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Fukushima's video (1986) shows a CNN that recognises handwritten digits [3], three years before LeCun's video (1989). CNN timeline taken from [5]: ★ 1969: Kunihiko Fukushima published rectified linear units or ReLUs [1] which are now extensively used in CNNs. ★ 1979: Fukushima published the basic CNN architecture with... show more
742,172 görüntüleme • 9 ay önce •via X (Twitter)
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It’s ironic that Schmidhuber credits Fukushima with inventing CNNs while overlooking several other deep-learning researchers—such as Galushkin—whose work dates back to the 1960s.
Galushkin was the pupil of Tsypkin whom I cite in the Annotated History of Modern AI and Deep Learning as follows: "See also Iakov Zalmanovich Tsypkin's even earlier work on gradient descent-based on-line learning for non-linear systems [GDa-b]." [GDa] Y. Z. Tsypkin (1966). Adaptation, training and self-organization automatic control systems, Avtomatika I Telemekhanika, 27, 23-61. On gradient descent-based on-line learning for non-linear systems. [GDb] Y. Z. Tsypkin (1971). Adaptation and Learning in Automatic Systems, Academic Press, 1971. On gradient descent-based on-line learning for non-linear systems.
@predict_addict Jürgen Schmidhuber never betrays the truth and history 🔥 Even if it seems that's the case, he will correct himself; my observation of him is he does his best to attribute the right part of history to the right people & doesn't have religious beliefs about "popular" attributions
@SchmidhuberAI Indeed, he is also great person in real life.
2026 update: 1st backprop-CNN for vision (Wei Zhang et al, 1988)
Schmiduber always brings receipts
Yapp Le Cum saw the VHS and didn’t think anyone would get this footage on internet years later…. L move tbh
In your view, does this call Yan LeCun's Turing Award into question?
I’ve done a few talks on the history of AI and always came to the same conclusions you did you again😋 ever since you came out with your history book I’ve been trying to convert peeps…albeit not here enough…I mean isn’t backprop from Leibniz?😉 We’re all part of the tapestry of knowledge here…some folks are just bigger patches and stronger twine than others🥳😎🦾
Thank you! I see the same video published so many times. I built a simple network in 1984 as a high school student, that recognised the hand-written alphabet, on a 386… from a one-page paragraph describing neural nets in new scientist (basically, nodes, weights, input and outputs. I used 16.16 fp, no bias nodes (they weren’t mentioned) and a monochrome cmos chip camera to capture the images. It took 38 training cycles to converge to 100% accuracy (I only had one set of images) but it was remarkably efficient and fast for the time (and my lack of knowledge 🤣). It wasn’t a CNN, it was fully connected, and trained through a crude approximation of back-propagation.
I have repeatedly said why is there no other major contribution from @ylecun all these years 35 years. This is the answer. It was not his original work.
LeCun made CNN work reliably and practically. It is, if you wish, what Steve Jobs did with Apple. This is no small feat. Discovery and product creation is always a process. Takes lots of people.
This needs to be fully recognized. It's the same pattern over and over again. Enough already.
Its not fair that LeCun gets all the credit. Its also not fair that the Fukushima reactor named after him blew up in 2012 😔
Just my humble response to this: Thanks for bringing up these references; personally, I wasn’t even aware of them. I 'grew up' with LeCun’s networks when I took my first deep learning course in 2006 with Geoff Hinton at UofT. I agree that we need to foster a much healthier environment for research. Ultimately, we are all standing on the shoulders of giants. There is often a distinction between who did it first and who communicated it in a way that the industry adopted at the right time! For example, Gerry Tesauro ( one of the best scientists I know) developed the first Deep RL for Backgammon (TD-Gammon) in the early 90s and achieved world-class performance. So AlphaGo, created by David Silver, who is also a friend Gerry, was not strictly the first. But Gerry definitely didn’t mind, AFIK, the credit going to David, as David scaled it and applied it to Computer Go. You have good points, and it’s important to bring these things up, but it would be nice if it came across more as adding historical context. It might not always seem fair to the pioneers, but that seems to be how the history of technology and science is written.
LeCun’s key contribution was showing that you could train the whole thing end to end with backpropagation on 1980s–1990s hardware and get state of the art practical performance. That made CNNs explode in the West and eventually led to the deep learning revolution.
How did this guy not receive the Turing Award instead of those grifters :(
Thank you for demystifying the history.
Classic. First, I saw Fukushima and was waiting to see the earthquake. Then I saw CNN and thought there was something going on about the news platform I usually watch. Then I read more carefully. I am an idiot!
I was always told wrong
No wonder Lecun always sounded like a joke.
