a central government that asserts its authority and sovereignty.”

Omar Abu Layla

12,164 Aufrufe • vor 9 Monaten