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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

Mark Poiler

1,496 subscribers

103,685 просмотров • 1 месяц назад •via X (Twitter)

Наука и технологии Образование

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Комментарии: 12

Фото профиля Rob Sneiderman

Rob Sneiderman1 месяц назад

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.

Фото профиля Sonny from Laguna

Sonny from Laguna1 месяц назад

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

Фото профиля Yagami

Yagami1 месяц назад

@grok Give me the gist of this article

Фото профиля JÜRGEN

JÜRGEN1 месяц назад

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

Фото профиля Michael

Michael1 месяц назад

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?

Фото профиля ᛕᎥᕼᗷᗴᖇᑎᗴ丅Ꭵᑕᔕ

ᛕᎥᕼᗷᗴᖇᑎᗴ丅Ꭵᑕᔕ1 месяц назад

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:

Фото профиля Huysolo

Huysolo1 месяц назад

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

Фото профиля michael

michael1 месяц назад

Is this why metronomes fall into synchronicity after a period?

Фото профиля PeteSK

PeteSK1 месяц назад

Thanks for posting these I have them filed for watchin!

Фото профиля AFX LAB

AFX LAB1 месяц назад

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.

Фото профиля ራስ ባሪያው Rass Bariaw

ራስ ባሪያው Rass Bariaw1 месяц назад

🗝️

Фото профиля DanNoaHided

DanNoaHided1 месяц назад

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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A mathematician who won the Fields Medal, the Abel Prize, and the Wolf Prize spends an hour on the equals sign. This is Pierre Deligne at the Institute for Advanced Study, September 2018, speaking at the memorial conference for Vladimir Voevodsky. The title is a question: what do we mean by equal. Deligne proved the last of the Weil conjectures. He holds every major honour mathematics gives. He chose to talk about the symbol every schoolchild learns first. The problem is real. Two objects can be the same in a way that carries information about how they are the same, and set theory throws that information away. Voevodsky spent his last years rebuilding foundations around exactly this. Watch the board. Set theory's definition of equality on one line, homotopy groups underneath, and the gap between them opening as he writes. A research engineer I know rewatched the setup twice and said type systems stopped feeling like an implementation detail. Free on YouTube from the Institute for Advanced Study, chalk only. Equality looked settled for 2,000 years. It was a placeholder.

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Eron

67,782 просмотров • 9 дней назад

This soundbite from the CEO of the SFA shows exactly why the officiating standards are where they are, and why they will never improve. It also shows why Collum will remain in situ after a mortifying season of gaffs and the reputation of Scottish football is in the toilet as a result. Not a single Scottish official at the World Cup. It's defensive and laughably blames the notion that "we live in a country that forensically examines every decision" we have to presume there's zero intention to look at things with a critical eye and even admit that we need vast change and improvement. I've said it before and this is the proof - the culture of low standards and low accountability starts at the top of the SFA and SPFL and is embedded right down. Nothing is going to change while these people are in charge. A footballing backwater.

This soundbite from the CEO of the SFA shows exactly why the officiating standards are where they are, and why they will never improve. It also shows why Collum will remain in situ after a mortifying season of gaffs and the reputation of Scottish football is in the toilet as a result. Not a single Scottish official at the World Cup. It's defensive and laughably blames the notion that "we live in a country that forensically examines every decision" we have to presume there's zero intention to look at things with a critical eye and even admit that we need vast change and improvement. I've said it before and this is the proof - the culture of low standards and low accountability starts at the top of the SFA and SPFL and is embedded right down. Nothing is going to change while these people are in charge. A footballing backwater.

Andy

78,741 просмотров • 3 месяцев назад

Anthropic engineer: "If you want to be in the top 1% of AI engineers, build a harness that improves itself Harnesses and graphs" In just 41 minutes, she explains how Anthropic builds with AI internally and why harnesses and graphs sit at the center of it Prompts → Agents → Harness → Loops → Graphs A harness makes each unit of work correct without you A graph decides which units should exist and turns every accepted result into a rule for the next run That's what "self-improving" actually means And it's the part almost nobody is building Watch it today Then save the full harness engineering guide below before everyone starts catching up ↓

Anthropic engineer: "If you want to be in the top 1% of AI engineers, build a harness that improves itself Harnesses and graphs" In just 41 minutes, she explains how Anthropic builds with AI internally and why harnesses and graphs sit at the center of it Prompts → Agents → Harness → Loops → Graphs A harness makes each unit of work correct without you A graph decides which units should exist and turns every accepted result into a rule for the next run That's what "self-improving" actually means And it's the part almost nobody is building Watch it today Then save the full harness engineering guide below before everyone starts catching up ↓

