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This is a great lecture at MIT by David Shirokoff on Markov Chains. He covers the fundamentals of Markov Chains using a simple particle movement example. He starts by explaining how a particle moves between two positions, A & B, with different probabilities. From there, the talk converts the... show more

𝗿𝗮𝗺𝗮𝗸𝗿𝘂𝘀𝗵𝗻𝗮— 𝗲/𝗮𝗰𝗰

12,867 subscribers

142,027 просмотров • 4 месяцев назад •via X (Twitter)

Образование Наука и технологии

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Комментарии: 8

Фото профиля Apollo

Apollo4 месяцев назад

Calling the probability matrix also A when there is a particle called A is bad notation.

Фото профиля Anton Korzhenkov

Anton Korzhenkov4 месяцев назад

ShirokOV gives a lecture about MarkOV, and KorzhenkOV likes it. This is the real Russia, I hope to see one day. Not the terrorist state, that attacked Kiev tonight.

Фото профиля 𝗿𝗮𝗺𝗮𝗸𝗿𝘂𝘀𝗵𝗻𝗮— 𝗲/𝗮𝗰𝗰

𝗿𝗮𝗺𝗮𝗸𝗿𝘂𝘀𝗵𝗻𝗮— 𝗲/𝗮𝗰𝗰4 месяцев назад

Russians mathematicians had contributed a lot to this world.

Фото профиля Two Cats

Two Cats4 месяцев назад

yeah on a chalkboard - bang up job

Фото профиля 𝗿𝗮𝗺𝗮𝗸𝗿𝘂𝘀𝗵𝗻𝗮— 𝗲/𝗮𝗰𝗰

𝗿𝗮𝗺𝗮𝗸𝗿𝘂𝘀𝗵𝗻𝗮— 𝗲/𝗮𝗰𝗰4 месяцев назад

That's just good old days now. Best way a teacher can teach. 😍

Фото профиля jhonk

jhonk4 месяцев назад

Very Nice! thnks

Фото профиля Stevarino1020

Stevarino10204 месяцев назад

All of this is a product of the Heisenberg described measurement problem. Inordinate dabbling in probability rather than simple exact measurement.

Фото профиля Axon

Axon4 месяцев назад

Does Shirokoff show why the steady state is unique, or just that it converges from the dominant eigenvector?

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Another Chinese student made $400K for last 30 days He used Markov Chains and Claude Even without little knowledge you can make money on it How it works : > Markov Property : future depends only on the current state > State Space : the set of all possible states in the system > Transition Probability : the chance of moving from one state to another > Transition Matrix : a table of all transition probabilities between states > Steady State : the long-run equilibrium distribution as steps → infinity Just using these 5 rules you will be able to trade with 55%-65% wr Chinese trader has 71% wr and $418k PNL Account : Top formulas for trading so : 1) The Markov Property RSI = 100 - (100 / (1 + RS)) RS = Avg Gain / Avg Loss (N=14) RSI > 70 → Sell | RSI < 30 → Buy 2) State Distribution SMA = (P1+P2+...+PN) / N EMA = P×k + EMA_prev×(1-k), k = 2/(N+1) Fast MA crosses UP → Buy (Golden Cross) Fast MA crosses DOWN → Sell (Death Cross) 3) Expected Value EV = (p × W) − (q × L) p = win prob q = 1 − p W = profit if win L = loss if lose My result is 66,3% and $780 profit for last week I keep on learning trading every day and going to make my first $10k Trade with me :

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11,242 просмотров • 5 месяцев назад

$In 1906, the world-renowned Russian mathematician Andrey A. Markov asked a heretic question of that time: if randomness has memory, do averages still behave or does probability theory collapse? His answer was a new kind of dependence where the next step only remembers the present, P(X_{n+1}=j | X_n=i, X_{n-1}, …) = P_ij, and yet the law of large numbers survives. The path stays noisy, while long-run state frequencies converge to a fixed profile pi = pi P. Markov’s idea is now the workhorse behind estimates by random-walks. MCMC (Markov Chain Monte Carlo) builds a chain whose stationary distribution is the target (posteriors, partition functions, constrained geometries), then uses time-averages as estimates. The same Markov structure shows up in hidden Markov models for time series, in biophysics as channels switching between states, and in control/RL as Markov decision processes. #ProbabilityTheory #Markov #MCMC$

