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Which shark is this? A. (๐ฆ-ck)+(๐ช-๐คฟ) ๐ฆ B. (๐ฅช-๐ง)+(๐ซ-๐ฌ) ๐ฆ C. (๐-๐) + (0๏ธโฃ+โฝ๏ธ) ๐ฆ D. ๐โช๏ธ๐ฆ E. Both A & C are correct. F. ๐๐งธ๐ฆ
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![Ever seen a chord that swings both ways? In this excerpt from Schubert, a seventh chord [F-A-C-Eโญ] first functions as a dominant of Bโญ, then is respelled with a Dโฏ instead of Eโญ and functions as a German Augmented Sixth chord in A minor.](https://image.24vids.com/tw-1849508156550103282/ext_tw_video_thumb/1458195579511656458/pu/img/VxEHdy8UNWOukgUJ.jpg)
![[Discrete Fourier Transform] by Hand โ๏ธ In signal processing, the Discrete Fourier Transform (DFT) is no doubt the most important method. But the math involved is extremely complex, literally, involving a summation over a complex number term e^(-iwt). I developed this exercise to demonstrate that underneath such complexity, DFT is just a series of matrix multiplications you can calculate by hand. โ๏ธ Once you see that, it should not surprise you that a deep neural network, which is also a series of matrix multiplications, with activation functions in-between, can learn to perform DFT to process and analyze signals so effectively. How does DFT work? [1] Given โณ Signals A, B, and C in the ๐ง frequency domain: โฆ A = cos(w) + 2cos(2w) โฆ B = cos(w) + cos(3w) + cos(4w) โฆ C = -cos(2w) + cos(3w) โฆ Each signal is a weighed sum of four cosine waves at frequencies 1w, 2w, 3w, and 4w. โฆ We will apply Inverse DFT to convert the signals to time domain representations, and then demonstrate DFT can convert back to their original frequency domain representations. โณ Signal X in the ๐ฉ time domain. X is sampled at 10 time points 1t, 2t, โฆ, 10t: โฆ X = [-2.5, -1.8, 3, -0.7, -1.0, -0.7, 3, -1.8, -2.5, 5] โฆ Suppose X is also a weighted sum of the same four cosine waves, but we donโt already know their weights. We will apply DFT to discover them. [2] ๐ง Frequency Matrix (F) โณ Write the coefficients of A, B, C as a matrix F. Each signal is a row. Each frequency is a column. โณ A โ [1, 2, 0, 0] โณ B โ [1, 0, 1, 1] โณ C โ [0, 1-, 1, 0] [3] Cosine โ Discrete โณ Sample from the continuous cosine waves at discrete time points 1t, 2t, 3t, to 10t. [4] Cosine Matrix (W) โณ Write the samples as a matrix, Each frequency is a row. Each time point is a column. [5] Inverse DFT: ๐ง Frequency โ ๐ฉ Time โณ Multiply the frequency matrix F and the cosine matrix W. โณ The meaning of this multiplication is to linearly combine the four cosine waves (rows in W) into time-domain signals (rows in T) using the weights specified in F. โณ The result is matrix T, which are signals A, B, C converted to the time domain. Each signal is a row. Each time point is a column. [6] Transpose โณ Transpose T, converting each signalโs time domain representation from a row to a column. [7] DFT: ๐ฉ Time โ ๐ง Frequency โณ Multiply the cosine matrix W with the transpose of matrix T. โณ The purpose of this multiplication is to take a dot-product between each time-domain signal (columns in the transpose of T) and each cosine wave (rows in W), which has the effect of projecting the signal onto a cosine wave to determine how much they are correlated. Zero means not correlated at all. โณ The result is an intermediate version of the โrecoveredโ frequency matrix where each column corresponds to a signal and each row corresponds to a frequency. โณ Compared to the original frequency matrix F, this intermediate matrix has non-zero weights in the correct places, but scaled up by a factor of 5 (n/2, n=10). For example, signal A, originally [1,2,0,0], is recovered at [5,10,0,0]. [8] Scale โณ Multiply each value by 2/n = 1/5 to scale down the intermediate matrix to match the