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SVM by hand ✍️ ~ 19 steps walkthrough below (Linear vs RBF) Support Vector Machines reigned supreme in machine learning before the deep learning revolution. An SVM predicts with dot products, the same matrix multiplication every model uses. What it does not do is train by backpropagation: it is... fitted by convex optimization, so there is no matrix-multiplication backward pass for a GPU to accelerate. I drew and calculated two SVMs by hand: a linear one (top) and an RBF one (bottom), classifying the same two test vectors. Goal: turn six training vectors and their learned coefficients into a prediction, and see what changing the kernel actually changes. = 1. Given = Six training vectors, their labels, and the coefficients and bias already learned. A coefficient of zero means that vector is not a support vector: too far from the boundary to matter. = 2. Linear kernel, test vector 1 = Let us take the dot product of the test vector with every training vector. The dot product stands in for cosine similarity, and the column of results is the first column of the kernel matrix K. = 3. Linear kernel, test vector 2 = We do the same for the second, and K is complete. = 4. Signed weights = Let us multiply each coefficient by its label. The second training vector drops out here, because its coefficient is 0. = 5. Weighted combination = We multiply the signed weights through K and add the bias b. The result is a signed distance to the decision boundary: 17 and 5. = 6. Classify = Let us take the sign. Both are positive. = 7 to 11. RBF kernel, test vector 1 = Now the same picture with a different kernel, in five moves: square the differences, sum them, take the square root for the L2 distance, multiply by minus gamma, and raise e to that power. The negation is what turns a distance into a similarity, and gamma controls how far a single training vector's influence reaches. = 12 to 16. RBF kernel, test vector 2 = We repeat all five. The numbers change, the moves do not. = 17 to 19. Decision boundary, again = Signed weights, weighted combination, sign. Identical arithmetic to steps 4 through 6, on a K that was built a completely different way. The outputs: Linear K, first column = [13, 25, 12, 15, 19, 27] Linear decision values = 17 and 5, both positive RBF decision values = -2 and 1, so negative and positive The takeaway: the kernel is the only thing that changed, and it changed the answer. The linear SVM calls both test vectors positive; the RBF one splits them. Everything after the kernel matrix, the signed weights and the weighted combination and the sign, is the same page of arithmetic twice. 💾 Save this post!show more

Tom Yeh

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Self Attention by hand ✍️ ~ 9 steps walkthrough below Self-attention is what enables LLMs to understand context. How does it work? So I drew and calculated one entirely by hand. Goal: turn four 6D features into four 3D attention weighted features, filling in every cell yourself. = 1. Given = Four feature vectors, six dimensions each, one per position. = 2. Query, key, value = Let us multiply the features by WQ, WK and WV. Queries, keys and values all come out of the same four features, and that is what the word "self" is doing in self-attention. = 3. Prepare for MatMul = We copy the queries across the top and the transposed keys down the side. Lining the two up is half the work. = 4. MatMul = Let us multiply K transpose by Q. Every cell is the dot product of one key with one query, which we use as a matching score. That works because the dot product is the numerator of cosine similarity: it is how alike two vectors are, before anyone divides by their lengths. = 5. Scale = We divide by the square root of dk, the dimension of a key vector, here 3. Without it the scores grow with the dimension and a 64-wide head would swamp the softmax. To keep the page doable in pen, the drawing approximates dividing by root 3 with halving. = 6. e to the power = Let us raise e to the power of each score. This is the first half of softmax, and the drawing uses 3 in place of e, which is close enough to do in your head. = 7. Sum = We add up each column: 16, 6, 7 and 12. = 8. Normalize = Let us divide every cell by its column sum. That gives the attention weight matrix in yellow, and each of its four columns is now a probability distribution over the four positions. The decimals are nudged as they are rounded, so every column still sums to exactly 1. = 9. MatMul = We multiply the value vectors by those weights. Each output is a blend of all four values, mixed in the proportion the attention matrix just decided, and it goes to the position-wise feed forward network in the next layer: the FFN box at the bottom of the page. The outputs: Attention weights (A), by column = [.2, .6, 0, .2], [.2, .4, .2, .2], [.4, .2, 0, .4], [.1, .7, .1, .1] Attention weighted features (Z) = [8, 2, 6], [8, 4, 4], [16, 4, 2], [4, 2, 7] The takeaway: attention is a weighted average, and everything before step 9 exists to decide the weights. Compare every position with every other, turn the scores into one distribution per position, then blend. 💾 Save this post!

Self Attention by hand ✍️ ~ 9 steps walkthrough below Self-attention is what enables LLMs to understand context. How does it work? So I drew and calculated one entirely by hand. Goal: turn four 6D features into four 3D attention weighted features, filling in every cell yourself. = 1. Given = Four feature vectors, six dimensions each, one per position. = 2. Query, key, value = Let us multiply the features by WQ, WK and WV. Queries, keys and values all come out of the same four features, and that is what the word "self" is doing in self-attention. = 3. Prepare for MatMul = We copy the queries across the top and the transposed keys down the side. Lining the two up is half the work. = 4. MatMul = Let us multiply K transpose by Q. Every cell is the dot product of one key with one query, which we use as a matching score. That works because the dot product is the numerator of cosine similarity: it is how alike two vectors are, before anyone divides by their lengths. = 5. Scale = We divide by the square root of dk, the dimension of a key vector, here 3. Without it the scores grow with the dimension and a 64-wide head would swamp the softmax. To keep the page doable in pen, the drawing approximates dividing by root 3 with halving. = 6. e to the power = Let us raise e to the power of each score. This is the first half of softmax, and the drawing uses 3 in place of e, which is close enough to do in your head. = 7. Sum = We add up each column: 16, 6, 7 and 12. = 8. Normalize = Let us divide every cell by its column sum. That gives the attention weight matrix in yellow, and each of its four columns is now a probability distribution over the four positions. The decimals are nudged as they are rounded, so every column still sums to exactly 1. = 9. MatMul = We multiply the value vectors by those weights. Each output is a blend of all four values, mixed in the proportion the attention matrix just decided, and it goes to the position-wise feed forward network in the next layer: the FFN box at the bottom of the page. The outputs: Attention weights (A), by column = [.2, .6, 0, .2], [.2, .4, .2, .2], [.4, .2, 0, .4], [.1, .7, .1, .1] Attention weighted features (Z) = [8, 2, 6], [8, 4, 4], [16, 4, 2], [4, 2, 7] The takeaway: attention is a weighted average, and everything before step 9 exists to decide the weights. Compare every position with every other, turn the scores into one distribution per position, then blend. 💾 Save this post!