In 1906, the world-renowned Russian mathematician Andrey A. Markov asked a heretic question of that time: if randomness has memory, do averages still behave or does probability theory collapse? His answer was a new kind of dependence where the next step only remembers the present, P(X_{n+1}=j | X_n=i, X_{n-1}, …) = P_ij, and yet the law of large numbers survives. The path stays noisy, while long-run state frequencies converge to a fixed profile pi = pi P. Markov’s idea is now the workhorse behind estimates by random-walks. MCMC (Markov Chain Monte Carlo) builds a chain whose stationary distribution is the target (posteriors, partition functions, constrained geometries), then uses time-averages as estimates. The same Markov structure shows up in hidden Markov models for time series, in biophysics as channels switching between states, and in control/RL as Markov decision processes. #ProbabilityTheory #Markov #MCMC

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Markov Proved that Randomness With Memory Can Still Settle Into Law Markov’s 1906 result was a major turning point in Probability Theory. He showed that even when successive events are dependent, long-run averages can still stabilize. That broke the old habit of tying the Law of Large Numbers only to independence, and opened the door to a much broader Mathematics of Stochastic Systems. From there, the whole subject of Markov Chains grew into one of the central languages of modern probability. It feeds directly into Ergodic Theory, Statistical Physics, Queueing Systems, Population Models, Monte Carlo methods, Hidden Markov models, Speech Recognition, Finance, Genetics, and even web search. The idea is simple and deep at once. It shows that local randomness can still produce global regularity. The paper: А. А. Марков (1906). Распространение закона больших чисел на величины, зависящие друг от друга (Extension of The Law of Large Numbers to Dependent Quantities).

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Zyphor

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This Russian mathematician from 1906 just dropped the cheat code that’s printing millions on Polymarket right now. Markov Chains aren’t some dusty theory - they’re exactly how the market actually works. Current price = the only state that matters. Past is already priced in. No ego, no anchoring, no career risk. Just pure, lightning-fast updates as new info hits. The smartest traders stopped "guessing" years ago. They read states and size up on the highest-probability outcomes. Exhibit A: [bobe2] -> one of the OGs on Polymarket (joined April 2023). > 1,317 predictions. > $7,500,000 positions. > $1,619,000+ all-time profit. He doesn’t chase moonshots. He stacks $7-8M on 97-99¢ obvious plays (FOMC, Biden term, Trump picks, Zelensky, BTC reserve). Classic Markov execution: the current state already contains everything. He just capitalizes. His profile -> This is how the real quants own Polymarket. Want the exact same edge without staring at charts 24/7? Copy him + the rest of the top Markov whales automatically with Just add his wallet [0xed107a85a4585a381e48c7f7ca4144909e7dd2e5] here and track/copy him. Stop guessing the future. Start copying the states that are already winning. Save him and this alpha at all.

slash1s

54,119 просмотров • 5 месяцев назад

a quant at citadel told me something that broke my entire trading framework "we don't predict markets. we model the state machine" he explained markov cains in 90 seconds the market is never random - it always exists in one of three states trending up, trending down, ranging - each has a fixed probability of shifting to another build the transition matrix from real price data: trending up -> 68% stays trending, 21% flips to range, 11% reverses ranging -> 54% stays range, 28% breaks up, 18% breaks down trending down -> 61% stays falling, 24% flips to range, 15% reverses now you're not guessing, you're playing probability identify current state, enter with the 68% edge, size with kelly criterion based on that probability the formula is public - andrei markov published it in 1906 hedge funds use it every day, the math costs nothing what costs you is asking the wrong question "where is price going" is random "what state am I in right now" has an answer transition matrix built from 10 years of data is your edge Bookmark and use in your own strategy 37 years without a losing year, 66% annual returns paper is free, data is free, implementation is 200 lines of python they kept you staring at candles while they ran probability tables

