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AlphaFold by hand✍️ Excel ~ I designed this exercise to show (1) MSA multi-head attention, (2) Pair triangular update, two key components of the EvoFormer architecture.👇Join the AI Math community. Download xlsx.
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![[LSTM] by Hand ✍️ LSTMs have been the most effective architecture to process long sequences of data, until our world was taken over by the Transformers. LSTMs belong to the broader family of recurrent neural network (RNNs) that process data sequentially in a recurrent manner. Transformers, on the other hand, abandon recurrence and use self-attention instead to process data concurrently in parallel. Recently, there is renewed interest in recurrence as people realized self-attention doesn’t scale to extremely long sequences, like hundreds of thousands of tokens. Mamba is a good example to bring back recurrence. All of a sudden, it is cool to study LSTMs. How do LSTMs work? [1] Given ↳ 🟨 Input sequence X1, X2, X3 (d = 3) ↳ 🟩 Hidden state h (d = 2) ↳ 🟦 Memory C (d = 2) ↳ Weight matrices Wf, Wc, Wi, Wo Process t = 1 [2] Initialize ↳ Randomly set the previous hidden state h0 to [1, 1] and memory cells C0 to [0.3, -0.5] [3] Linear Transform ↳ Multiply the four weight matrices with the concatenation of current input (X1) and the previous hidden state (h0). ↳ The results are feature values, each is a linear combination of the current input and hidden state. [4] Non-linear Transform ↳ Apply sigmoid σ to obtain gate values (between 0 and 1). • Forget gate (f1): [-4, -6] → [0, 0] • Input gate (i1): [6, 4] → [1, 1] • Output gate (o1): [4, -5] → [1, 0] ↳ Apply tanh to obtain candidate memory values (between -1 and 1) • Candidate memory (C’1): [1, -6] → [0.8, -1] [5] Update Memory ↳ Forget (C0 .* f1): Element-wise multiply the current memory with forget gate values. ↳ Input (C’1 .* o1): Element-wise multiply the “candidate” memory with input gate values. ↳ Update the memory to C1 by adding the two terms above: C0 .* f1 + C’1 .* o1 = C1 [6] Candiate Output ↳ Apply tanh to the new memory C1 to obtain candidate output o’1. [0.8, -1] → [0.7, -0.8] [7] Update Hidden State ↳ Output (o’1 .* o1 → h1): Element-wise multiply the candidate output with the output gate. ↳ The result is updated hidden state h1 ↳ Also, it is the first output. Process t = 2 [8] Initialize ↳ Copy previous hidden state h1 and memory C1 [9] Linear Transform ↳ Repeat [3] [10] Update Memory (C2) ↳ Repeat [4] and [5] [11] Update Hidden State (h2) ↳ Repeat [6] and [7] Process t = 3 [12] Initialize ↳ Copy previous hidden state h2 and memory C2 [13] Linear Transform ↳ Repeat [3] [14] Update Memory (C3) ↳ Repeat [4] and [5] [15] Update Hidden State (h3) ↳ Repeat [6] and [7]](https://image.24vids.com/tw-1806285518126317877/ext_tw_video_thumb/1806285245903482880/pu/img/9-DANs-lC9ghBLvA.jpg)
![[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)