Tom Yeh

25,913 просмотров • 4 дней назад

[CLIP] by Hand ✍️ The CLIP (Contrastive Language–Image Pre-training) model, a groundbreaking work by OpenAI, redefines the intersection of computer vision and natural language processing. It is the basis of all the multi-modal foundation models we see today. How does CLIP work? Goal: 🟨 Learn a shared embedding space for text and image [1] Given ↳ A mini batch of 3 text-image pairs ↳ OpenAI used 400 million text-image pairs to train its original CLIP model. Process 1st pair: "big table" [2] 🟪 Text → 2 Vectors (3D) ↳ Look up word embedding vectors using word2vec. [3] 🟩 Image → 2 Vectors (4D) ↳ Divide the image into two patches. ↳ Flatten each patch [4] Process other pairs ↳ Repeat [2]-[3] [5] 🟪 Text Encoder & 🟩 Image Encoder ↳ Encode input vectors into feature vectors ↳ Here, both encoders are simple one layer perceptron (linear + ReLU) ↳ In practice, the encoders are usually transformer models. [6] 🟪 🟩 Mean Pooling: 2 → 1 vector ↳ Average 2 feature vectors into a single vector by averaging across the columns ↳ The goal is to have one vector to represent each image or text [7] 🟪 🟩 -> 🟨 Projection ↳ Note that the text and image feature vectors from the encoders have different dimensions (3D vs. 4D). ↳ Use a linear layer to project image and text vectors to a 2D shared embedding space. 🏋️ Contrastive Pre-training 🏋️ [8] Prepare for MatMul ↳ Copy text vectors (T1,T2,T3) ↳ Copy the transpose of image vectors (I1,I2,I3) ↳ They are all in the 2D shared embedding space. [9] 🟦 MatMul ↳ Multiply T and I matrices. ↳ This is equivalent to taking dot product between every pair of image and text vectors. ↳ The purpose is to use dot product to estimate the similarity between a pair of image-text. [10] 🟦 Softmax: e^x ↳ Raise e to the power of the number in each cell ↳ To simplify hand calculation, we approximate e^□ with 3^□. [11] 🟦 Softmax: ∑ ↳ Sum each row for 🟩 image→🟪 text ↳ Sum each column for 🟪 text→ 🟩 image [12] 🟦 Softmax: 1 / sum ↳ Divide each element by the column sum to obtain a similarity matrix for 🟪 text→🟩 image ↳ Divide each element by the row sum to obtain a similarity matrix for 🟩 image→🟪 text [13] 🟥 Loss Gradients ↳ The "Targets" for the similarity matrices are Identity Matrices. ↳ Why? If I and T come from the same pair (i=j), we want the highest value, which is 1, and 0 otherwise. ↳ Apply the simple equation of [Similarity - Target] to compute gradients of for both directions. ↳ Why so simple? Because when Softmax and Cross-Entropy Loss are used together, the math magically works out that way. ↳ These gradients kick off the backpropagation process to update weights and biases of the encoders and projection layers (red borders).

[CLIP] by Hand ✍️ The CLIP (Contrastive Language–Image Pre-training) model, a groundbreaking work by OpenAI, redefines the intersection of computer vision and natural language processing. It is the basis of all the multi-modal foundation models we see today. How does CLIP work? Goal: 🟨 Learn a shared embedding space for text and image [1] Given ↳ A mini batch of 3 text-image pairs ↳ OpenAI used 400 million text-image pairs to train its original CLIP model. Process 1st pair: "big table" [2] 🟪 Text → 2 Vectors (3D) ↳ Look up word embedding vectors using word2vec. [3] 🟩 Image → 2 Vectors (4D) ↳ Divide the image into two patches. ↳ Flatten each patch [4] Process other pairs ↳ Repeat [2]-[3] [5] 🟪 Text Encoder & 🟩 Image Encoder ↳ Encode input vectors into feature vectors ↳ Here, both encoders are simple one layer perceptron (linear + ReLU) ↳ In practice, the encoders are usually transformer models. [6] 🟪 🟩 Mean Pooling: 2 → 1 vector ↳ Average 2 feature vectors into a single vector by averaging across the columns ↳ The goal is to have one vector to represent each image or text [7] 🟪 🟩 -> 🟨 Projection ↳ Note that the text and image feature vectors from the encoders have different dimensions (3D vs. 4D). ↳ Use a linear layer to project image and text vectors to a 2D shared embedding space. 🏋️ Contrastive Pre-training 🏋️ [8] Prepare for MatMul ↳ Copy text vectors (T1,T2,T3) ↳ Copy the transpose of image vectors (I1,I2,I3) ↳ They are all in the 2D shared embedding space. [9] 🟦 MatMul ↳ Multiply T and I matrices. ↳ This is equivalent to taking dot product between every pair of image and text vectors. ↳ The purpose is to use dot product to estimate the similarity between a pair of image-text. [10] 🟦 Softmax: e^x ↳ Raise e to the power of the number in each cell ↳ To simplify hand calculation, we approximate e^□ with 3^□. [11] 🟦 Softmax: ∑ ↳ Sum each row for 🟩 image→🟪 text ↳ Sum each column for 🟪 text→ 🟩 image [12] 🟦 Softmax: 1 / sum ↳ Divide each element by the column sum to obtain a similarity matrix for 🟪 text→🟩 image ↳ Divide each element by the row sum to obtain a similarity matrix for 🟩 image→🟪 text [13] 🟥 Loss Gradients ↳ The "Targets" for the similarity matrices are Identity Matrices. ↳ Why? If I and T come from the same pair (i=j), we want the highest value, which is 1, and 0 otherwise. ↳ Apply the simple equation of [Similarity - Target] to compute gradients of for both directions. ↳ Why so simple? Because when Softmax and Cross-Entropy Loss are used together, the math magically works out that way. ↳ These gradients kick off the backpropagation process to update weights and biases of the encoders and projection layers (red borders).