magnitude of the original frequency matrix F. [9] Transpose โณ Transpose the recovered frequency matrix back to the same orientation of the original frequency matrix F. โณ Like magic ๐ช, the result is identical to the original F, which means DFT successfully recovered the frequency components of signals A, B, C. [10] Apply DFT to X: ๐ฉ Time โ ๐ง Frequency โณ Now that we have some confidence in DFTโs ability to recover frequency components, we apply DFT to Xโs time-domain representation by multiplying W with X. โณ The result is the an intermediate matrix. [11] Scale โณ Similarly, we scale down by a factor of 5 to obtain the recovered frequency components of X (a column). [12] Transpose โณ Similarly, we transpose the recovered column to row to match the orientation of the frequency matrix. โณ Using the coefficients [0,0,3,2], we can write the equation of X as 3cos(3w) + 2cos(4w). Notes: I hope this by hand exercise helps you understand the essence of DFT. But there is more technical details, such as: โข Sine: The complete DFT math also includes sine waves that follow a similar calculation process. โข Phase: Here, we assume all the cosine waves are aligned at the origin, namely, phase is 0. If a phase p is added, for example, cos(w+p), we will need to calculate the sine component and use their ratio to figure out what p is. โข Magnitude: If phase is not zero, the magnitude will need to be calculated by combining both cosine and sine terms.](https://image.24vids.com/tw-1805694219031662799/ext_tw_video_thumb/1805693963464327168/pu/img/kOcpELm1j7wfqJlR.jpg)
![[Graph Convolutional Network] by hand โ๏ธ 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 -- ๐ช Graph Convolutional Network (GCN) 1. GCN1(4,3) 2. GCN2(3,3) ๐ฆ Fully Connected Network (FCN) 1. Linear1(3,5) 2. ReLU 3. Linear2(5,1) 4. Sigmoid Simplications: โข Adjacent matrices are not normalized. โข ReLU is applied to messages directly. -- Walkthrough -- [1] Given โณ A graph with five nodes A, B, C, D, E [2] ๐ฉ Adjacency Matrix: Neighbors โณ Add 1 for each edge to neighbors โณ Repeat in both directions (e.g., A->C, C->A) โณ Repeat for both GCN layers [3] ๐ฉ Adjacency Matrix: Self โณ Add 1's for each self loop โณ Equivalent to adding the identity matrix โณ Repeat for both GCN layers [4] ๐ช GCN1: Messages โณ Multiply the node embeddings ๐จ with weights and biases โณ Apply ReLU (negatives โ 0) โณ The result is one message per node [5] ๐ช GCN1: Pooling โณ Multiply the messages with the adjacent matrix โณ The purpose is the pool messages from each node's neighbors as well as from the node itself. โณ The result is a new feature per node [6] ๐ช GCN1: Visualize โณ For node 1, visualize how messages are pooled to obtain a new feature for better understanding โณ [3,0,1] + [1,0,0] = [4,0,1] [7] ๐ช GCN2: Messages โณ Multiply the node features with weights and biases โณ Apply ReLU (negatives โ 0) โณ The result is one message per node [8] ๐ช GCN2: Pooling โณ Multiply the messages with the adjacent matrix โณ The result is a new feature per node [9] ๐ช GCN2: Visualize โณ For node 3, visualize how messages are pooled to obtain a new feature for better understanding โณ [1,2,4] + [1,3,5] + [0,0,1] = [2,5,10] [10] ๐ฆ FCN: Linear 1 + ReLU โณ Multiply node features with weights and biases โณ Apply ReLU (negatives โ 0) โณ The result is a new feature per node โณ Unlike in GCN layers, no messages from other nodes are included. [11] ๐ฆ FCN: Linear 2 โณ Multiply node features with weights and biases [12] ๐ฆ FCN: Sigmoid โณ Apply the Sigmoid activation function โณ The purpose is to obtain a probability value for each node โณ One way to calculate Sigmoid by hand โ๏ธ is to use the approximation below: โข >= 3 โ 1 โข 0 โ 0.5 โข <= -3 โ 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)
