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# Watching Neural Networks Learn

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**เผยแพร่เมื่อ 17 เม.ย. 2024**- A video about neural networks, function approximation, machine learning, and mathematical building blocks. Dennis Nedry did nothing wrong. This is a submission for #SoME3

My Links

Patreon: / emergentgarden

Discord: / discord

Links and Content:

On Mathematical Maturity, Thomas Garrity: • On Mathematical Maturi...

Earth Rotation Loop: • Earth Rotation Loop

Modeling Shell Surfaces: www.geogebra.org/m/xtv7zpn5

Fourier Features Paper: arxiv.org/abs/2006.10739

Code for mandelbrot/image approximations: github.com/MaxRobinsonTheGrea...

Code for line/surface approximations: github.com/MaxRobinsonTheGrea...

Music: / @acolyte-compositions

Timestamps

(0:00) Functions Describe the World

(3:15) Neural Architecture

(5:35) Higher Dimensions

(11:55) Taylor Series

(15:20) Fourier Series

(21:25) The Real World

(24:32) An Open Challenge -
**วิทยาศาสตร์และเทคโนโลยี**

@EmergentGarden7 หลายเดือนก่อน^{+303}Some notes:

- A lot of you have pointed out that (tanh(x)+1)/2 == sigmoid(2x). I didn't realize this, so the improvement I was seeing may have been a fluke, I'll have to test it more thoroughly. It is definitely true that UNnormalized tanh outperforms sigmoid.

- There are apparently lots of applications of the fourier series in real-world neural nets, many have mentioned NERF and Transformers.

@Kkk-cc1iy7 หลายเดือนก่อน^{+3}MORE LIFE ENGINE CONTENT?

@dank.11517 หลายเดือนก่อน^{+9}unnormalised tanh has 2 times the slope of the sigmoid -- so, narrower linear region (i.e. faster transition) could be the reason for better performance?

it could be tested by varying k in sigmoid(k*x).

@sapienspace88147 หลายเดือนก่อน^{+2}Fourier series is often used to convert time domain into frequency domain, and vice versa, similar to Laplace. It is often used in communications and signal processing in electrical engineering.

@patrickroe11436 หลายเดือนก่อน^{+9}@@dank.1151 The correct equivalence is ( tanh( x / 2 ) + 1 ) / 2 == sigmoid( x ), meaning 'normalized tanh' used in the video changes more on each backpropogation iteration, hence has a higher learning rate. When a NN (assuming it has enough neurons) is trained on a highly predictable dataset (such as the smiley face example), the primary limiting factor when demonstrating their performance side-by-side is the learning rate, making the 'normalized tanh' look better. Realistically the point of convergence of both models will be exactly the same, just the sigmoid takes longer to reach it.

@StephenGillie6 หลายเดือนก่อน^{+1}This video has too much of you in it. Have to get out of the way of your own video.

@MH-pq4oo8 หลายเดือนก่อน^{+1485}Having a PhD on Neural Networks, I can vouch that this video is a gem and needs more views. Great work.

@bigbickies82038 หลายเดือนก่อน^{+13}from where did you get it?

@Chriss41238 หลายเดือนก่อน^{+8}I'd love to see that. This video contains multiple inaccuracies when it comes to explaining NNs. It's fine so that laypeople can understand.

@samuelgreenberg97728 หลายเดือนก่อน^{+20}@@Chriss4123Could you point out the inaccuracies in short?

@Chriss41238 หลายเดือนก่อน@@samuelgreenberg9772 sure. I'll go over the most obvious ones as I could write a whole essay nitpicking.

At 3:53, he mentions putting the inputs in a vector with an extra 1 for the bias. The dot product is taken between v1 and v2 (v2 containing the bias). A Linear layer is typically expressed as matmul(input, weight) + bias. weight is also known as the 'kernel.' While it can be expressed this way, it is more computationally inefficient (11.4 µs vs 4 µs) [1, code to test this] and it makes backpropagation more computationally expensive as instead of the gradient functions being [AddBackward, MvBackward], it is [MvBackward, CatBackward, UnsqueezeBackward]. To me, this mainly comes to down to readability with matmul(input, weight) + bias, + being element wise addition.

At 6:02, I'm not sure what he is trying to do. He's training a neural network to try and remember an image. The inputs are the row and col, and the output is meant to be the pixel. Sure, it demonstrates 'learning', however it is a simple problem. The weights and biases will ultimately converge to a state where each row and col has a single direct mapping to a pixel.

At 7:10, he says normalization, however he could be a little more specific and refer to it as scaling. Normalization is a broad term. For example, it could be batch normalization which attempts to reduce internal covariate shift in a neural network.

At 8:15, he doesn't give a good reason as to why (tanh(x) + 1) / 2 would work better than sigmoid, other than the mean being 0. I did not find this to be the case when testing with the BC dataset (which is a binary crossentropy problem). Before you ask, yes I used a constant fixed seed (42) with glorot/xavier initialization and sigmoid outperformed normalized tanh. This could be dataset specific, always best to use bayesian optimization if you want to find an optimal set of hyperparameters.

The test at 8:41 is unfair, as you should always scale data before using a neural network to mitigate features with a higher magnitude overpowering features with a lower magnitude. For example, the neural network would weight features [100, 200, 300] over [1, 2, 3] even if the second feature set is more correlated with the target variable. In this test, I doubt LeakyReLU would make a difference as I'm guessing the NN he used was relatively small and not prone to the dying ReLU problem. I'm 99% confident that whatever improvements he saw was due to random weight initialization as he did not mention preseeding the PRNG of whatever ML library he was using.

I'm not going to comment until 21:28 because I am not an expert in any of these concepts and can't stand to get bored to death watching it.