Tom Yeh

67,834 просмотров • 2 лет назад

[RLHF] by Hand ✍️ Yesterday, Jan Leike (Jan Leike) announced he is joining #Anthropic to lead their "super-alignment" mission. He is the co-inventor of Reinforcement Learning with Human Feedback (#RLHF). How does RLHF work? [1] Given ↳ Reward Model (RM) ↳ Large Language Model (LLM) ↳ Two (Prompt, Next) Pairs 🟪 TRAIN RM Goal: Learn to give higher rewards to winners [2] Preferences ↳ A human reviews the two pairs and picks a "winner" ↳ (doc is, him) Embeddings ↳ This prompt has never received human feedback directly ↳ [S] is the special start symbol [11] Transformer ↳ Attention (yellow) ↳ Feed Forward (4x2 weight and bias matrix) ↳ Output: 3 "transformed" feature vector, one per position ↳ More details in my previous post 8. Transformer [] [12] Output Probabilities ↳ Apply a linear layer to map each transformed feature vector to a probability distribution over the vocabulary. [13] Sample ↳ Apply the greedy method, which is to pick the word with the highest score ↳ For output 1 and 2, the model accurately predicts the next word ↳ For 3rd output position, the model's predicts "him" [14] Reward Model ↳ The new pair (CEO is, him) is fed to the reward model ↳ The process is same as [3]-[6] ↳ Output: Reward = 3 [15] Loss Gradient ↳ We set the loss as the negative of the reward. ↳ The loss gradient is simply a constant -1. ↳ Run backpropagation and gradient descent to update LLM's weights and biases (red border)

[RLHF] by Hand ✍️ Yesterday, Jan Leike (Jan Leike) announced he is joining #Anthropic to lead their "super-alignment" mission. He is the co-inventor of Reinforcement Learning with Human Feedback (#RLHF). How does RLHF work? [1] Given ↳ Reward Model (RM) ↳ Large Language Model (LLM) ↳ Two (Prompt, Next) Pairs 🟪 TRAIN RM Goal: Learn to give higher rewards to winners [2] Preferences ↳ A human reviews the two pairs and picks a "winner" ↳ (doc is, him) Embeddings ↳ This prompt has never received human feedback directly ↳ [S] is the special start symbol [11] Transformer ↳ Attention (yellow) ↳ Feed Forward (4x2 weight and bias matrix) ↳ Output: 3 "transformed" feature vector, one per position ↳ More details in my previous post 8. Transformer [] [12] Output Probabilities ↳ Apply a linear layer to map each transformed feature vector to a probability distribution over the vocabulary. [13] Sample ↳ Apply the greedy method, which is to pick the word with the highest score ↳ For output 1 and 2, the model accurately predicts the next word ↳ For 3rd output position, the model's predicts "him" [14] Reward Model ↳ The new pair (CEO is, him) is fed to the reward model ↳ The process is same as [3]-[6] ↳ Output: Reward = 3 [15] Loss Gradient ↳ We set the loss as the negative of the reward. ↳ The loss gradient is simply a constant -1. ↳ Run backpropagation and gradient descent to update LLM's weights and biases (red border)

Tom Yeh

79,758 просмотров • 2 лет назад

Backpropagation by hand ✍️ ~ 11 steps walkthrough below Backpropagation is the algorithm that actually trains a neural network, and it is where most people stop following along. It is not calculus you cannot do. It is matrix multiplication, working backward, one layer at a time. So I drew and calculated one entirely by hand. Goal: push the loss gradient back through a 3-layer network and land on a new value for every weight and bias. = 1. Given = A 3-layer perceptron, an input X, predictions Ypred = [0.5, 0.5, 0], and the truth Ytarget = [0, 1, 0]. = 2. Backprop gradient cells = Let us draw empty cells for every gradient we are about to compute. The shape of the answer comes first. = 3. Layer 3 softmax = We get dL/dz3 straight from Ypred minus Ytarget = [0.5, -0.5, 0]. No chain rule needed, and that shortcut is the whole reason softmax and cross-entropy are paired. = 4. Layer 3 weights and biases = Let us multiply dL/dz3 by [a2 | 1]. One multiplication gives the gradient for W3 and b3 together. = 5. Layer 2 activations = We multiply dL/dz3 by W3 to get dL/da2. The gradient moves back across a layer the same way the signal moved forward. = 6. Layer 2 ReLU = Let us pass it through the gate: keep the gradient where the activation was positive, zero it everywhere else. = 7. Layer 2 weights and biases = We multiply dL/dz2 by [a1 | 1]. The same figure as step 4, one layer up. = 8. Layer 1 activations = Let us multiply dL/dz2 by W2. = 9. Layer 1 ReLU = We apply the same gate again, now on a1. = 10. Layer 1 weights and biases = Let us multiply dL/dz1 by [x | 1], and every weight in the network now has a gradient. = 11. Update = We subtract, and the network has learned. In practice a learning rate scales this step. The gradients: dL/dz3 = [0.5, -0.5, 0] dL/da1 = [1, -2, 2, -1] dL/dz1 = [0, -2, 2, -1] The takeaway: matrix multiplication is all you need. Just like the forward pass, backpropagation is matrix multiplications end to end. You can do every one by hand, slowly and imperfectly, which is exactly why a GPU's ability to do them fast mattered so much to deep learning. 💾 Save this post!