Neural networks don't beep like they used to. :|
So @ylecun . Did you steal the credit for something that was achieved many years prior?
Shunichi Amari for the win!!
why are we hearing about this only now?
Why did not Japan become an AI superpower? If they had Fukushima?
"The simplest technique is a direct-convolution realization using the relation NI-I Nz-1 y(n1, n2) = c c h(m1, mz)x(n1 - m1, n2 - mz) (5) m1=0 mp=o where x(n1, nz) is the input to the filter and y(nl, fl2) is the filter output." -- 1972
my dad did text only systems recognition is the defining line here the use of a visual CNN? cause retrieval algo is a RNN too? my dad made a fortran search algo that returned hallucinated text as well off of deweyDD tag information NO VLM THO so idk if that counts
Fukushima created the architecture Zhang created the first modern supervised CNN LeCun standardized it, pushed it into practice and owned the narrative All three matter. Only one became famous.
Please don't stare in rear mirror for too long, we need the brightest people to look ahead!
Ich hoffe das überzeugt @kimmonismus zur richtigen Seite zu wechseln.
LeCunt
Are you saying these are cases of multiple discovery or plagiarism?
he called it neocognitron
@HacKanCuBa.
![Dropout by hand ✍️ ~ 10 steps walkthrough below Dropout is the simplest trick in deep learning that actually works: during training you randomly switch neurons off, so the network cannot lean on any one of them. It is two lines of code and almost nobody has worked through what those lines do to the numbers. So I drew and calculated one entirely by hand. Goal: train one pass through a small network with two dropout layers, then run inference with dropout switched off. The network: Linear(2,4), ReLU, Dropout(0.5), Linear(4,3), ReLU, Dropout(0.33), Linear(3,2). = 1. Given = A training set of two examples, X1 and X2, and the weight matrices for all three linear layers. = 2. Draw the first random numbers = Let us draw 4 random numbers, one per neuron in the first hidden layer. Above 0.5 we keep (◯), below we drop (╳). Here that gives [◯, ╳, ◯, ╳]. = 3. Build the first dropout matrix = We turn that pattern into a diagonal matrix. The scaling factor is 1/(1-p) = 2, so a kept neuron gets 2 and a dropped one gets 0. Multiplying by it does both jobs at once: it deletes the 2nd and 4th neurons and doubles the two that survive. = 4. Draw the second random numbers = Let us do it again for the 3 neurons in the next layer, this time against p = 0.33. The result is [◯, ◯, ╳]. = 5. Build the second dropout matrix = We set the diagonal to 1.5 where kept and 0 where dropped. Only the 3rd neuron goes. = 6. Feed forward = Let us run the whole thing top to bottom: one matrix multiplication per layer, ReLU setting the negatives to zero, and the two dropout matrices doing their work in between. The outputs Y come out at the bottom. = 7. MSE loss gradients = We compare Y against the targets Y', subtract, and multiply each element by 2. That is the whole gradient of the mean squared error. = 8. Update the weights = Let us push those gradients back through the network and update the weights (marked in light red). = 9. Deactivate dropout = Training is over, so we set both dropout matrices to the identity. Every neuron is back, and nothing is scaled. = 10. Feed forward again = One more pass, this time on unseen data, to make the prediction. You have just trained and run a network with dropout by hand. ✍️ The outputs: Training outputs Y = [-6, 9; 13, 4] Loss gradients = [-4, 4; 6, -2] Inference outputs = [13, 13; 4, 3] 💾 Save this post! #AIbyHand #Dropout #DeepLearning #NeuralNetworks](https://image.24vids.com/tw-2078846104460783996/ext_tw_video_thumb/2078846092997763074/pu/img/pljAsdaJjvGEEjPE.jpg)
![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!](https://image.24vids.com/tw-2079208495094677511/ext_tw_video_thumb/2079208483669315584/pu/img/dDD0x6lHYHuV1POQ.jpg)
![AGI? One day, but not yet. The only AI that works well right now is the one behind the screen [12-17]. But passing the Turing Test [9] behind a screen is easy compared to Real AI for real robots in the real world. No current AI-driven robot could be certified as a plumber [13-17]. Hence, the Turing Test isn't a good measure of intelligence (and neither is IQ). And AGI without mastery of the physical world is no AGI. That’s why I created the TUM CogBotLab for learning robots in 2004 [5], co-founded a company for AI in the physical world in 2014 [6], and had teams at TUM, IDSIA, and now KAUST work towards baby robots [4,10-11,18]. Such soft robots don't just slavishly imitate humans and they don't work by just downloading