KAN: Kolmogorov-Arnold Networks
ฝัง
- เผยแพร่เมื่อ 22 พ.ค. 2024
- Paper: arxiv.org/abs/2404.19756
Spline Video: • B-Splines
My notes: drive.google.com/file/d/1twcI...
00:00 Intro
00:45 MLPs and Intuition
05:12 Splines
19:02 KAN Formulation
28:00 Potential Downsides to KANs
32:09 Results
Thank you so much for such great explanation!
via Pi
"The Kolmogorov-Arnold Network (KAN) is a fascinating type of neural network that takes a unique approach to learning compared to traditional multilayer perceptrons (MLPs). Here are some key points about KANs:
* KANs have learnable activation functions on edges (or "weights"), while MLPs have fixed activation functions on nodes ("neurons").
* This design choice is based on the Kolmogorov-Arnold representation theorem, which states that any continuous multivariate function can be represented as a superposition of continuous functions of a single variable.
* KANs can achieve better accuracy with fewer parameters compared to MLPs.
* They also offer advantages in terms of model accuracy and interpretability.
* The concept of KANs was introduced in the paper "KAN: Kolmogorov-Arnold Networks" and has since been implemented in various libraries and frameworks, including the PyKAN Python package.
Overall, KANs provide an intriguing alternative to MLPs, with their theoretical foundations in the Kolmogorov-Arnold representation theorem and practical advantages in terms of model performance and interpretability."
Thanks for the explanation Grabriel!
Thank you for this clear and comprehensive explanation. Also your insights about the downsides of the model was quite accurate and helpful.
Wow! The extractability of solutions really catches my eye.
what surprises me is that nobody tried to used splines before (they probably did).. seems like an obvious way to approximate functions
they have, but they would solve for one symbolic set of splines, instead of continously learning them. The closest thing to this is physics informed neural networks
@@slavrine Hmmm, no, Kolmogorov formula also used to design Neural Networks. Just search for Kolmogorov Spline Netwotks. People generally like to exaggerate things, to reinvent the wheel. Nothing very special for KAN.
Well, MoE in general aren't so far from splines.
@@user-uc2qy1ff2z you mean Mixture of Experts?
There are three successful applications of splines before this article all published. Find references on the articles on my video, I can't post links, system filters them. Splines is representation of the functions, training is completely different subject. We train models in a completely different way, MIT work is different from our, but splines were used KAN before.
Good explanation!
13:13 btw that order n Bazier curve recursion is just a linear interpolation of two order (n-1) bazier curves.
Thank you very much. You explained very clear
Great explanation
nice explanation
thank you for the explanation
Great explanation! However, instead of "activation matrix," you should probably say "activation function matrix." An activation is the output of an activation function, which is not what this matrix contains.
Summary: while current RNs work as a universal approximator (probability machines), Kolmogorov networks allow you to leverage a more deterministic function.
End point: the axiom of the universal approximator as a pillar of current networks could be replaced by networks of variable geometry. Think about the difference between searching or decoding information in 1 dimension, and now adding 2 dimensions. A data point vs a line of data.
Like go from 480p to 8KHD resolution.
Someone who made the acronym knew that it will remind us on KAM - Kolmogorov-Arnold-Moser.
Greta video! Any chance you can cover xlstms? It just released an hr ago lol
I can probably cover that one!
Thanks for the greatest video. I have some questions, I can't understand how that algorithm update the c (learnable terms)? back propagation? or something else
It is tree of functions. All functions are presented via splines. You can initialize and compute target Z. It is compared to actual and you have DELTA Z to propagate. Since all functions are splines, they have derivatives and you can implement one of many descents, such as Newton or gradient or mixture. They refereed theirs in article. Imagine z = f [ g(x) + q(y) ] if f, g, q are splines, x, y arguments and you need to update splines to modify z.
"Forest has been lost behind trees"
I haven't ever heard something more true
Your forest comment seems quite random, I'll talk about it to my nearest neighbors
Thank you for this very helpful video.
One thing that I’m curious about is whether the assumption of a spline relationship is a valid assumption for all use cases or not.
No. It is only best way to approximate functions listed in articles. The observation of real systems have errors. They are not small, 10% is usual case. There are uncertainties, such as not accounted features, not observed. All that makes accuracies of 0.1% absurd. I use piecewise linear models and it more than enough. I even used piecewise constant and it worked well on approximate data. There is no such thing as exact solution when you train model to detect credit card fraud or predict stock price.
Will be interesting to see if splines generalize to all methods like MLPs. They seem to be particularly good as smaller problems, but I'm guessing they are going to run into problems with large models as splines have more local behavior.
There’s something to be said about General models always seeming to be (shitty) POCs that people end up fine tuning so that it preforms better on specialized tasks.
The idea of combining the splines for the activation function is in any way similar to RBF networks ?
I've actually seen someone use RBF to approximate B-splines for KANs!
github.com/ZiyaoLi/fast-kan
MLP can't learn this, but their method KAN =)
How about liquid neural network ?