Backpropagation by hand ✍️ ~ 11 steps walkthrough below Backpropagation is the algorithm that actually trains a neural network, and it is where most people stop following along. It is not calculus you cannot do. It is matrix multiplication, working backward, one layer at a time. So I drew and calculated one entirely by hand. Goal: push the loss gradient back through a 3-layer network and land on a new value for every weight and bias. = 1. Given = A 3-layer perceptron, an input X, predictions Ypred = [0.5, 0.5, 0], and the truth Ytarget = [0, 1, 0]. = 2. Backprop gradient cells = Let us draw empty cells for every gradient we are about to compute. The shape of the answer comes first. = 3. Layer 3 softmax = We get dL/dz3 straight from Ypred minus Ytarget = [0.5, -0.5, 0]. No chain rule needed, and that shortcut is the whole reason softmax and cross-entropy are paired. = 4. Layer 3 weights and biases = Let us multiply dL/dz3 by [a2 | 1]. One multiplication gives the gradient for W3 and b3 together. = 5. Layer 2 activations = We multiply dL/dz3 by W3 to get dL/da2. The gradient moves back across a layer the same way the signal moved forward. = 6. Layer 2 ReLU = Let us pass it through the gate: keep the gradient where the activation was positive, zero it everywhere else. = 7. Layer 2 weights and biases = We multiply dL/dz2 by [a1 | 1]. The same figure as step 4, one layer up. = 8. Layer 1 activations = Let us multiply dL/dz2 by W2. = 9. Layer 1 ReLU = We apply the same gate again, now on a1. = 10. Layer 1 weights and biases = Let us multiply dL/dz1 by [x | 1], and every weight in the network now has a gradient. = 11. Update = We subtract, and the network has learned. In practice a learning rate scales this step. The gradients: dL/dz3 = [0.5, -0.5, 0] dL/da1 = [1, -2, 2, -1] dL/dz1 = [0, -2, 2, -1] The takeaway: matrix multiplication is all you need. Just like the forward pass, backpropagation is matrix multiplications end to end. You can do every one by hand, slowly and imperfectly, which is exactly why a GPU's ability to do them fast mattered so much to deep learning. 💾 Save this post!

Tom Yeh

944,407 просмотров • 6 дней назад

[Self-Attention] by Hand ✍️ Self-attention is what enables LLMs to understand context. How does it work? This exercise demonstrates how to calculate a 6-3 attention head by hand. Note that if we have two instances of this, we get 6-6 attention (i.e., multi-head attention, n=2). -- 𝗚𝗼𝗮𝗹 -- Transform [6D Features 🟧] to [3D Attention Weighted Features 🟦] -- 𝗪𝗮𝗹𝗸𝘁𝗵𝗿𝗼𝘂𝗴𝗵 -- [1] Given ↳ A set of 4 feature vectors (6-D): x1,x2,x3,x4 [2] Query, Key, Value ↳ Multiply features x's with linear transformation matrices WQ, WK, and WV, to obtain query vectors (q1,q2,q3,q4), key vectors (k1,k2,k3,k4), and value vectors (v1,v2,v3,v4). ↳ "Self" refers to the fact that both queries and keys are derived from the same set of features. [3] 🟪 Prepare for MatMul ↳ Copy query vectors ↳ Copy the transpose of key vectors [4] 🟪 MatMul ↳ Multiply K^T and Q ↳ This is equivalent to taking dot product between every pair of query and key vectors. ↳ The purpose is to use dot product as an estimate of the "matching score" between every key-value pair. ↳ This estimate makes sense because dot product is the numerator of Cosine Similarity between two vectors. [5] 🟨 Scale ↳ Scale each element by the square root of dk, which is the dimension of key vectors (dk=3). ↳ The purpose is to normalize the impact of the dk on matching scores, even if we scale dk to 32, 64, or 128. ↳ To simplify hand calculation, we approximate [ □/sqrt(3) ] with [ floor(□/2) ]. [6] 🟩 Softmax: e^x ↳ Raise e to the power of the number in each cell ↳ To simplify hand calculation, we approximate e^□ with 3^□. [7] 🟩 Softmax: ∑ ↳ Sum across each column [8] 🟩 Softmax: 1 / sum ↳ For each column, divide each element by the column sum ↳ The purpose is normalize each column so that the numbers sum to 1. In other words, each column is a probability distribution of attention, and we have four of them. ↳ The result is the Attention Weight Matrix (A) (yellow) [9] 🟦 MatMul ↳ Multiply the value vectors (Vs) with the Attention Weight Matrix (A) ↳ The results are the attention weighted features Zs. ↳ They are fed to the position-wise feed forward network in the next layer.

[Self-Attention] by Hand ✍️ Self-attention is what enables LLMs to understand context. How does it work? This exercise demonstrates how to calculate a 6-3 attention head by hand. Note that if we have two instances of this, we get 6-6 attention (i.e., multi-head attention, n=2). -- 𝗚𝗼𝗮𝗹 -- Transform [6D Features 🟧] to [3D Attention Weighted Features 🟦] -- 𝗪𝗮𝗹𝗸𝘁𝗵𝗿𝗼𝘂𝗴𝗵 -- [1] Given ↳ A set of 4 feature vectors (6-D): x1,x2,x3,x4 [2] Query, Key, Value ↳ Multiply features x's with linear transformation matrices WQ, WK, and WV, to obtain query vectors (q1,q2,q3,q4), key vectors (k1,k2,k3,k4), and value vectors (v1,v2,v3,v4). ↳ "Self" refers to the fact that both queries and keys are derived from the same set of features. [3] 🟪 Prepare for MatMul ↳ Copy query vectors ↳ Copy the transpose of key vectors [4] 🟪 MatMul ↳ Multiply K^T and Q ↳ This is equivalent to taking dot product between every pair of query and key vectors. ↳ The purpose is to use dot product as an estimate of the "matching score" between every key-value pair. ↳ This estimate makes sense because dot product is the numerator of Cosine Similarity between two vectors. [5] 🟨 Scale ↳ Scale each element by the square root of dk, which is the dimension of key vectors (dk=3). ↳ The purpose is to normalize the impact of the dk on matching scores, even if we scale dk to 32, 64, or 128. ↳ To simplify hand calculation, we approximate [ □/sqrt(3) ] with [ floor(□/2) ]. [6] 🟩 Softmax: e^x ↳ Raise e to the power of the number in each cell ↳ To simplify hand calculation, we approximate e^□ with 3^□. [7] 🟩 Softmax: ∑ ↳ Sum across each column [8] 🟩 Softmax: 1 / sum ↳ For each column, divide each element by the column sum ↳ The purpose is normalize each column so that the numbers sum to 1. In other words, each column is a probability distribution of attention, and we have four of them. ↳ The result is the Attention Weight Matrix (A) (yellow) [9] 🟦 MatMul ↳ Multiply the value vectors (Vs) with the Attention Weight Matrix (A) ↳ The results are the attention weighted features Zs. ↳ They are fed to the position-wise feed forward network in the next layer.