the web like LLMs/VLMs. No. Instead, they exploit the principles of Artificial Curiosity to improve their neural World Models (two terms I used back in 1990 [1-4]). These robots work with lots of sensors, but only weak actuators, such that they cannot easily harm themselves [18] when they collect useful data by devising and running their own self-invented experiments. Remarkably, since the 1970s, many have made fun of my old goal to build a self-improving AGI smarter than myself and then retire. Recently, however, many have finally started to take this seriously, and now some of them are suddenly TOO optimistic. These people are often blissfully unaware of the remaining challenges we have to solve to achieve Real AI. My 2024 TED talk [15] summarises some of that. REFERENCES (easy to find on the web): [1] J. Schmidhuber. Making the world differentiable: On using fully recurrent self-supervised neural networks (NNs) for dynamic reinforcement learning and planning in non-stationary environments. TR FKI-126-90, TUM, Feb 1990, revised Nov 1990. This paper also introduced artificial curiosity and intrinsic motivation through generative adversarial networks where a generator NN is fighting a predictor NN in a minimax game. [2] J. S. A possibility for implementing curiosity and boredom in model-building neural controllers. In J. A. Meyer and S. W. Wilson, editors, Proc. of the International Conference on Simulation of Adaptive Behavior: From Animals to Animats, pages 222-227. MIT Press/Bradford Books, 1991. Based on [1]. [3] J.S. AI Blog (2020). 1990: Planning & Reinforcement Learning with Recurrent World Models and Artificial Curiosity. Summarising aspects of [1][2] and lots of later papers including [7][8]. [4] J.S. AI Blog (2021): Artificial Curiosity & Creativity Since 1990. Summarising aspects of [1][2] and lots of later papers including [7][8]. [5] J.S. TU Munich CogBotLab for learning robots (2004-2009) [6] NNAISENSE, founded in 2014, for AI in the physical world [7] J.S. (2015). On Learning to Think: Algorithmic Information Theory for Novel Combinations of Reinforcement Learning (RL) Controllers and Recurrent Neural World Models. arXiv 1210.0118. Sec. 5.3 describes an RL prompt engineer which learns to query its model for abstract reasoning and planning and decision making. Today this is called "chain of thought." [8] J.S. (2018). One Big Net For Everything. arXiv 1802.08864. See also patent US11853886B2 and my DeepSeek tweet: DeepSeek uses elements of the 2015 reinforcement learning prompt engineer [7] and its 2018 refinement [8] which collapses the RL machine and world model of [7] into a single net. This uses my neural net distillation procedure of 1991: a distilled chain of thought system. [9] J.S. Turing Oversold. It's not Turing's fault, though. AI Blog (2021, was #1 on Hacker News) [10] J.S. Intelligente Roboter werden vom Leben fasziniert sein. (Intelligent robots will be fascinated by life.) F.A.Z., 2015 [11] J.S. at Falling Walls: The Past, Present and Future of Artificial Intelligence. Scientific American, Observations, 2017. [12] J.S. KI ist eine Riesenchance für Deutschland. (AI is a huge chance for Germany.) F.A.Z., 2018 [13] H. Jones. J.S. Says His Life's Work Won't Lead To Dystopia. Forbes Magazine, 2023. [14] Interview with J.S. Jazzyear, Shanghai, 2024. [15] J.S. TED talk at TED AI Vienna (2024): Why 2042 will be a big year for AI. See the attached video clip. [16] J.S. Baut den KI-gesteuerten Allzweckroboter! (Build the AI-controlled all-purpose robot!) F.A.Z., 2024 [17] J.S. 1995-2025: The Decline of Germany & Japan vs US & China. Can All-Purpose Robots Fuel a Comeback? AI Blog, Jan 2025, based on [16]. [18] M. Alhakami, D. R. Ashley, J. Dunham, Y. Dai, F. Faccio, E. Feron, J. Schmidhuber. Towards an Extremely Robust Baby Robot With Rich Interaction Ability for Advanced Machine Learning Algorithms. Preprint arxiv 2404.08093, 2024.](https://image.24vids.com/tw-1925204946967801861/amplify_video_thumb/1925197727224250368/img/U-5TNxCzlWIlxyYf.jpg)