Gabriel if only you could start writing in a readable way some random postgrad at mit would solve the agi by now. love the vid, big thanks
working on it
It is not as complex as look like. I have working code piecewise linear code without using any library modeling KAN with 500 lines. My coauthor published spline version (different from MIT), which also not using any library about same length in MATLAB. MIT code is published. I found module, navigated code, everything is opened. Where the problem?
u so smart, we so dumb
@@leastofyourconcerns4615 install their library, find by command location of the module, navigate to code and find all there, it is 500 lines python code using splines and one version of Newton descent, all is taught in colleges and students practice that in labs.
😂
It looks like multi layer version of GAMs to me (Generalized Additive Models), which has problem when extrapolate
All models return garbage when used outside training. You only can introduce uncertainty level and make model return confidence interval.
@@andrewpolar1685 Fair point
Thanks for the video
what is the video about splines?
m.th-cam.com/video/qhQrRCJ-mVg/w-d-xo.html
Hello everyone, can you please tell me if I understood correctly that splines are built based on the input data, then the neuron calculates the activation function using these splines (which then go as weights into the matrix) and the result of the calculation is passed on to the other layers?
No, it is tree of functions. Lower level have features as args and pass function values to upper level as args. All functions are presented by splines and training is tuning them for data by related to Newton method.
I also did KAN, forgot to tell, you can find my video and code. There are also other KANs, I added few references. It is not as new as might look.
@@andrewpolar1685 Okay, but I would like to understand where the splines come from? This is the only thing I do not know, I will be glad if you explain it to me in more detail
@@andrewpolar1685 By the way, where can I find your video and code specifically?
@@algoritm3034 that is another video with over 4000 views. find yourself, I can't publish link, system treat it as ad and remove comment, in description you can find links to articles and code locations
I published c# KAN, which shows about same performance and accuracy, it does not use any 3rd party library. Interested?
Same performance and accuracy as an MLP? Sounds cool! Are you open sourcing the code?
I am not going to lie. I am unsure about the authors motivation of using a B-Spline and paper as a whole. The authors simply want to replace the linear combination of inputs with a non linear combination. So why not just replace a linear layer(with activations) of a pytorch network with a "recursive linear layer", where every weight of the linear layer is simply another linear layer.
Much simpler to implement, just a small 5-6 line for loop, and also fits within neural network literature perfectly. No need to prove, with regards to Kolmogorv-Arnold theorems, because our formulation is simply another neural network. Whats more this relieves the model from performance issues that the authors express.
What am I missing ?
Thats actually kind of what Transformers are doing: Key and Query vector decide together, how much of the value vector goes out. Runtime-Learned Weight sort of
You missed to share the code 🧏♂️
Hm, suppose you wanted to learn some function in a way that, using some kind of constraint or prior, would be expected to generalize to regions far from a ball bounding the training data (so, not just, “far from any point in the training data” as a consequence of the input data being high-dimensional, but far from a ball bounding the data.
(Not like with language modeling or whatever, where the input values should have bounded norm after embedding)
How might one impose constraints that would still apply outside that region?
Like, I’m imagining using some kind of algebraic property of the non-linear activation function, in order to express some bound on something or other, and like, put that into a loss?
Like, for ReLU, one can say some nice things about it, as, for positive inputs it is just the identity, of course.
And MLP network using RELU can in principle be expressed as a piecewise-affine function of the input, with finitely many regions (though the number of regions may be exponential in the number of “neurons” I think). The number of these regions is much too large for this to be practical, but, seeing as there are only finitely many of them,
the union over the ones of those regions that are finite, is finite. I wonder how large the number of infinite such regions is.
Well, I imagine that this depends more on the dimension of the first layer and the last layer?
Each of these regions is convex, and is defined by a system of finitely many inequalities.
Uh, any infinite region, there will be a ray which it contains.
(These regions are subsets of the space of possible inputs to the first layer, to be clear)
If we associate to each infinite region, the set of directions such that that region contains a ray with that direction,
well, I think that the different regions sets of directions will only overlap at the boundaries.
For any direction in the interior of the set of associated directions, I think that moving far enough in that direction will always land one in that region?
If there are few enough of them, maybe the loss could have terms based on the affine behavior in these regions, to encourage them to have particular properties?
Like maybe invertibility or something? Idk.
why don't you think it won't scale?
The authors mainly show that the spline assumption works at small scale, but for large scale splines may run into issues such as with the curse of dimensionality. Also KANs lose interoperability at scale.
quality content
I was thinking the same thats its gonna be time complex as it sounds, 😅 but they wrote initially that it requires fewer parameters &faster inference.
It's probably possible to create a more optimized version of their code with some type of Cuda kernel since they say they didn't optimize it very much. Recent implementations are looking pretty good.
Nice . Real AI channel. For engineers and not for blogwriters script kiddos
You completely lost me on the splines, not necessarily your fault :)
I went to ChatGPT to explain it to me as a 5 year old and I did haha
@@ProAndProudMLG just a core method that has existed for 70+ years
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Its MLP in disguise
Im sick of neral networks, we need to shut them down