Tom Yeh

101,010 просмотров • 2 лет назад

Scaled Dot-Product Attention by hand ✍️ I explained attention in my recent AI seminar as follows: Step 1 — Compare We take one token as a query and compare it against all keys using dot products. This gives us a grid of similarity scores: every query against every key. Step 2 — Scale We divide those scores by √dₖ to keep the values numerically stable. Step 3 — Normalize We apply softmax to turn raw scores into a probability distribution—values between 0 and 1 that sum to 1. These are the attention weights. Step 4 — Combine We use those weights to compute a weighted sum of the values. Each output token becomes a linear combination of all previous tokens—maybe 37% from one, 23% from another, and a little from the rest. No blackbox. Just dot products, scaling, softmax, and weighted sums—built one row at a time in Excel.

Scaled Dot-Product Attention by hand ✍️ I explained attention in my recent AI seminar as follows: Step 1 — Compare We take one token as a query and compare it against all keys using dot products. This gives us a grid of similarity scores: every query against every key. Step 2 — Scale We divide those scores by √dₖ to keep the values numerically stable. Step 3 — Normalize We apply softmax to turn raw scores into a probability distribution—values between 0 and 1 that sum to 1. These are the attention weights. Step 4 — Combine We use those weights to compute a weighted sum of the values. Each output token becomes a linear combination of all previous tokens—maybe 37% from one, 23% from another, and a little from the rest. No blackbox. Just dot products, scaling, softmax, and weighted sums—built one row at a time in Excel.

Tom Yeh

31,114 просмотров • 6 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 • 6 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 • 5 месяцев назад

LSTM by hand ✍️ ~ 15 steps walkthrough below Since Hochreiter and Schmidhuber introduced them in 1997, LSTMs were the most effective way to handle long sequences, right up until the Transformer wave. They are a recurrent network: they read one input at a time and carry a memory forward. Lately recurrence is back in fashion (Mamba), because attention does not scale to hundreds of thousands of tokens. So I drew and calculated one entirely by hand. Goal: run an LSTM cell over a sequence of three inputs, filling in every gate and memory cell yourself. 1. Given Three inputs X1, X2, X3, and four weight matrices: forget, input, candidate, and output. 2. Initialize Let us set the previous hidden state h0 and the memory cell C0 to their start values. 3. Linear transform We multiply the four weight matrices by the stack of the current input, the previous hidden state, and a 1. 4. The gates Let us squash three of those results with sigmoid, giving the forget, input, and output gates, each between 0 and 1. 5. Update the memory We forget part of the old memory (C0 times the forget gate) and add the new (the candidate times the input gate). That is the new memory C1. 6. Candidate output Let us apply tanh to the new memory. 7. Update the hidden state We multiply that candidate by the output gate. The result is h1. 8. Process X2 Copy h1 and C1 forward, then repeat the whole cell: linear transform, the gates, update memory to C2, output gate to h2. 9. Process X3 Once more. Copy h2 and C2 forward, repeat, and read off h3, the final hidden state. Now you can show off to your friends that you calculated an LSTM by hand. ✍️😉 💾 Save this post! #AIbyHand #LSTM #DeepLearning

LSTM by hand ✍️ ~ 15 steps walkthrough below Since Hochreiter and Schmidhuber introduced them in 1997, LSTMs were the most effective way to handle long sequences, right up until the Transformer wave. They are a recurrent network: they read one input at a time and carry a memory forward. Lately recurrence is back in fashion (Mamba), because attention does not scale to hundreds of thousands of tokens. So I drew and calculated one entirely by hand. Goal: run an LSTM cell over a sequence of three inputs, filling in every gate and memory cell yourself. 1. Given Three inputs X1, X2, X3, and four weight matrices: forget, input, candidate, and output. 2. Initialize Let us set the previous hidden state h0 and the memory cell C0 to their start values. 3. Linear transform We multiply the four weight matrices by the stack of the current input, the previous hidden state, and a 1. 4. The gates Let us squash three of those results with sigmoid, giving the forget, input, and output gates, each between 0 and 1. 5. Update the memory We forget part of the old memory (C0 times the forget gate) and add the new (the candidate times the input gate). That is the new memory C1. 6. Candidate output Let us apply tanh to the new memory. 7. Update the hidden state We multiply that candidate by the output gate. The result is h1. 8. Process X2 Copy h1 and C1 forward, then repeat the whole cell: linear transform, the gates, update memory to C2, output gate to h2. 9. Process X3 Once more. Copy h2 and C2 forward, repeat, and read off h3, the final hidden state. Now you can show off to your friends that you calculated an LSTM by hand. ✍️😉 💾 Save this post! #AIbyHand #LSTM #DeepLearning