a quant at citadel told me something that broke my entire trading framework "we don't predict markets. we model the state machine" he explained markov cains in 90 seconds the market is never random - it always exists in one of three states trending up, trending down, ranging - each has a fixed probability of shifting to another build the transition matrix from real price data: trending up -> 68% stays trending, 21% flips to range, 11% reverses ranging -> 54% stays range, 28% breaks up, 18% breaks down trending down -> 61% stays falling, 24% flips to range, 15% reverses now you're not guessing, you're playing probability identify current state, enter with the 68% edge, size with kelly criterion based on that probability the formula is public - andrei markov published it in 1906 hedge funds use it every day, the math costs nothing what costs you is asking the wrong question "where is price going" is random "what state am I in right now" has an answer transition matrix built from 10 years of data is your edge Bookmark and use in your own strategy 37 years without a losing year, 66% annual returns paper is free, data is free, implementation is 200 lines of python they kept you staring at candles while they ran probability tables

Livsun

19,808 просмотров • 1 месяц назад

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Mathelirium

96,504 просмотров • 7 месяцев назад

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Сarm1ne

422,200 просмотров • 5 месяцев назад

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics

Mathelirium

133,454 просмотров • 7 месяцев назад

renaissance technologies hasn't had a losing year since 1988 not because they're smarter or have better data they're asking a completely different question than everyone else while retail asks "will it go up?" quant desks ask something else entirely which state is the market in right now, and with what probability does it transition to the next tool for that: markov chains markets cycle through states - trending, ranging, volatile, reversing each transition has a historical probability you can calculate trending -> stays trending: 48% trending -> ranging: 34% trending -> reversing: 18% you don't pick direction. you calculate expected value across every possible transition enter only when the math is positive that's it. that's the entire edge the framework is 119 years old - andrei markov published it in 1906 it's in every statistics textbook on earth Bookmark it or lose last chance to be quant nobody in your trading group ever mentioned it

renaissance technologies hasn't had a losing year since 1988 not because they're smarter or have better data they're asking a completely different question than everyone else while retail asks "will it go up?" quant desks ask something else entirely which state is the market in right now, and with what probability does it transition to the next tool for that: markov chains markets cycle through states - trending, ranging, volatile, reversing each transition has a historical probability you can calculate trending -> stays trending: 48% trending -> ranging: 34% trending -> reversing: 18% you don't pick direction. you calculate expected value across every possible transition enter only when the math is positive that's it. that's the entire edge the framework is 119 years old - andrei markov published it in 1906 it's in every statistics textbook on earth Bookmark it or lose last chance to be quant nobody in your trading group ever mentioned it

Livsun

51,500 просмотров • 4 месяцев назад

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics

Mathelirium

66,892 просмотров • 8 месяцев назад

FINALLY finishing up a MASSIVE PR from hell for the Sega Dreamcast port of Grand Theft Auto 3! This is an actual hardware capture now of the DC version under a high load, which would've previously been a slideshow, between the dynamic lighting from the sirens, the amount of rigid bodies in the physics simulation from the cars, and the high-speed chase placing high-demands on asset streaming... I went through all of the low-level common math infrastructure in both the engine and at the RenderWare driver layer and made numerous optimizations, before slowly working my way up to optimizing individual algorithms at the application layer using the new math routines. Firstly, the common low-level floating-point math routines for everything from trig and inverse square root operations to floor(), ceiling(), and clamp(), were replaced with what was meticulously found (in Compiler Explorer) to be the optimal patterns for GCC 14.2.0, targeting our SH architecture (sometimes favoring C builtins, sometimes inline SH4 ASM). Next, in the layer above, with inline SH4 assembly, the common matrix math and linear algebra routines were accelerated using the Dreamcast's vector instructions. Some cleverness went down here, such as cramming matrix metadata into unused insignificant bits of an element, combining loading two matrices with multiplying them, fast transposes, fast quaternion multiplication using 4 dot products, etc. Once the foundation was laid, some of the Renderware code such as the calculations for the lighting, updating bounding volumes, and deriving UV coordinates for specular environment maps on the cars was made faster automatically. The main gainz were actually made rewriting a decent amount of the collision intersection and contact resolution code, though, from using C++-style overloaded operators for multiplying single 4D vectors by a 4x4 matrix to doing batches of 4D vectors by the same matrix. This SUBSTANTIALLY reduced the number of times the backing 4x4 matrix bank had to be reloaded and allowed me to keep it resident within registers while it was being used by the intersection algorithms!