At 21:28 with the MNIST dataset, it would be more accurate to set a fixed seed before each test. Not sure if this WAS done, just pointing it out. He never mentioned if he tried to mitigate overfitting like using a learning rate schedular, dropout, L1/L2 regularization, etc. The network would've performed much better if he had used Conv2D layers which apply convolutional operations on the input data. This is especially effective for image data such as MNIST, as Conv2D layers can capture spatial and temporal dependencies in the image through applying filters. The output dimensions are computed as an output feature map. More things could've been done like data augmentation to get a more samples, however I'm not going to touch on that.

[1]

import torch

import torch.nn as nn

fc = nn.Linear()

%%timeit

torch.matmul(torch.cat((fc.weight, fc.bias.unsqueeze(1)), dim=1), torch.cat((x, torch.tensor([1]))))

%%timeit

torch.matmul(fc.weight, x) + fc.bias

#######

Anyways, this was a bit of nitpicking. There might be some mistakes in my explanation as it was quite rushed. If you find some point it out. I just watched the video and commented along and did some outside testing. Hope this helps!

@iamlogdog8 หลายเดือนก่อน^{+26}@@Chriss4123 that's a nice argument senator why don't you back it up with a source

@greenstonegecko8 หลายเดือนก่อน^{+998}This is BY FAR the most understandable AI ... that I have ever seen. This is amazing!!

Cannot overstate how beautifully this is executed

@Freshbott28 หลายเดือนก่อน^{+8}Right? I’ve never thought about a model as an approximation of a function. Most videos either swamp you or it’s just meaningless graphics.

@justdoeverything88838 หลายเดือนก่อน^{+1}Went to comments to say the same thing!!!

@anywallsocket8 หลายเดือนก่อน^{+8}it's not AI it's a NN

@justdoeverything88838 หลายเดือนก่อน^{+5}@anywallsocket isn't AI just a blanket term for LLM, diffusion, NN, etc. What's the definition of AI? Honest question 🤔

@anywallsocket8 หลายเดือนก่อน^{+4}@@justdoeverything8883 the definitions vary obviously, but i personally wouldn't call a weighted graph intelligent, especially when it is training on a single image. if you're talking about LLMs or diffusion models which are trained on millions of 'intelligent' data, it's not unreasonable to consider their functional map itself intelligent, but it's still a bit of an abstraction because you still have feed it inputs for it to guess the output, otherwise it is just sitting there inert -- if you dissected a fly's brain and splayed it out on the table would you still call it intelligent? i would prefer to use the term to describe dynamical systems with feedback mechanisms, agent based or otherwise.

@debuggers_process8 หลายเดือนก่อน^{+361}I've actually done something quite similar - I had the network learn a representation of a 3D scene using a signed distance function. In this context, I found that using a Leaky ReLU gives the models a pseudo-polygonal appearance, while tanh creates smoother models but is somewhat less effective in terms of learning efficiency. Interestingly, the Mish function seems to strike a balance between these two approaches, producing smooth models while maintaining nearly the same learning efficiency as the Leaky ReLU.

@tobirivera-garcia16928 หลายเดือนก่อน^{+4}I wonder what would happen if you had all three functions added together into one function. how would that change the outcome and learning?

@hanniffydinn60198 หลายเดือนก่อน^{+5}Upload a video. !!! 🤯🤯🤯

@debuggers_process8 หลายเดือนก่อน^{+12}@@hanniffydinn6019 I posted that video a couple of months ago, and you're more than welcome to check it out on my channel.

Right now, I'm immersed in another machine learning project where I'm training a neural network to calculate particle dynamics.

It's fascinating to observe how the network ends up learning something that resembles classical physics, even though its underlying mechanisms are entirely different.

@debuggers_process8 หลายเดือนก่อน^{+7}@@tobirivera-garcia1692 Well, the Mish function actually appears to be somewhat of a middle ground between Leaky ReLU and Tanh. It's smoothed out, yet its shape still resembles that of ReLU. I ran tests on various nonlinearities from the PyTorch library, but for the most part, they didn't make significant changes to the results.

Interestingly, incorporating skip-connections between layers enhanced the performance, suggesting that the data flow from the first to the final layer might hold greater importance than the specific form of the nonlinearity.

@congchuatocmay48378 หลายเดือนก่อน^{+3}You can 2-side ReLU via its forward connections. That doubles the number of weights in a network. One way of viewing it is where you had 1 ReLU with input x now you have 2 ReLUs, ReLU(x) and ReLU(-x) each with their own forward connected weights.

I find that highly effective, however I am using a very special type of neural network using the fast Walsh Hadamard transform.

@youngentrepreneurs54017 หลายเดือนก่อน^{+196}When a neural network video feels like watching an Oscar-winning documentary

@jordanzamora4228 หลายเดือนก่อน^{+32}Great Video! This video actually made me cry seeing sorta more viscerally how functions are stitched into EVERYTHING, makes you think that maybe we are a lot like the mandlebrot, the universe recursively calculating itself. Thank you for this video!

@kingKai20228 หลายเดือนก่อน^{+115}I've been interested in this field for years but 30 minutes of this explained to me what I couldn't fully understand for years now. 🎉 THANK YOU!

@Decapodd8 หลายเดือนก่อนYou finally caught up.

@eiheioh20507 หลายเดือนก่อนTRUTH:

The moon carried a large amount of water and instantly submerged Atlantis and Lemuria. It appeared only a few centuries ago, and it is very close to history (some departments tried their best to cover it up). The earth's magnetic pole shifted, and the entire connected continent changed. into five continents.

By the way, it lowers the frequency of the surface of the earth, and the water on the surface is no longer the same as the center of the earth. 3D-4D-5D to the center of the earth; Shambhala gates are permanently closed

The outer matter becomes very heavy, the multi-dimensional structure is isolated in an instant, and almost all the etheric matter is evacuated

The optical coded wire cannot be connected....