Lunar

16,448 просмотров • 16 дней назад

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

124,305 просмотров • 1 месяц назад

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 просмотров • 1 месяц назад

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

60,347 просмотров • 1 месяц назад

A mathematician pulls a volunteer from the audience and tells them to corrupt his data on purpose. Then he finds every flipped bit. This is Marcus du Sautoy at the Royal Institution, 4 minutes on error-correcting codes. A grid of 0s and 1s on the screen and a live demonstration of how a machine catches a single flipped digit. The math is Hamming's. Every hard drive, satellite link, QR code, and model checkpoint transferred between GPUs depends on it working silently. Watch the moment the volunteer changes 1 bit and he locates it without being told where. No equations shown. The trick is the parity structure underneath. A backend engineer I know sent it to a junior who kept asking why checksums exist and closed the question in 4 minutes. Free on YouTube, Royal Institution stage, subtitles on. The bits flip constantly. You just never see it happen.

A mathematician pulls a volunteer from the audience and tells them to corrupt his data on purpose. Then he finds every flipped bit. This is Marcus du Sautoy at the Royal Institution, 4 minutes on error-correcting codes. A grid of 0s and 1s on the screen and a live demonstration of how a machine catches a single flipped digit. The math is Hamming's. Every hard drive, satellite link, QR code, and model checkpoint transferred between GPUs depends on it working silently. Watch the moment the volunteer changes 1 bit and he locates it without being told where. No equations shown. The trick is the parity structure underneath. A backend engineer I know sent it to a junior who kept asking why checksums exist and closed the question in 4 minutes. Free on YouTube, Royal Institution stage, subtitles on. The bits flip constantly. You just never see it happen.

obssnnn

44,484 просмотров • 1 месяц назад

Neural networks sometimes memorize training data perfectly for thousands of steps - then suddenly generalize. Physicists call this a phase transition. The same math that explains why water freezes explains why an LLM stops cheating and starts understanding. The key is weight decay acting like pressure on a thermodynamic system. Memorization requires large, fragile weights. A real rule requires smaller, more structured ones. As training continues, the balance shifts until the rule-based solution becomes cheaper to maintain - and the model snaps into it. 14 minutes. Bookmark & watch today. The physics behind the most mysterious thing AI does.

Neural networks sometimes memorize training data perfectly for thousands of steps - then suddenly generalize. Physicists call this a phase transition. The same math that explains why water freezes explains why an LLM stops cheating and starts understanding. The key is weight decay acting like pressure on a thermodynamic system. Memorization requires large, fragile weights. A real rule requires smaller, more structured ones. As training continues, the balance shifts until the rule-based solution becomes cheaper to maintain - and the model snaps into it. 14 minutes. Bookmark & watch today. The physics behind the most mysterious thing AI does.

Eron

26,222 просмотров • 24 дней назад

Most People Would Agree With Elon Musk That Watching The United States Is Like Watching Rome Collapse (But With Memes) Seen on Joe Rogan: We are witnessing the last stage behavior of all great civilizations before their collapse. Be ready. “What do you mean this happens at the end of every civilization's reign? And I think this happened with the Greeks and it happened with the Romans. It's one of the things that Douglas Murray, when I had him on my podcast, was discussing, he said one of the things that happens at the end of civilizations is they become obsessed with gender. It's a thing that happens where men start becoming women, women start becoming men, and it becomes like a big focus, like cross dressing and all this stuff becomes a big point of focus. And I was like, well, why is that? He goes, It seems to be that they're dissolving all boundaries and all norms and all societal structure. And that's a part of it, is like gender roles.”

Most People Would Agree With Elon Musk That Watching The United States Is Like Watching Rome Collapse (But With Memes) Seen on Joe Rogan: We are witnessing the last stage behavior of all great civilizations before their collapse. Be ready. “What do you mean this happens at the end of every civilization's reign? And I think this happened with the Greeks and it happened with the Romans. It's one of the things that Douglas Murray, when I had him on my podcast, was discussing, he said one of the things that happens at the end of civilizations is they become obsessed with gender. It's a thing that happens where men start becoming women, women start becoming men, and it becomes like a big focus, like cross dressing and all this stuff becomes a big point of focus. And I was like, well, why is that? He goes, It seems to be that they're dissolving all boundaries and all norms and all societal structure. And that's a part of it, is like gender roles.”

Wall Street Apes

1,698,616 просмотров • 2 лет назад