Tom Yeh

17,921 просмотров • 8 дней назад

Linux is the most widespread operating system, globally – but how is it built? Few people are better to answer this than Greg Kroah-Hartman (Greg K-H): Greg has been a Linux kernel maintainer for 25 years, and one of the 3 Linux Kernel Foundation Fellows (the other two are Linus Torvalds and Shuah Khan). Greg manages the Linux kernel’s stable releases, and is a maintainer of multiple kernel subsystems. We cover the inner workings of Linux kernel development, exploring everything from how changes get implemented to why its community-driven approach produces such reliable software. Greg shares insights about the kernel's unique trust model and makes a case for why engineers should contribute to open-source projects. Watch or listen: • YouTube: • Spotify: • Apple: Brought to you buy: • WorkOS — The modern identity platform for B2B SaaS. • Vanta — Automate compliance and simplify security with Vanta. Check out all of our current sponsor and their offerings: ---- One of the most surprising things I learned about Linux: although 4,000 developers contribute to the project from 500 companies per year, and has around 800 kernel maintainers, the project is run with email, git... and that's about it! No dedicated project managers, no regular meetings across the group. It's a truly fascinating and unique model (and only applies to the Linux kernel project: not to Linux distributions.) We go into more detail about this topic with Greg in the podcast as well.

Linux is the most widespread operating system, globally – but how is it built? Few people are better to answer this than Greg Kroah-Hartman (Greg K-H): Greg has been a Linux kernel maintainer for 25 years, and one of the 3 Linux Kernel Foundation Fellows (the other two are Linus Torvalds and Shuah Khan). Greg manages the Linux kernel’s stable releases, and is a maintainer of multiple kernel subsystems. We cover the inner workings of Linux kernel development, exploring everything from how changes get implemented to why its community-driven approach produces such reliable software. Greg shares insights about the kernel's unique trust model and makes a case for why engineers should contribute to open-source projects. Watch or listen: • YouTube: • Spotify: • Apple: Brought to you buy: • WorkOS — The modern identity platform for B2B SaaS. • Vanta — Automate compliance and simplify security with Vanta. Check out all of our current sponsor and their offerings: ---- One of the most surprising things I learned about Linux: although 4,000 developers contribute to the project from 500 companies per year, and has around 800 kernel maintainers, the project is run with email, git... and that's about it! No dedicated project managers, no regular meetings across the group. It's a truly fascinating and unique model (and only applies to the Linux kernel project: not to Linux distributions.) We go into more detail about this topic with Greg in the podcast as well.

Gergely Orosz

50,945 просмотров • 1 год назад

You start with a plain old complex polynomial p(z) = a₀ + a₁z + … + aₙzⁿ. Instead of fixing the coefficients, you treat them as variables in a controlled experiment and modulate them. Maybe you draw each aₖ from some distribution, or you let them drift along a parameter path. Do that 50,000 times, say, and you no longer have a polynomial. You have an ensemble. For each realization, the goal is the same Find its roots. Instead of attacking p(z)=0 directly with something like fsolve, you rewrite the polynomial as a matrix problem. Specifically, you build its Companion Matrix. For a monic polynomial zⁿ + bₙ₋₁zⁿ⁻¹ + … + b₀ the companion matrix is an n×n matrix whose last column contains the coefficients (with minus signs), and whose subdiagonal is filled with ones. Everything else is zero. It’s sparse, structured and deterministic. The theorem that makes this worth doing is simple and beautiful The eigenvalues of the companion matrix are exactly the roots of the polynomial. So each polynomial sample becomes a matrix. Each matrix becomes an eigenvalue problem. And eigenvalues are something numerical linear algebra is very, very good at computing.

You start with a plain old complex polynomial p(z) = a₀ + a₁z + … + aₙzⁿ. Instead of fixing the coefficients, you treat them as variables in a controlled experiment and modulate them. Maybe you draw each aₖ from some distribution, or you let them drift along a parameter path. Do that 50,000 times, say, and you no longer have a polynomial. You have an ensemble. For each realization, the goal is the same Find its roots. Instead of attacking p(z)=0 directly with something like fsolve, you rewrite the polynomial as a matrix problem. Specifically, you build its Companion Matrix. For a monic polynomial zⁿ + bₙ₋₁zⁿ⁻¹ + … + b₀ the companion matrix is an n×n matrix whose last column contains the coefficients (with minus signs), and whose subdiagonal is filled with ones. Everything else is zero. It’s sparse, structured and deterministic. The theorem that makes this worth doing is simple and beautiful The eigenvalues of the companion matrix are exactly the roots of the polynomial. So each polynomial sample becomes a matrix. Each matrix becomes an eigenvalue problem. And eigenvalues are something numerical linear algebra is very, very good at computing.

Mathelirium

15,566 просмотров • 4 месяцев назад

New short course on Building Applications with Vector Databases, taught by Pinecone’s Tim Tully! At the heart of a vector database is the ability to store a collection of vectors and then query against that, meaning input a new vector and find similar ones. This is useful for many AI applications. In this course, you'll learn how to use vector databases to build: (i) Semantic Search: Create a text search tool that goes beyond keyword matching, and instead focuses on the meaning of content. (ii) RAG (retrieval augmented generation): Enhance your LLM output by incorporating context from sources the model wasn't trained on. (iii) Recommender System: Combine semantic search and RAG to recommend topics, and demonstrate it with a news article recommender. (iv) Hybrid Search: Build an application that finds items using both images and descriptive text -- by combining both sparse and dense vector representations of the data -- using an eCommerce dataset as an example. (v) Image Similarity: Use image vector embeddings to create an app to compare facial features, using a database of public figures to determine the likeness between them. (vi) Anomaly Detection: Build an anomaly detection app that identifies unusual patterns in network communication logs. I hope you’ll enjoy learning how to build all these types of applications! Please sign up here:

New short course on Building Applications with Vector Databases, taught by Pinecone’s Tim Tully! At the heart of a vector database is the ability to store a collection of vectors and then query against that, meaning input a new vector and find similar ones. This is useful for many AI applications. In this course, you'll learn how to use vector databases to build: (i) Semantic Search: Create a text search tool that goes beyond keyword matching, and instead focuses on the meaning of content. (ii) RAG (retrieval augmented generation): Enhance your LLM output by incorporating context from sources the model wasn't trained on. (iii) Recommender System: Combine semantic search and RAG to recommend topics, and demonstrate it with a news article recommender. (iv) Hybrid Search: Build an application that finds items using both images and descriptive text -- by combining both sparse and dense vector representations of the data -- using an eCommerce dataset as an example. (v) Image Similarity: Use image vector embeddings to create an app to compare facial features, using a database of public figures to determine the likeness between them. (vi) Anomaly Detection: Build an anomaly detection app that identifies unusual patterns in network communication logs. I hope you’ll enjoy learning how to build all these types of applications! Please sign up here:

Andrew Ng

137,091 просмотров • 2 лет назад

Did you know that in Blender you can easily make the rotation of one object drive the rotation of another? ✨ 1. Select your object 2. Hover over one of its rotation fields and press CTRL+D 3. In the popup select the rotation of another object 4. You can write "smoothstep(0, tau, var) * 2" to both limit the rotation and make it non-linear. Tau is 180 degrees, and it's the end angle for the wheel, and 2 is the end angle for the gate leaf in radians (3 is approx. 180, so 2 is approx 2/3 of that!) 5. To copy the animation to the second leaf, select both, press CTRL+L, and choose 'Link Animation Data'. To invert the angle, add a minus sign, and you're done! 👌 #b3d

Did you know that in Blender you can easily make the rotation of one object drive the rotation of another? ✨ 1. Select your object 2. Hover over one of its rotation fields and press CTRL+D 3. In the popup select the rotation of another object 4. You can write "smoothstep(0, tau, var) * 2" to both limit the rotation and make it non-linear. Tau is 180 degrees, and it's the end angle for the wheel, and 2 is the end angle for the gate leaf in radians (3 is approx. 180, so 2 is approx 2/3 of that!) 5. To copy the animation to the second leaf, select both, press CTRL+L, and choose 'Link Animation Data'. To invert the angle, add a minus sign, and you're done! 👌 #b3d

passivestar

107,138 просмотров • 3 лет 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

114,524 просмотров • 1 год назад

Filesystems vs Vector search is the new MCP vs CLI. Claude uses agentic search. And Dens Sumesh at mintlify like filesystems too. but Retrieval is still used by Notion, Cursor, and others. We hit the cafes of SF to see what people want. - The debate is hot. Filesystems won by 1 point - filesystems feel very simple and intuitive. - RAG requires embedding, vector search and other stuff filesystems won. Introducing SMFS - Supermemory Filesystem we brought the best of these worlds into one single product - it's a filesystem, but agent can also do semantic search using grep. live today. Try it! Works with any sandbox, you mount and sync with cloud, it has a sync engine built in, all filetypes supported - even images and videos can be grepped. So, what do you choose - Filesystems, or vector search? we have both!

Filesystems vs Vector search is the new MCP vs CLI. Claude uses agentic search. And Dens Sumesh at mintlify like filesystems too. but Retrieval is still used by Notion, Cursor, and others. We hit the cafes of SF to see what people want. - The debate is hot. Filesystems won by 1 point - filesystems feel very simple and intuitive. - RAG requires embedding, vector search and other stuff filesystems won. Introducing SMFS - Supermemory Filesystem we brought the best of these worlds into one single product - it's a filesystem, but agent can also do semantic search using grep. live today. Try it! Works with any sandbox, you mount and sync with cloud, it has a sync engine built in, all filetypes supported - even images and videos can be grepped. So, what do you choose - Filesystems, or vector search? we have both!

Dhravya Shah

153,119 просмотров • 2 месяцев назад

IN 1999 MIT FILMED A MATH LECTURE THAT QUIETLY BECAME THE FOUNDATION OF EVERY AI MODEL YOU'VE EVER USED AND ALMOST NO ONE WAS TAUGHT TO SEE IT THAT WAY 39 minutes from Gilbert Strang, who taught this at MIT for over 60 years -- the linear algebra course an entire generation of engineers and data scientists grew up on. -> The shift it creates: you stop seeing matrices as boring grids of numbers and start seeing them as the language of space, data, and motion itself. School drilled you to crunch matrices by hand and never told you why. Strang shows you what they actually mean. Every neural net, every embedding, every model you prompt is linear algebra running underneath. The math you skipped is the engine of the thing you use all day. Memorizing the steps was never the skill -> seeing what the numbers do is. This is where it finally clicks. Most people fear linear algebra and move on. The ones who watched this see straight into how AI actually works. Bookmark & Watch it today, this one's a legend ↓

IN 1999 MIT FILMED A MATH LECTURE THAT QUIETLY BECAME THE FOUNDATION OF EVERY AI MODEL YOU'VE EVER USED AND ALMOST NO ONE WAS TAUGHT TO SEE IT THAT WAY 39 minutes from Gilbert Strang, who taught this at MIT for over 60 years -- the linear algebra course an entire generation of engineers and data scientists grew up on. -> The shift it creates: you stop seeing matrices as boring grids of numbers and start seeing them as the language of space, data, and motion itself. School drilled you to crunch matrices by hand and never told you why. Strang shows you what they actually mean. Every neural net, every embedding, every model you prompt is linear algebra running underneath. The math you skipped is the engine of the thing you use all day. Memorizing the steps was never the skill -> seeing what the numbers do is. This is where it finally clicks. Most people fear linear algebra and move on. The ones who watched this see straight into how AI actually works. Bookmark & Watch it today, this one's a legend ↓