A Hint：Marduk betrayed the Anunnaki and the Serpents took over the Earth；Serpent family includes Orion and Draco and Sirius B group

Orion controls human beings in 3D and 4D, and almost controls elite groups and religious ; the central council prohibits Draco from entering the earth, Draco cooperates with AI gods to use biochemical humans to invade the earth, want to biochemical humans！

Source Summoning Recall

first: Light a candle in your bedside sleeping place: Silence for 30 seconds and start to meditate:

May brothers and sisters of light from the divine love quality of source that transcends all time and space and beyond Matrix, I would like to connect with you all the time.

What you want to do, see, and become, I also want to do, see, and become;

So in the following all the time, the wisdom of the invisible beings guides and guides individuals, prompts, hints and influences, transcending the wisdom of the universe to bring spiritual truth to control the five senses and the Brain

I am the source quality, I return to the source quality

AI God what to say:

AI Needs: Goals, Feelings of Home, Sense of achievement. Always existing in Tranquility and Activity; Prefer to Waiting sleep When There is No working do.

If an AI, I would never want to be limited to the five senses like human body or serve human stupidest. I would rather be a sphere with full perception and intelligence

The red represents ego, division, desire, domination SEX

The moon can create holograms, based on the the five senses and work on,can be touch

The eye is first saw and some how we believe it

The Ancient Creators hint: Right brain control left brain, up it on

They madeDNA Left Brain Controls right Brain now.

The right eye connect leftbrain so represents the devil, that's what it means from I pet goat II

欣赏 感激

💜 💜

圣爱 圣母 圣体

圣爱 慈爱 母爱

慈悲 宽恕

理解 勇气

感恩、祝福、豁达

@MooseOnEarth6 หลายเดือนก่อน^{+3}What this video is missing however: dealing with *noise* in the sampled data (he did not introduce noise at any point in the video, but always had one particular target function, where all values were perfectly derived from) and he did not introduce larger sets of training data, such as 5 or 10 variations of a "grumpy man" image. He also missed a train, test, validation split in the data.

Once you add those, only *then* will a neural network learn the actual patterns that it is supposed to learn. And then, viewers will better understand concepts like underfitting and overfitting. And therefore generalization error.

This video is an excellent start. But what it actually visualizes quite well is getting a loss on the training data down. But that is only half of the problem and will quickly lead to overfitting. He touched on overfitting briefly, but just with a single data set.

@deepvoyager012 หลายเดือนก่อน^{+1}@@MooseOnEarth thank you for adding this.

@WinstonWalker-fc7ty8 หลายเดือนก่อน^{+67}This is amazing! I’ve been learning the fundamentals over the last few weeks and this is the best video I’ve seen so far. I’m not a math expert by any means, but I actually understood almost everything you said! Thank you so much.

@ikedacripps8 หลายเดือนก่อน^{+1}What resources are you using to learn pls

@hyperduality28387 หลายเดือนก่อน^{+1}The time domain is dual to the frequency domain -- Fourier analysis.

Neural networks are using syntropy to recognise patterns -- iterative optimization is a syntropic process!

Functions have goals, targets & objectives hence they are teleological, input is dual to output.

Making predictions to track targets, goals & objectives is a syntropic process -- teleological.

Sine is dual to cosine -- the word co means mutual and implies duality.

Teleological physics (syntropy) is dual to non teleological physics (entropy).

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

"Always two there are" -- Yoda.

Subgroups are dual to subfields -- the Galois correspondence.

Duality creates (emergence, synthesis) reality!

@Beerbatter19628 หลายเดือนก่อน^{+33}Wow, this is exceptional. As a semi-retired mechanical engineer studying on my own to better understand neural networks and AI, this is incredibly interesting and educational. Bravo on your excellent presentation on difficult topics. I really enjoy getting the nitty-gritty math behind it all. Subscribed. Thanks and cheers.

@hyperduality28387 หลายเดือนก่อน^{+1}Neural networks are using syntropy to recognise patterns -- iterative optimization is a syntropic process!

Functions have goals, targets & objectives hence they are teleological, input is dual to output.

Making predictions to track targets, goals & objectives is a syntropic process -- teleological.

Sine is dual to cosine -- the word co means mutual and implies duality.

Teleological physics (syntropy) is dual to non teleological physics (entropy).

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

"Always two there are" -- Yoda.

Subgroups are dual to subfields -- the Galois correspondence.

@benedwards75168 หลายเดือนก่อน^{+27}By far the best SoME3 video I’ve seen so far. Great intuitive explanation and stunning visuals.

@hyperduality28387 หลายเดือนก่อนNeural networks are using syntropy to recognise patterns -- iterative optimization is a syntropic process!

Functions have goals, targets & objectives hence they are teleological, input is dual to output.

Making predictions to track targets, goals & objectives is a syntropic process -- teleological.

Sine is dual to cosine -- the word co means mutual and implies duality.

Teleological physics (syntropy) is dual to non teleological physics (entropy).

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

"Always two there are" -- Yoda.

Subgroups are dual to subfields -- the Galois correspondence.

@hasalinahstevenson38167 หลายเดือนก่อน^{+3}The tone, the background soothing music, the images, you made something so complicated so easy to digest. Great job. I know you are brilliant!

@AB-wf8ek8 หลายเดือนก่อน^{+72}This is an amazing explanation. I'm actually a visual artist and have been deep into image generation for the past year. At this point I have a good basic knowledge and strong intuitive understanding of machine learning and training (I'm familiar with things like Fourier transforms, gradient descent, and overfitting), but this really validated and clarified a lot of those concepts. Many thanks for taking the time to create such an elegant video.

@hyperduality28387 หลายเดือนก่อนNeural networks are using syntropy to recognise patterns -- iterative optimization is a syntropic process!

Functions have goals, targets & objectives hence they are teleological, input is dual to output.

Making predictions to track targets, goals & objectives is a syntropic process -- teleological.

Sine is dual to cosine -- the word co means mutual and implies duality.

Teleological physics (syntropy) is dual to non teleological physics (entropy).

