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#NFL Injury Updates #Dolphins De'Von Achane - LP. Unless setback occurs, data strongly favors playing. Re-injury risk likely low (~10%) given that it took only 2 wks to return to practice #Ravens Isaiah Likely - Out. Wk 2-4 return = likely. Data suggests 4 game ramp up 1/10
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![[VAE] by Hand ✍️ A Variational Auto Encoder (VAE) learns the structure (mean and variance) of hidden features and generates new data from the learned structure. In contrast, GANs only learn to generate new data to fool a discriminator; they may not necessarily know the underlying structure of the data. The International Conference on Learning Representations (ICLR) this year announced its first ever "Test of Time Award" to recognizes the VAE paper, published 10 years ago. This exercise demonstrates how to calculate a VAE by hand. [1] Given: ↳ Three training examples X1, X2, X3 ↳ Copy training examples to the bottom ↳ The purpose is to train the network to reconstruct the training examples. ↳ Since each target is a training example itself, we use the Greek word "auto" which means "self." This crucial step is what makes an autoencoder "auto." [2] Encoder: Layer 1 + ReLU ↳ Multiply inputs with weights and biases ↳ Apply ReLU, crossing out negative values (-1 -> 0) [3] Encoder: Mean and Variance ↳ Multiply features with two sets of weights and biases ↳ 🟩 The first set predicts the means (𝜇) of latent distributions ↳ 🟪 The second set predicts the standard deviation (𝜎) of latent distributions [4] Reparameterization Trick: Random Offset ↳ Sample epsilon ε from the normal distribution with mean = 0 and variance = 1. ↳ The purpose is to randomly pick a offset away from the mean. ↳ Multiply the standard deviation values with epsilon values. ↳ The purpose is to scale the offset by the standard deviation. [5] Reparameterization Trick: Mean + Offset ↳ Add the sampled offset to predicted mean ↳ The result are new parameters or features 🟨 as inputs to the Decoder. [6] Decoder: Layer 1 + ReLU ↳ Multiply input features with weights and biases ↳ Apply ReLU, crossing out negative values. Here, -4 is crossed out. [7] Decoder: Layer 2 ↳ Multiply features with weights and biases ↳ The output is Decoder's attempt to reconstruct the input data X from reparameterized distributions described by 𝜇 and 𝜎. [8]-[10] KL Divergence Loss [8] Loss Gradient: Mean 𝜇 ↳ We want 𝜇 to approach 0. ↳ A lot of math called SGVB simplifies the calculation of loss gradients to simply 𝜇 [9,10] Loss Gradient: Stdev 𝜎 ↳ We want 𝜎 to approach 1. ↳ A lot of math simplifies the calculation to 𝜎 - (1/ 𝜎) [11] Reconstruction Loss ↳ We want the reconstructed data Y (dark 🟧) to be the same as the input data X. ↳ Some math involving Mean Square Error simplifies the calculation to Y - X.](https://image.24vids.com/tw-1817192583191564496/ext_tw_video_thumb/1817192009591091200/pu/img/XAVt6rAbRS292En_.jpg)
![[Dropout] by hand ✍️ Dropout is a simple yet effective way of reducing overfitting and improving generalization. This by-hand exercise lets students practice calculating dropout, thereby gaining insight into its inner workings. As an additional bonus, students get to practice calculating the gradients of the Mean Square Error (MSE) loss. After the practice, students are often surprised by how simple it is. -- 𝗡𝗲𝘁𝘄𝗼𝗿𝗸 𝗔𝗿𝗰𝗵𝗶𝘁𝗲𝗰𝘁𝘂𝗿𝗲 -- 1. Linear(2,4) 2. ReLU 3. Dropout(0.5) 4. Linear(4,3) 5. ReLU 6. Dropout(0.33) 7. Linear(3,2) -- 𝗪𝗮𝗹𝗸𝘁𝗵𝗿𝗼𝘂𝗴𝗵 -- 🏋️ Training [1] Given ↳ A training set of 2 examples X1, X2 [2] 🟧 Random: p > 0.5 ↳ Draw 4 random numbers ↳ For each random number, if it is above 0.5, we keep and denote it as ◯. Otherwise we drop and denote it as ╳. ↳ The result is [◯, ╳, ◯, ╳] [3] 🟧 Dropout: Matrix ↳ Calculate the scaling factor: 1 / (1-p) = 2 ↳ Set the diagonal based on [◯, ╳, ◯, ╳], where ◯ = 2 and ╳ = 0 ↳ The purpose is to drop the 2nd and the 4th nodes, and scale the remaining two nodes by 2. [4] 🟦 