FINALLY finishing up a MASSIVE PR from hell for the Sega Dreamcast port of Grand Theft Auto 3! This is an actual hardware capture now of the DC version under a high load, which would've previously been a slideshow, between the dynamic lighting from the sirens, the amount of rigid bodies in the physics simulation from the cars, and the high-speed chase placing high-demands on asset streaming... I went through all of the low-level common math infrastructure in both the engine and at the RenderWare driver layer and made numerous optimizations, before slowly working my way up to optimizing individual algorithms at the application layer using the new math routines. Firstly, the common low-level floating-point math routines for everything from trig and inverse square root operations to floor(), ceiling(), and clamp(), were replaced with what was meticulously found (in Compiler Explorer) to be the optimal patterns for GCC 14.2.0, targeting our SH architecture (sometimes favoring C builtins, sometimes inline SH4 ASM). Next, in the layer above, with inline SH4 assembly, the common matrix math and linear algebra routines were accelerated using the Dreamcast's vector instructions. Some cleverness went down here, such as cramming matrix metadata into unused insignificant bits of an element, combining loading two matrices with multiplying them, fast transposes, fast quaternion multiplication using 4 dot products, etc. Once the foundation was laid, some of the Renderware code such as the calculations for the lighting, updating bounding volumes, and deriving UV coordinates for specular environment maps on the cars was made faster automatically. The main gainz were actually made rewriting a decent amount of the collision intersection and contact resolution code, though, from using C++-style overloaded operators for multiplying single 4D vectors by a 4x4 matrix to doing batches of 4D vectors by the same matrix. This SUBSTANTIALLY reduced the number of times the backing 4x4 matrix bank had to be reloaded and allowed me to keep it resident within registers while it was being used by the intersection algorithms!

Falco Girgis

114,621 просмотров • 1 год назад

Day 9/90 of Inference Engineering I built a visualization tool for CUDA kernels and C code to view the buffers for each step through. Check out Yesterday, I was terramogged by a simple implementation of a basic matmul between three matrices in 1D vector form. There were essentially three 1D vectors, each vector representing a corresponding matrix (A, B, C). In the implementation, each matrix is just just a pointer to the first element of a 1D array. You are to compute the output matrix through A^T @ B and store it in C. This morning I built a visualizer to help me see and understand why the triple for-loops work the way they do. In the coming days, I’m confident I can make sense as well as backpropagation! As for it does not include a built-in compiler. It's just an interpreter; the visualization shows exactly what the code does, bugs included. The main thing I wanted to visualize is matmul with a triple for-loop between a sequential program to that of a parallel one. Expect bugs and other inconsistencies within I’ll polish this mini-project in the coming days. As for the rest of today, I’ll be trying to understand backpropagation in C and CUDA .

Day 9/90 of Inference Engineering I built a visualization tool for CUDA kernels and C code to view the buffers for each step through. Check out Yesterday, I was terramogged by a simple implementation of a basic matmul between three matrices in 1D vector form. There were essentially three 1D vectors, each vector representing a corresponding matrix (A, B, C). In the implementation, each matrix is just just a pointer to the first element of a 1D array. You are to compute the output matrix through A^T @ B and store it in C. This morning I built a visualizer to help me see and understand why the triple for-loops work the way they do. In the coming days, I’m confident I can make sense as well as backpropagation! As for it does not include a built-in compiler. It's just an interpreter; the visualization shows exactly what the code does, bugs included. The main thing I wanted to visualize is matmul with a triple for-loop between a sequential program to that of a parallel one. Expect bugs and other inconsistencies within I’ll polish this mini-project in the coming days. As for the rest of today, I’ll be trying to understand backpropagation in C and CUDA .