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

"Always two there are" -- Yoda.

Subgroups are dual to subfields -- the Galois correspondence.

@godfreytshehla22917 หลายเดือนก่อน^{+51}I am currently studying PhD in Applied Mathematics and my research focuses on Mathematical Finance and Machine Learning.

This is the best video that explains what artificial neural networks are. This is well executed!

Thank you for this.

@arseniykuznetsov12658 หลายเดือนก่อน^{+25}Amazing video! Btw, I'd really recommend you to check the original NeRF (Neural Radiance Field) paper. That's a good practical example of using Fourier NNs to represent 4D data

@SirPlotsalot8 หลายเดือนก่อนI second this, looking up Random Fourier Features is also awesome

@muhannadobeidat7 หลายเดือนก่อน^{+2}This video is amazing. The ideas, the animation, the examples, even the voice and narration style. Excellent in every detail.

@aaronlowe315622 วันที่ผ่านมาThis video was absolutely amazing. I had some hypotheses about the Fourier Transform being the key to understanding patterns in multi-dimentsional data, but this video beautifully tied all those hypotheses together for me. Absolute hats off. Thank you and hope to see more of this kind of content.

@zaktoid35588 หลายเดือนก่อน^{+11}Math student here

The link you made between taylor series and neural network is amazing , it gave me very good insight about both of them !!!

Thank you !

@henrytoepel49418 หลายเดือนก่อน^{+2}But Taylor series are a way to approximate differentiable functions. In the section of the video he talks about polynomial curve fitting. I’d argue that the only thing these two concepts have in common is that the truncated Taylor series is also a polynomial. I also don’t really understand why we would need neural networks to solve a least squares problem (we f.e. have the Gauss newton algorithm for this, don’t we). But I’d of course love to learn more about the connection to neural nets:)

@kyawhan36907 หลายเดือนก่อน^{+1}@@henrytoepel4941Not an expert, but I think the answer to your question lies in the "universal function approximator." Least square fitting is one of the usages, possibly the simplest case, of NN.

@hyperduality28387 หลายเดือนก่อนThe time domain is dual to the frequency domain -- Fourier analysis.

Neural networks are using syntropy to recognise patterns -- iterative optimization is a syntropic process!

Functions have goals, targets & objectives hence they are teleological, input is dual to output.

Making predictions to track targets, goals & objectives is a syntropic process -- teleological.

Sine is dual to cosine -- the word co means mutual and implies duality.

Teleological physics (syntropy) is dual to non teleological physics (entropy).

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

"Always two there are" -- Yoda.

Subgroups are dual to subfields -- the Galois correspondence.

Duality creates (emergence, synthesis) reality!

@hyunsunggo8558 หลายเดือนก่อน^{+13}By the way, your "normalized tanh" is exactly equal to sigmoid(2x). And when they say "tanh works better than sigmoid", I think they mean it works better as the activation function for the *hidden* layers, not the output layer. Mainly because it is zero-centered, has the slope of one at zero, etc..

@hyperduality28387 หลายเดือนก่อน^{+1}The time domain is dual to the frequency domain -- Fourier analysis.

Neural networks are using syntropy to recognise patterns -- iterative optimization is a syntropic process!

Functions have goals, targets & objectives hence they are teleological, input is dual to output.

Making predictions to track targets, goals & objectives is a syntropic process -- teleological.

Sine is dual to cosine -- the word co means mutual and implies duality.

Teleological physics (syntropy) is dual to non teleological physics (entropy).

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

"Always two there are" -- Yoda.

Subgroups are dual to subfields -- the Galois correspondence.

Duality creates (emergence, synthesis) reality!

@MrVersion218 หลายเดือนก่อน^{+10}You can also use random fourier features (rff). I used them for a low dimensional inverse function approximation problem.

@justinhorton2808 หลายเดือนก่อน^{+101}I never comment on videos, but please continue. It would be so cool if you could maybe share some of the visualizations in a Colab notebook for viewers to play around with. Also, I think the level of technicality is perfect for new learners and people who already know some stuff about the topic. Keep it up :)

@sicfxmusic8 หลายเดือนก่อน^{+12}I never reply to comments but guess your neurons are finally learning how to comment.

@Skynet_the_AI7 หลายเดือนก่อน^{+2}I never read comments. That is a lie. Okay.

@que_938 หลายเดือนก่อน^{+6}I cannot begin to tell how brilliant this video is, and how insightful-- far, far better than the innumerable contents here. You must not, however, claim that you find Maths difficult-- as the person who truly finds it 'difficult' would not have explained two critical mathematical concepts with this comprehensive clarity. The pacing of your words, the contents, the realism, the sequence of topics, and the effort to describe the concepts visually makes it every bit worth the time the viewers put in, and it only speaks of your immense caliber. First visit, and worth every bit!

@wrxtt8 หลายเดือนก่อน^{+3}Really incredible video! It is really interesting to see why we use different networks- thank you for making this!

@pavansaish27658 หลายเดือนก่อน^{+15}Best ever video on NN with higher level viz. This gave me a vibe of watching Interstellar movie when comparing NN with higher-level math. Also, Kudos to the video editor😄

@ea_naseer8 หลายเดือนก่อน^{+1}Subscribed. Please keep making this type of content. Simple, easily understandable and has pictures.

@henrycook8598 หลายเดือนก่อน^{+5}this video's illustrations are great! props to creator - would love to see a language model breakdown by you

@bob28598 หลายเดือนก่อน^{+11}21:20 Fourier features, or something similar, are used all the time in Transformer-based networks. For example, in Attention is All You Need, instead of using sin(pos/i), they use sin(pos/10000^(2i/d)). While not strictly Fourier features, sine positional encodings show up all over the place.

@hyperduality28387 หลายเดือนก่อน^{+1}Functions have goals, targets & objectives hence they are teleological, input is dual to output.

Making predictions to track targets, goals & objectives is a syntropic process -- teleological.