Random: p > 0.33 ↳ Draw 3 random numbers ↳ For each random number, if it is above 0.33, we keep and denote it as ◯. Otherwise we drop and denote it as ╳. ↳ The result is [◯, ◯, ╳] [5] 🟦 Dropout: Matrix ↳ Calculate the scaling factor: 1 / (1-p) = 1.5 ↳ Set the diagonal based on [◯, ◯, ╳], where ◯ = 1.5 and ╳ = 0 ↳ The purpose is to drop the 3rd node, and scale the remaining two nodes by 1.5. [6] Feed Forward ↳ Now we have all the matrices ready across the layers, perform the feed forward pass by calculating a series of matrix multiplications from the top to the bottom ↳ ReLU activation function is applied along the way to set negative feature values to zeroes, denoted by ╳. ↳ The outputs are Y. [7] 🟥 Loss Gradients of Mean Square Error (MSE) ↳ The formula is 2 * (Y - Y') ↳ First we calculate Outputs (Y) - Targets (Y') ↳ Second we multiply each element by 2 [8] Update Weights ↳ Use loss gradients to start back propagation ↳ Update some weights (light red) ↳ The values of the new weights are for demonstration purpose only, not based on real calculation. 🔍 Inference [9] Deactivate Dropout ↳ We set both dropout matrices to identity matrices ↳ The effect is to keep all the features as is. [10] Feed Forward ↳ Take the forward pass to make predictions about unseen data.](https://image.24vids.com/tw-1836044959020900392/ext_tw_video_thumb/1836044944273735680/pu/img/HAksFrOwf0Ct8B5_.jpg)
![[Graph Convolutional Network] by hand ✍️ Graph Convolutional Networks (GCNs), introduced by Thomas Kipf and Max Welling in 2017, have emerged as a powerful tool in the analysis and interpretation of data structured as graphs. This exercise demonstrates how GCN works in a simple application: binary classification. -- Goal -- Predict if a node in a graph is X. -- Architecture -- 🟪 Graph Convolutional Network (GCN) 1. GCN1(4,3) 2. GCN2(3,3) 🟦 Fully Connected Network (FCN) 1. Linear1(3,5) 2. ReLU 3. Linear2(5,1) 4. Sigmoid Simplications: • Adjacent matrices are not normalized. • ReLU is applied to messages directly. -- Walkthrough -- [1] Given ↳ A graph with five nodes A, B, C, D, E [2] 🟩 Adjacency Matrix: Neighbors ↳ Add 1 for each edge to neighbors ↳ Repeat in both directions (e.g., A->C, C->A) ↳ Repeat for both GCN layers [3] 🟩 Adjacency Matrix: Self ↳ Add 1's for each self loop ↳ Equivalent to adding the identity matrix ↳ Repeat for both GCN layers [4] 🟪 GCN1: Messages ↳ Multiply the node embeddings 🟨 with weights and biases ↳ Apply ReLU (negatives → 0) ↳ The result is one message per node [5] 🟪 GCN1: Pooling ↳ Multiply the messages with the adjacent matrix ↳ The purpose is the pool messages from each node's neighbors as well as from the node itself. ↳ The result is a new feature per node [6] 🟪 GCN1: Visualize ↳ For node 1, visualize how messages are pooled to obtain a new feature for better understanding ↳ [3,0,1] + [1,0,0] = [4,0,1] [7] 🟪 GCN2: Messages ↳ Multiply the node features with weights and biases ↳ Apply ReLU (negatives → 0) ↳ The result is one message per node [8] 🟪 GCN2: Pooling ↳ Multiply the messages with the adjacent matrix ↳ The result is a new feature per node [9] 🟪 GCN2: Visualize ↳ For node 3, visualize how messages are pooled to obtain a new feature for better understanding ↳ [1,2,4] + [1,3,5] + [0,0,1] = [2,5,10] [10] 🟦 FCN: Linear 1 + ReLU ↳ Multiply node features with weights and biases ↳ Apply ReLU (negatives → 0) ↳ The result is a new feature per node ↳ Unlike in GCN layers, no messages from other nodes are included. [11] 🟦 FCN: Linear 2 ↳ Multiply node features with weights and biases [12] 🟦 FCN: Sigmoid ↳ Apply the Sigmoid activation function ↳ The purpose is to obtain a probability value for each node ↳ One way to calculate Sigmoid by hand ✍️ is to use the approximation below: • >= 3 → 1 • 0 → 0.5 • <= -3 → 0 -- Outputs -- A: 0 (Very unlikely) B: 1 (Very likely) C: 1 (Very likely) D: 1 (Very likely) E: 0.5 (Neutral)](https://image.24vids.com/tw-1828407238408438145/ext_tw_video_thumb/1828407079696039936/pu/img/mjnHB1FbGJngAUIw.jpg)