max fu

12,746 просмотров • 2 месяцев назад

a russian mathematician solved trading in 1906 wall street found the paper 70 years later, built billion-dollar funds around it, never mentioned it to retail andrei markov proved markets aren't random - they're state machines trending, ranging, reversing - each state has a fixed historical probability of shifting to the next build a transition matrix from real data: trending -> stays trending: 68% trending -> flips to range: 21% trending -> reverses: 11% now you're not predicting direction. you're entering on 68% historical completion identify your state, size with kelly, take the trade when math says yes that's the edge. the whole edge renaissance has run this since 1988. 37 years of 66% annual returns paper is free, data is free, implementation is 200 lines of python Bookmark and use in your own strategy they kept you staring at candles while they ran probability tables

a russian mathematician solved trading in 1906 wall street found the paper 70 years later, built billion-dollar funds around it, never mentioned it to retail andrei markov proved markets aren't random - they're state machines trending, ranging, reversing - each state has a fixed historical probability of shifting to the next build a transition matrix from real data: trending -> stays trending: 68% trending -> flips to range: 21% trending -> reverses: 11% now you're not predicting direction. you're entering on 68% historical completion identify your state, size with kelly, take the trade when math says yes that's the edge. the whole edge renaissance has run this since 1988. 37 years of 66% annual returns paper is free, data is free, implementation is 200 lines of python Bookmark and use in your own strategy they kept you staring at candles while they ran probability tables

Livsun

376,877 просмотров • 3 месяцев назад

A 91-year-old professor is why Nvidia is worth $4 trillion. His name is Gilbert Strang. He teaches linear algebra at MIT. Every AI model on Earth runs on his course. The course has been free on YouTube since 2005. The videos have earned him nothing. MIT 18.06 opens with "The Geometry of Linear Equations." No advanced math. Strang takes a system of two equations, draws it two ways, and shows the class that a matrix is a picture, not an abstraction. The row picture is two lines that cross. The column picture is two arrows that sum to a target. Every neural network on Earth operates on the column picture. Strang first taught linear algebra at MIT in 1962. He wrote the textbook in 1976. It is on every serious engineer's shelf. Every quant fund, every ML lab, every rendering engine at Pixar is running his math. His central insight is that most people are taught matrices as bookkeeping. That is the first thing to unlearn. A matrix is a linear transformation. A linear transformation is a way of moving space. Once you see the space move, the math stops being algebra and becomes geometry. The Kalman filter is a linear system. PCA is a linear system. Every gradient step in a neural net is a matrix-vector product. GPT is a stack of matrix-vector products, each one a scene from MIT 18.06 running on a Blackwell GPU. He retired in 2023 after 61 years at MIT. The course is still up. Watched tens of millions of times. The chip is $40,000. Strang never asked for a royalty.

A 91-year-old professor is why Nvidia is worth $4 trillion. His name is Gilbert Strang. He teaches linear algebra at MIT. Every AI model on Earth runs on his course. The course has been free on YouTube since 2005. The videos have earned him nothing. MIT 18.06 opens with "The Geometry of Linear Equations." No advanced math. Strang takes a system of two equations, draws it two ways, and shows the class that a matrix is a picture, not an abstraction. The row picture is two lines that cross. The column picture is two arrows that sum to a target. Every neural network on Earth operates on the column picture. Strang first taught linear algebra at MIT in 1962. He wrote the textbook in 1976. It is on every serious engineer's shelf. Every quant fund, every ML lab, every rendering engine at Pixar is running his math. His central insight is that most people are taught matrices as bookkeeping. That is the first thing to unlearn. A matrix is a linear transformation. A linear transformation is a way of moving space. Once you see the space move, the math stops being algebra and becomes geometry. The Kalman filter is a linear system. PCA is a linear system. Every gradient step in a neural net is a matrix-vector product. GPT is a stack of matrix-vector products, each one a scene from MIT 18.06 running on a Blackwell GPU. He retired in 2023 after 61 years at MIT. The course is still up. Watched tens of millions of times. The chip is $40,000. Strang never asked for a royalty.