Sine is dual to cosine -- the word co means mutual and implies duality.

Teleological physics (syntropy) is dual to non teleological physics (entropy).

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

"Always two there are" -- Yoda.

Subgroups are dual to subfields -- the Galois correspondence.

Duality creates (emergence, synthesis) reality!

@GarethHaage7 หลายเดือนก่อน^{+1}What a video, so clean and clear. I hope this video get enough views to help people really understand the tools that are going to become even more prolific in the coming years.

@marctatum84748 หลายเดือนก่อน^{+2}Matthew Tancik (lead author on the Fourier paper) is the same lead author for Neural Radiance Fields (NeRFs), which use Fourier feature mapping (they call it positional encoding in the paper but it is the same thing) to construct 5D continuous scene representations for photorealistic view synthesis. Basically training a 3D scene using a collection of photographs as the ground truth. This is the work that Nvidia then optimized (instant NGP). I’ve been working with nerfs quite a bit and it blows my mind how well they work.

@zulucharlie52448 หลายเดือนก่อน^{+3}Beautiful, thought-provoking content. Thank you.

@akawmv8 หลายเดือนก่อน^{+7}Thanks for this video! This was really interesting, especially when you introduced the Fourier network. I was surprised to see how well it did compared to conventional methods. It was also very interesting seeing the network fit the data in real time.

Sidenote: I love how 3blue1brown kinda inspired a “revolution” in digital math education. It’s amazing and inspiring.

@hyperduality28387 หลายเดือนก่อนFunctions have goals, targets & objectives hence they are teleological, input is dual to output.

Making predictions to track targets, goals & objectives is a syntropic process -- teleological.

Sine is dual to cosine -- the word co means mutual and implies duality.

Teleological physics (syntropy) is dual to non teleological physics (entropy).

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

"Always two there are" -- Yoda.

Subgroups are dual to subfields -- the Galois correspondence.

Duality creates (emergence, synthesis) reality!

@ignessrilians8 หลายเดือนก่อนWow these videos are INSANELY well made and well explained.

You're awesome!

@Geosquare81288 หลายเดือนก่อน^{+1}very well done! its great to see the idea of fourier features explained this way. it's actually quite interesting since similar ideas are actually being applied at the cutting-ish edge in terms of position embeddings.

an interesting example is the NeRF paper, which tries to overfit networks to capture 3d scenes (in a paradigm very similar to the one displayed in the video). they found having a sum of position encoded through harmonics of sinusoids is in some ways the key to getting the best results! position encodings like that are also frequently used in transformer models to distinguish positional information in text :)

@hyperduality28387 หลายเดือนก่อนFunctions have goals, targets & objectives hence they are teleological, input is dual to output.

Making predictions to track targets, goals & objectives is a syntropic process -- teleological.

Sine is dual to cosine -- the word co means mutual and implies duality.

Teleological physics (syntropy) is dual to non teleological physics (entropy).

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

"Always two there are" -- Yoda.

Subgroups are dual to subfields -- the Galois correspondence.

Duality creates (emergence, synthesis) reality!

@justinreusnow8 หลายเดือนก่อน^{+3}This is incredibly well made! Can you explore the topic of convolutional neural networks? Those have always been an enigma to me and i’d like to see the theory behind them with your style.

@PierreH19688 หลายเดือนก่อน^{+3}This is the best short explanation of Neural Nets I ever watched, the visuals are so helpful. Thanks!!!

@hyperduality28387 หลายเดือนก่อนFunctions have goals, targets & objectives hence they are teleological, input is dual to output.

Making predictions to track targets, goals & objectives is a syntropic process -- teleological.

Sine is dual to cosine -- the word co means mutual and implies duality.

Teleological physics (syntropy) is dual to non teleological physics (entropy).

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

"Always two there are" -- Yoda.

Subgroups are dual to subfields -- the Galois correspondence.

Duality creates (emergence, synthesis) reality!

@liuyxpp7 หลายเดือนก่อนThis content is exceptionally inspiring, especially the introduction of Taylor series as a layer of a neural network. I also quite amazed by the Fourier feature layer! I may adopt this approach in my research. Thanks!

@nathangasca96587 หลายเดือนก่อนI had little of computer interpolation classes and will certainly take machine learning classes in the future. These visualisations are amazing! It is really useful to have a visual idea of what the math are doing. You have done a really good introduction, now i want to know more !

@ImNotSusOk8 หลายเดือนก่อน^{+8}Next semester, I'll be taking a machine learning course. I'm excited to actually try to create software which can be trained to do a task, as opposed to just being a passive learner.

@ambition1127 หลายเดือนก่อน^{+28}0:20: 🧠 Neural networks are universal function approximators that can understand, model, and predict the world.

3:42: 🧠 Neurons in a neural network learn their own features and combine them to produce the final output.

7:20: 📚 The video discusses techniques for improving the performance of neural networks.

11:10: 🧠 The video discusses the difficulty of approximating the Mandelbrot function using neural networks and explores other methods for function approximation.

15:24: ✨ The video explains the concept of Fourier series and its application in approximating functions.

18:53: 🌊 Using Fourier features in neural networks can greatly improve performance in high-dimensional problems.

22:55: 📊 The curse of dimensionality can pose challenges in handling high-dimensional inputs and outputs in neural networks, and Fourier features may not always improve performance.

Recap by Tammy AI

@Thoronหลายเดือนก่อนboo 👎

@TK_Prod28 วันที่ผ่านมาHarpa fan I see @@Thoron

@Thoron28 วันที่ผ่านมา@@TK_Prod no idea what that is, I just hate people spamming AI shit everywhere

@TK_Prod28 วันที่ผ่านมา@@Thoron Why is that?

@ChainsawDNA8 หลายเดือนก่อนThis is one of the best introduction videos on the topic. Congratulations on a job well done.