![Everybody is talking about recursive self-improvement (RSI) and meta learning. Here is my old 2020 talk about this [1]. It has aged well. Example: humans still define the starts & ends of trials of many modern meta learners. My RSI systems since 1994 LEARN to (re)define them [2]! [1] Meta Learning Machines in a Single Lifelong Trial (talk for workshops at ICML 2020 and NeurIPS 2021, based on earlier talks since 1994). Abstract: the most widely used machine learning algorithms were designed by humans and thus are hindered by our cognitive biases and limitations. Can we also construct meta learning algorithms that can learn better learning algorithms so that our self-improving AIs have no limits other than those inherited from computability and physics? This question has been a main driver of my research since I wrote a thesis on it in 1987 [2]. Here I summarize our work on meta reinforcement learning with self-modifying policies in a single lifelong trial (since 1994), and mathematically optimal meta-learning through the self-referential Gödel Machine (since 2003). Many additional publications on meta-learning since 1987 can be found in the RSI overview [2]. [2] J. Schmidhuber (AI Blog, 2020-2025). 1/3 century anniversary of first publication on recursive self-improvement (RSI) and meta learning machines that learn to learn (1987). For its cover I drew a robot that bootstraps itself. 1992-: gradient descent-based neural meta learning. 1994-: meta reinforcement learning with self-modifying policies. 1997: meta RL plus artificial curiosity and intrinsic motivation. 2002-: asymptotically optimal meta learning for curriculum learning. 2003-: mathematically optimal Gödel Machine. 2020-: new stuff!](https://image.24vids.com/tw-2031397968511717584/amplify_video_thumb/2031397324404985860/img/FjgqUEEDb0gBsKbU.jpg)
![U-Net by hand ✍️ ~ 17 steps walkthrough below I consider U-Net as a key milestone in deep learning, the first image-to-image model that really worked! It came out of medical imaging, an unusual place, not from NeurIPS or CVPR or ACL. Now it is the backbone of diffusion models, which you see in almost all modern image generation models. I drew the network as a C so the matrix multiplication flows naturally down. Tilt your head to the right and it is a U again. 🤣 Goal: push a 3 x 16 image down to a 2 x 4 bottleneck and back out again, filling in every cell yourself. = 1. Given = An image of three channels, R, G and B, sixteen pixels wide, and every kernel the network will use. = 2. Convolution 1 = Let us slide the first kernel over the image. Each output is one multiply-and-add over a 2 x 3 window, and the result is the green feature map. = 3. Find the maxima = We circle the largest value in each 1 x 2 window. Circling first is worth the extra step: it is the pooling decision, made before anything is written down. = 4. Max pool 1 = Let us copy those maxima down. Sixteen columns become eight, and half the detail is gone for good. = 5. Convolution 2 = We convolve again with the second kernel, deeper into the contracting path. The feature map is blue now. = 6. Find the maxima again = Same move as step 3, on the blue map. = 7. Max pool 2 = Eight columns become four. = 8. The bottleneck = Let us convolve once more. This is the bottom of the U, a 2 x 4 block that is everything the network kept. = 9. Spread it out = We start back up. The transposed convolution writes each bottleneck value into a wider grid, leaving gaps between them. = 10. Transposed convolution 1 = Let us fill those gaps by convolving over the spread-out grid. Four columns become eight. = 11. The first skip = We copy the encoder's matching row straight across. This is the skip connection, and it is the whole reason a U-Net can recover detail that pooling threw away. = 12. Convolution with the skip = Let us convolve the upsampled features together with the copied ones. = 13. Spread it out again = Same as step 9, one level up. = 14. Transposed convolution 2 = Eight columns become sixteen, back to the width we started at. = 15. The second skip = The encoder's first feature map comes across, the one made before any pooling happened. = 16. Convolution and ReLU = We convolve, then cross out every negative and set it to zero. = 17. Output convolution = Let us apply the last kernel. Out comes R', G' and B', an image the same size as the one we started with. The outputs: R' = [3, 0, 7, 0, 7, 0, 17, 0, 3, 0, 9, 0, 2, 0, 6, 0] G' = [1, 20, 1, 10, 1, 12, 1, 19, 2, 5, 1, 11, 1, 3, 1, 7] B' = [4, 20, 8, 10, 8, 12, 18, 19, 5, 5, 10, 11, 3, 3, 7, 7] Congrats! You just calculated a U-Net by hand. 💾 Save this post!](https://image.24vids.com/tw-2081382845243691256/ext_tw_video_thumb/2081382802667249664/pu/img/n2gZcnKRYe9G781o.jpg)
![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](https://image.24vids.com/tw-2077396566864773414/ext_tw_video_thumb/2077396555410051073/pu/img/JX9U_Rk-wmgd1S-x.jpg)
![Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!](https://image.24vids.com/tw-2096604730453577857/ext_tw_video_thumb/2096604718290120705/pu/img/Uxa9xebSM4EBcEzP.jpg)
![Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!](https://image.24vids.com/tw-2083194764485153128/ext_tw_video_thumb/2083194751193403393/pu/img/xPTCxxe0pHHx3B7I.jpg)