Ochob

130,880 просмотров • 1 месяц назад

Graph Convolutional Network by hand ✍️ ~ 12 steps walkthrough below Graph Convolutional Networks (GCNs), introduced by Thomas Kipf and Max Welling in 2017, are the tool for data shaped like a graph: social networks, recommendations, biological networks, drug discovery, molecular chemistry. I drew and calculated a simple GCN entirely by hand. Goal: run a two-layer GCN, then a small classifier, on a five-node graph, filling in every cell yourself. 1. Given A graph of five nodes, A to E, with edges between some of them. 2. Adjacency matrix (neighbors) Put a 1 wherever two nodes share an edge, in both directions. 3. Adjacency matrix (self) Add 1s down the diagonal, one self-loop per node. That is just adding the identity matrix. 4. Messages Multiply each node's embedding by the weights and biases, then ReLU. Negatives become 0. 5. Pooling Multiply the messages by the adjacency matrix. Each node gathers the messages of its neighbours and itself. 6. Visualize Node A pools [3,0,1] + [1,0,0] = [4,0,1]. 7. Second GCN layer Messages again: weights, biases, ReLU. 8. Pooling again Pool over each node and its neighbours, once more. 9. Visualize Node C pools [1,2,4] + [1,3,5] + [0,0,1] = [2,5,10]. 10. Fully connected layer Weights, biases, ReLU. This time there are no neighbours to pool, just the node itself. 11. Linear layer One more: weights and biases. 12. Sigmoid Squash each score to a probability (≥ 3 → 1, 0 → 0.5, ≤ -3 → 0). That is the classification for each node. You have just classified every node in the graph by hand. ✍️ The outputs: A: 0 (very unlikely) B: 1 (very likely) C: 1 (very likely) D: 1 (very likely) E: 0.5 (neutral) The takeaway: a GCN layer is two parts. The top part pools each node with its neighbours through the adjacency matrix. The bottom part is an MLP that transforms each node on its own. A transformer layer has the same two parts, with an attention matrix where the adjacency matrix was. Both matrices do one job, mixing across positions: attention over tokens, adjacency over nodes. In my class I call the GCN the transformer's little cousin: a bit more stubborn, because its attention is fixed by the graph rather than computed from Q, K, and V. Draw the two side by side and the resemblance is hard to miss. 💾 Save this post! #AIbyHand #GraphNeuralNetworks #DeepLearning

Graph Convolutional Network by hand ✍️ ~ 12 steps walkthrough below Graph Convolutional Networks (GCNs), introduced by Thomas Kipf and Max Welling in 2017, are the tool for data shaped like a graph: social networks, recommendations, biological networks, drug discovery, molecular chemistry. I drew and calculated a simple GCN entirely by hand. Goal: run a two-layer GCN, then a small classifier, on a five-node graph, filling in every cell yourself. 1. Given A graph of five nodes, A to E, with edges between some of them. 2. Adjacency matrix (neighbors) Put a 1 wherever two nodes share an edge, in both directions. 3. Adjacency matrix (self) Add 1s down the diagonal, one self-loop per node. That is just adding the identity matrix. 4. Messages Multiply each node's embedding by the weights and biases, then ReLU. Negatives become 0. 5. Pooling Multiply the messages by the adjacency matrix. Each node gathers the messages of its neighbours and itself. 6. Visualize Node A pools [3,0,1] + [1,0,0] = [4,0,1]. 7. Second GCN layer Messages again: weights, biases, ReLU. 8. Pooling again Pool over each node and its neighbours, once more. 9. Visualize Node C pools [1,2,4] + [1,3,5] + [0,0,1] = [2,5,10]. 10. Fully connected layer Weights, biases, ReLU. This time there are no neighbours to pool, just the node itself. 11. Linear layer One more: weights and biases. 12. Sigmoid Squash each score to a probability (≥ 3 → 1, 0 → 0.5, ≤ -3 → 0). That is the classification for each node. You have just classified every node in the graph by hand. ✍️ The outputs: A: 0 (very unlikely) B: 1 (very likely) C: 1 (very likely) D: 1 (very likely) E: 0.5 (neutral) The takeaway: a GCN layer is two parts. The top part pools each node with its neighbours through the adjacency matrix. The bottom part is an MLP that transforms each node on its own. A transformer layer has the same two parts, with an attention matrix where the adjacency matrix was. Both matrices do one job, mixing across positions: attention over tokens, adjacency over nodes. In my class I call the GCN the transformer's little cousin: a bit more stubborn, because its attention is fixed by the graph rather than computed from Q, K, and V. Draw the two side by side and the resemblance is hard to miss. 💾 Save this post! #AIbyHand #GraphNeuralNetworks #DeepLearning

Tom Yeh

16,800 просмотров • 2 месяцев назад