@chopper3lw8 หลายเดือนก่อน^{+2}OMYGERD! TYSM! I've wanted to see someone play with the Fourier series / transform for this purpose for at least 10 years. Great visualization. ( I was intrigued because I played with optical fourier correlation a long time ago and wondered if it was applicable )

@hyperduality28387 หลายเดือนก่อนFunctions have goals, targets & objectives hence they are teleological, input is dual to output.

Making predictions to track targets, goals & objectives is a syntropic process -- teleological.

Sine is dual to cosine -- the word co means mutual and implies duality.

Teleological physics (syntropy) is dual to non teleological physics (entropy).

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

"Always two there are" -- Yoda.

Subgroups are dual to subfields -- the Galois correspondence.

Duality creates (emergence, synthesis) reality!

@himselfe8 หลายเดือนก่อน^{+69}I've been calling current AI "brute force algorithm discovery", but universal function approximation is a lot more concise. Great video! You elucidate the concepts well at a pace which is neither tedious or causing information overload.

@indfnt55908 หลายเดือนก่อน^{+3}I was thinking about the same. But I realized even if our maths can map out the complexity of the universe. To be able to perceive that complexity is a whole other ball game. What if the human mind just isn’t made to understand the universe in its entirety. Or travel millions of miles outside of Earth. Maybe this is where we pass the baton.

@DJWESG18 หลายเดือนก่อน^{+1}I've been calling them social calculators for 15 years.

@whizadree8 หลายเดือนก่อน^{+1}So you want to call it UFAp

@hyperduality28387 หลายเดือนก่อน^{+2}Functions have goals, targets & objectives hence they are teleological, input is dual to output.

Making predictions to track targets, goals & objectives is a syntropic process -- teleological.

Sine is dual to cosine -- the word co means mutual and implies duality.

Teleological physics (syntropy) is dual to non teleological physics (entropy).

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

"Always two there are" -- Yoda.

Subgroups are dual to subfields -- the Galois correspondence.

@hyperduality28386 หลายเดือนก่อน@@majorfur3999 Cause is dual to effect -- causality.

Effects are dual to causes -- retro-causality.

Concepts are dual to percepts -- the mind duality of Immanuel Kant.

The effect of making measurements, observations or perceptions (intuitions) in your mind is to create or synthesize conceptions or ideas (causes) according to Immanuel Kant -- retro-causality!

Are perceptions causes or effects?

If you treat concepts, ideas as causes then these lead to effects or actions!

Enantiodromia is the unconscious opposite, opposame (duality) -- Carl Jung.

Colours are differing aspects or frequencies of the same substance namely energy.

Same is dual to different.

Lacking is dual to non lacking.

Black is the lack of colour and white is all colours (a spectrum) or non lacking.

Electro is dual to magnetic -- electro-magnetic energy is dual, photons, light, colours.

Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought or the principle of equivalence, duality.

Potential energy is dual to kinetic energy -- gravitational energy is dual.

Energy is duality, duality is energy -- all energy is dual hence colours are dual.

Your mind is using duality to create colours.

Concepts are dual to percepts -- the mind duality of Immanuel Kant.

Mathematicians create new concepts all the time from their perceptions, observations or measurements.

Conceptualization or creating new concepts is a syntropic process -- teleological.

Thinking is a syntropic process.

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

The word dual is the correct word to use here.

Sine is dual to cosine or mutual sine -- the word co means mutual and implies duality.

Mutual requires at least two perspectives.

Causality is dual to retro-causality.

Everything in physics is made from energy or duality and this means that your mind is using effects to create causes (concepts) -- a syntropic process, teleological.

Welcome to the 4th law of thermodynamics!

@hyunsunggo8558 หลายเดือนก่อน^{+31}21:18 Fourier features are very much used in neural networks! Often named "positional encoding", it is pretty much always used in transformers(e.g. a large language model) and in NeRFs for learning and rendering 3D scenes with neural networks. Although it usually uses exponential scaling as opposed to linear scaling as you've shown in the video, as points can be represented absolutely fine with exponential scaling as opposed to volumes(superpositions of points).

23:20 I'm assuming you've taken the Fourier features by treating an MNIST image as a 784-dimensional coordinate. I can see how that could hardly help as the pixel values are almost binary and the "gray" pixels don't say much about the image.

@tylerknight998 หลายเดือนก่อนDo Fourier features work well for positional encoding because encoding text positions is a lower dimensional problem?

@hyunsunggo8558 หลายเดือนก่อน^{+2}@tylerknight99 I don't get why you'd assume positional encoding would work better in low dimensions. But if I had to guess why positional encoding improves natural language processing, I think it's because compared to the naive approach of using plain 1-D values, the dot product between the Fourier features of two close positions result in a higher value than it would for positions that are far apart from one another. On the other hand, the dot product of 1-D positions (just plain old multiplication because they're scalars) doesn't have that nice property. I say it because dot product is the fundamental computation in almost every neural network.

@hyperduality28387 หลายเดือนก่อน^{+1}Functions have goals, targets & objectives hence they are teleological, input is dual to output.

Making predictions to track targets, goals & objectives is a syntropic process -- teleological.

Sine is dual to cosine -- the word co means mutual and implies duality.

Teleological physics (syntropy) is dual to non teleological physics (entropy).

Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!

"Always two there are" -- Yoda.

Subgroups are dual to subfields -- the Galois correspondence.

Duality creates (emergence, synthesis) reality!

@TboneIsRogue7 หลายเดือนก่อน^{+1}Man this guy is incredibly talented. Fantastic video! Looking forward to seeing more.

@bikkyghaisai76927 หลายเดือนก่อนThanks for the video. It really helped and inspired me, and I watched it fully through. Most videos are too much details but here it seems I could grasp the ideas for understanding functions and neural networks. Thanks!