slash1s

432,172 просмотров • 1 месяц назад

Tokenization -- turning text into a sequence of integers -- is a key part of generative AI, and most API providers charge per million tokens. How does tokenization work? Learn the details of tokenization and RAG optimization in Retrieval Optimization: From Tokenization to Vector Quantization, created in collaboration with Qdrant and taught by its Developer Relations Lead, Kacper Łukawski. This course focuses on Retrieval augmented generation (RAG), which has two steps: First, a retriever finds relevant information; then, the generator uses what’s retrieved as context to produce a response. You’ll learn to optimize the first step (the retriever) by understanding how tokenization works and how it impacts the relevance of your search. In addition, you will also learn to measure and improve retrieval quality, speed, and memory. In detail, you’ll: - Learn about the internal workings of the embedding models and how your text turns into vectors. - Understand how several tokenizers, such as Byte-Pair Encoding, WordPiece, Unigram, and SentencePiece work. - Explore common challenges with tokenizers, such as unknown tokens, domain-specific identifiers, and numerical values, that can negatively affect your vector search. - Understand how to measure the quality of your search across relevance, ranking, and score-related metrics. - Understand how the main parameters in "HNSW", a graph-based algorithm, affect the relevance and speed of vector search, and how to tune its parameters. - Experiment with the three major quantization methods – product, scalar, and binary – and learn how they impact memory requirements, search quality, and speed. By the end of this course, you’ll have a solid understanding of how tokenization functions and how to optimize vector search in your RAG systems. Please sign up here!

Tokenization -- turning text into a sequence of integers -- is a key part of generative AI, and most API providers charge per million tokens. How does tokenization work? Learn the details of tokenization and RAG optimization in Retrieval Optimization: From Tokenization to Vector Quantization, created in collaboration with Qdrant and taught by its Developer Relations Lead, Kacper Łukawski. This course focuses on Retrieval augmented generation (RAG), which has two steps: First, a retriever finds relevant information; then, the generator uses what’s retrieved as context to produce a response. You’ll learn to optimize the first step (the retriever) by understanding how tokenization works and how it impacts the relevance of your search. In addition, you will also learn to measure and improve retrieval quality, speed, and memory. In detail, you’ll: - Learn about the internal workings of the embedding models and how your text turns into vectors. - Understand how several tokenizers, such as Byte-Pair Encoding, WordPiece, Unigram, and SentencePiece work. - Explore common challenges with tokenizers, such as unknown tokens, domain-specific identifiers, and numerical values, that can negatively affect your vector search. - Understand how to measure the quality of your search across relevance, ranking, and score-related metrics. - Understand how the main parameters in "HNSW", a graph-based algorithm, affect the relevance and speed of vector search, and how to tune its parameters. - Experiment with the three major quantization methods – product, scalar, and binary – and learn how they impact memory requirements, search quality, and speed. By the end of this course, you’ll have a solid understanding of how tokenization functions and how to optimize vector search in your RAG systems. Please sign up here!

Andrew Ng

146,313 просмотров • 1 год назад

Traditional data pipelines don't work for RAG applications. There are 3 issues with them: 1. Traditional data engineering solutions are optimized to handle structured data. RAG applications rely primarily on unstructured data. 2. The connector ecosystem to load data from unstructured data sources is very immature. 3. Traditional solutions do not offer any way to transform unstructured data into an optimized vector search index. The goal of a RAG Pipeline is to solve these problems. The number one objective is to create a reliable vector search index using factual knowledge and relevant context. This sounds easy, but it's one of the biggest challenges we face when building RAG applications. At a high level, there are four different stages in the architecture of a RAG pipeline: 1. Ingestion: Here is where the pipeline loads the information from the data source. 2. Extraction: Where the pipeline processes the input data and decides how to retrieve the text contained inside them. 3. Transform: Where the pipeline chunks the data and generates document embeddings. 4. Load: Where the pipeline creates a search index in a vector database and loads the document embeddings. There are different rabbit holes at each one of these stages. Here are three of them: 1. Ingesting data once is simple. The hard part is refreshing the vector database whenever the original data source changes. 2. Extracting the content of a plain text document is simple. The hard part is to extract content from complex documents containing tables, images, or cross-references. 3. A simple continual chunking strategy with an overlap is simple. The hard part is to find the optimal strategy for your specific knowledge base and the way you are planning to query it. In the attached video, I'll show you how you can build an enterprise-grade RAG Pipeline that solves every one of the above problems. I'll use Vectorize. They partnered with me on this post. You can use them to build RAG pipelines optimized for accurate context retrieval. If you have a few documents lying around, set up a free account and give it a try.

Traditional data pipelines don't work for RAG applications. There are 3 issues with them: 1. Traditional data engineering solutions are optimized to handle structured data. RAG applications rely primarily on unstructured data. 2. The connector ecosystem to load data from unstructured data sources is very immature. 3. Traditional solutions do not offer any way to transform unstructured data into an optimized vector search index. The goal of a RAG Pipeline is to solve these problems. The number one objective is to create a reliable vector search index using factual knowledge and relevant context. This sounds easy, but it's one of the biggest challenges we face when building RAG applications. At a high level, there are four different stages in the architecture of a RAG pipeline: 1. Ingestion: Here is where the pipeline loads the information from the data source. 2. Extraction: Where the pipeline processes the input data and decides how to retrieve the text contained inside them. 3. Transform: Where the pipeline chunks the data and generates document embeddings. 4. Load: Where the pipeline creates a search index in a vector database and loads the document embeddings. There are different rabbit holes at each one of these stages. Here are three of them: 1. Ingesting data once is simple. The hard part is refreshing the vector database whenever the original data source changes. 2. Extracting the content of a plain text document is simple. The hard part is to extract content from complex documents containing tables, images, or cross-references. 3. A simple continual chunking strategy with an overlap is simple. The hard part is to find the optimal strategy for your specific knowledge base and the way you are planning to query it. In the attached video, I'll show you how you can build an enterprise-grade RAG Pipeline that solves every one of the above problems. I'll use Vectorize. They partnered with me on this post. You can use them to build RAG pipelines optimized for accurate context retrieval. If you have a few documents lying around, set up a free account and give it a try.

Santiago

40,441 просмотров • 1 год назад