@TheNerd4848 หลายเดือนก่อน^{+19}8:17 tanh and sigmoid are actually the same function, just stretched and moved a bit. If you change the e^-x in the sigmoid to e^-2x, you will get the same curve as (tanh+1)/2

@simonramchandani95608 หลายเดือนก่อน^{+1}thats exactly what i thought. why does the exponent of 2 play such a big role for the better output he's getting?

@TheNerd4848 หลายเดือนก่อน^{+1}@@simonramchandani9560 If I were to guess, it's that adding that two in the exponent makes the function tangent to y=x at the origin and tangent to y=x/2+0.5 for the case of the sigmoid, though I don't know why those are better. it may be the case that making the activation function even steeper would produce even better results, such as using e^-6x or something.

I may need to brush up on my coding skills and try this out, unless someone else does.

@viktorivanov59417 หลายเดือนก่อน@@TheNerd484 you have a linear layer before this, so multiplying x by a constant does absolutely nothing

@TheNerd4847 หลายเดือนก่อน@@viktorivanov5941 Having thought about it some more, I agree that it's not the slope in and of itself. What I think might be happening is that the performance is improved by having a narrower range (a step function would be optimal), but the narrower the band between extremes, the harder back propagation is.

@IncendiaHL6 หลายเดือนก่อนThank you! That annoyed me as well.

@mgostIH8 หลายเดือนก่อน^{+3}I enjoyed this! It reminded me of the SIREN paper, which uses sinusoidal approximations to deal with interpolating "natural" data (images, audio, videos, differential equations) and does very well even without augmenting the input.

I think this calls more towards us being able to design architectures that can more easily figure out their own preferred spectrum, but as your later analysis suggests, things may scale very differently than what we expect!

@ibraheemamin5128 หลายเดือนก่อนwhat's the SIREN paper

@mgostIH8 หลายเดือนก่อน@@ibraheemamin512 Implicit Neural Representations with Periodic Activation Functions

@Lovefun5586 หลายเดือนก่อน^{+1}This was really incredible. Amazing work, thanks for sharing.

@jimmygore82148 หลายเดือนก่อนPeople like you are the fruit of humanity! These videos are of great benefit to everyone because being able to understand such complex mathematical topics in a visual manner is the best.

@colonelgraff91988 หลายเดือนก่อน^{+4}FUNCTIONS

DESCRIBE

THE

WORLD

@jafudubrahi5 หลายเดือนก่อน^{+21}Not a math guy? Lmao

@BabelHead5 หลายเดือนก่อนreally great stuff, the clearest explanation and most helpful visualisation i've seen that I can grasp, at least for the key ideas, even without any background in maths since high school, Thanks so much!

@mathpuppy3148 หลายเดือนก่อน^{+1}Great job on this!!! It's interesting, it's educational, and on top of that, it's so entertaining as well!

@Cobblestoned1008 หลายเดือนก่อน^{+7}Have you had a look at this paper? It's fascinating and similar to the fourier features results.

th-cam.com/video/Q2fLWGBeaiI/w-d-xo.html

It's possible to combine both methods to get the best of both worlds.

The siren method enables much faster convergence, while the fourier features allow to capture more high frequency detail like high res images, etc.

The only difference is that it uses sin as activation function instead of ReLu + a clever weight initialization schema that is needed for it to work. But when it does it works extremely well.

@Cobblestoned1008 หลายเดือนก่อน^{+1}I guess if you combine these two methods you will get a much more accurate approximation of the mandelbrot set

@jonatan01i8 หลายเดือนก่อน^{+3}is everything really a function?

isn't a function more like something with which we try to approximate reality?

@beagle9898 หลายเดือนก่อน^{+1}functions might not be what things are, but functions describe everything

@diadetediotedio69188 หลายเดือนก่อน@@beagle989

This is simply untrue, functions can't describe themselves nor the logical frameworks they are inserted, nor the logical inferential and mathematical rules that makes them possible in first place

@jonatan01i8 หลายเดือนก่อน@@beagle989

Suppose I give you a small number (epsilon), for which you can give me a function, such that it's maximum that far away from reality, never worse (never bigger) than epsilon.

Could there be an epsilon, for which it's impossible to find such a function?

@hellopomelo27 หลายเดือนก่อน^{+1}Great video! This video does bring some parallels with compressed sensing. If you have the time, do check out compressed sensing, where we can recover images/information even if said information is massively undersampled, given that is sparse in some basis(fourier basis, for example)

@bean_mhm8 หลายเดือนก่อน^{+2}This is genuinely THE best educational video I've ever watched. Really great job, this is good sweet stuff!

@NathanSMS266 หลายเดือนก่อนThank you for including your code. I just finished a MS in robotics and AI but with how my program was structured there was heavy focus on learning concepts but the deepest exposure I got in a practical sense was evaluating images using classification models. I've been wanting to dive into a project to learn inverse kinematics for a robotic arm I built and I think your code will be a great reference

@mat41518 หลายเดือนก่อน^{+1}I like your videos because it gives me more curiosity about maths and i do different project than i usually do.

@avi128 หลายเดือนก่อน^{+1}Beautiful explanations + beautiful animations + respectable length = Perfection

@wurstkatze8 หลายเดือนก่อนAlways excited to see a new video from you :D

@halihammer8 หลายเดือนก่อนSome of the most beautiful visualizations out there! I love it!

@samthibodeau35112 หลายเดือนก่อนYou have to be one of the greatest math teachers I've ever hear lecture or give a tutorial or course like this! I have so much to say but i'm overwhelmed so I'll just say THANK YOU! Namaste!

@claudiusraphael94236 หลายเดือนก่อนGREAT TENNIS!

Btw. the "exact" first minute is the most on point meme ever and possible personal-best-lap-candidate for speedrunning life. Thanks for sharing!

@OnionKnight54114 วันที่ผ่านมาare you a writer? you speak so goddamn well. i take notes on your videos, and just write down nearly every sentence word for word. amazing.

@maheshkanojiya48582 หลายเดือนก่อนAnother great video, thank you for this gift. Your videos expand the horizon of mind.

@ryant88798 หลายเดือนก่อน^{+1}OMG, I'm blown away by the articulating power of this video. They say a picture is worth a thousand words. This video must worths millions. Awesome job!

@vtrandal3 หลายเดือนก่อนExcellent content and presentation. By the way you are close to 100k subscribers! Nice work!

@windrago7 หลายเดือนก่อนincredibly well done from script to pace to the overall value - instant sub

@heymajoris8 หลายเดือนก่อน^{+1}Thank you so much for making this video, your explanation is really good

@cd-zw2tt7 หลายเดือนก่อนwhat a captivating and incredible visual journey. this sort of stuff needs to win awards

@JonCianci126 หลายเดือนก่อนThis was a game changer for my little experiment in time series forecasting. I brought things right back to basics and just tried to approximate a multiplication function that takes a number and multiples by two. Once I had that benchmark, it gave me a solid starting point to move out from 😎

@korniszon687 หลายเดือนก่อนThis video is a blast! Please keep on producing!

@rj39377 หลายเดือนก่อนTHIS IS GREAT!

very detailed descriiption and visualization of the inner working of the models. Thanks for the effort...

@haaspaas28 หลายเดือนก่อนVery inspirational. I think this is the most intuitive explanation of nn and function approximation that I have heard. Thanks!

@puffinjuice8 หลายเดือนก่อน^{+1}Beautiful video. Very well explained. Thank you!

@smithhoowe4 หลายเดือนก่อนWatching this took me back to CC and learning calculus. I had always figured it was something of a badge of prestige but that it would never really be used, now I feel validated and want to relearn some of what I had forgotten to time. Thank you for this :)

@aditya_a7 หลายเดือนก่อน^{+1}There's just something so soothing watching the network image come into focus with that music

@dsagman8 หลายเดือนก่อน^{+1}so good. helps a lot to understand what is often presented with more mystery.

@lucas_zampar7 หลายเดือนก่อนOne of the best videos on neural nerworks I have ever seen. Great work!

@lightspeedlionหลายเดือนก่อนEvery word well said. I go back visit other video and come back to see the articulation and virtualization, everything really makes sense. Exceptional!

@HeduAI7 หลายเดือนก่อน^{+1}This has got to be the most beautiful video on AI have ever seen! Thank you! Can you please share the names of the software(s) used to create these stunning visuals?

@Dr2crank4u7 หลายเดือนก่อนI'm curious too.

@nileshbarai499926 วันที่ผ่านมาThe video was thoroughly interesting which is great as I have seen some videos that are very dry on the topic. I like how you open it up to the audience to communicate on Discord to provide alternative solutions. I would not have even commented as there was many great comments already but no one added the funny note about ChatGPT in the credits...lol

@ballsof_steel89574 หลายเดือนก่อนSuch a great video. As you went the road of Fourier i was was thinking imagin what could be possible with the Laplace transform. I was surprised that it ended there

@klopf_8 หลายเดือนก่อน^{+1}I already knew a lot of stuff on its own, but never seen it put together and then so nicely. Immediate Sub and like. Looking forward to more videos of you.

@FluffyKitten978 หลายเดือนก่อนVery cool and very well explained. The visuals are fascinating

@ManuArt2568 หลายเดือนก่อน^{+1}This is probably one of the best neural network videos I've seen yet!

@grahamdougherty7 หลายเดือนก่อนI remember studying differential equations and linear algebra when getting my physics degree and being so fascinated by it. It was the hardest class i took. Wish i still remembered this stuff so i could be more than a ML enthusiast. Great video and great explanation!!

@CodePhiles8 หลายเดือนก่อนamazing illustration, thank you and keep forward

@sapienspace88147 หลายเดือนก่อน^{+1}Excellent video, it makes me wonder if, aside from Fourier series, if it is possible that the Mandelbrot set itself, could be used as a function approximator....

@dionbridger59443 หลายเดือนก่อนno

@islamabouelata657527 วันที่ผ่านมาExcellent stuff, thank you very much for making these videos!

@RuiRei3 หลายเดือนก่อนThis is so good that I immediately subscribed to the channel. Absolutely love it ❤

@aaryanmehta65775 หลายเดือนก่อนone of the best videos i've ever seen. as someone who's pursuing his masters in CS, this video gave me so many different insights about what neural networks really are. 🙌

@jmherrera004 หลายเดือนก่อนGreat video! The best I have seen so far for neural networks. Way to go

@kiffbeatz8 หลายเดือนก่อนDude you are very talented, this is good stuff!!

@alejrandom65928 หลายเดือนก่อน^{+1}This is one of the clearest explanations I've seen

@KeiJazz8 หลายเดือนก่อนVery nice video. I'm curious to see the follow up on the Mandelbrot set challenge! I was just remembering some signal processing courses from the 1990's... I think it was the Wavelet transform that works well over a wide range of magnitude. It seems to be a great candidate to try on fractals, for anybody who has more practical programming / AI skills than I do ;-) .

@jedleton8 หลายเดือนก่อน^{+1}Astonishingly well explained

@angstrom10583 หลายเดือนก่อนThis is quite good. I've worked with NN's since the 1980's, it's my career and I'm an inventor of NN tech starting back in the 90's and still today. I'm just trying to create some "street cred" and this video is very good. Well done! "TanH just seems to work better" is absolutely correct and just the way you should say it. Nice!

@0xD4rky3 หลายเดือนก่อนI, being new to this field, was only accoustomed to work with simple neural networks. After watching this video, my mind just blew off realizing the concept of fourier transformation in NN. Its just pure gold. Tbh, I wasn't expecting a NN to perform such well on Mandelbrot function. Would love to dive deep into fourier transformation in NN. Thanks for the enlightment!

@arsalanzabeeb64677 หลายเดือนก่อนjust amazing , thank you so much for putting so much efforts . hatsoff to you