I can't help but think of orthogonal frequency-division multiplexing and it's use in encoding data on multiple carrier frequencies, and it makes me wonder what other parallels we will discover between digital transmission technology for cross-domain stuff like this.
Trigonometric polynomials are also polynomials. And linear spaces are all "the same". That is what the definition is for. Even the transpose-mapping is linear.
I have this strange sensation that I can't put into words that somehow we are on the brink of unveiling an entirely new paradigm of AIs or perhaps even of combining AI with classical algorithms in a way to rapidly iterate between each other (and sensor data) that will instantly 10x or 100x current capabilities.
I think part of it is the feeling of false understanding that comes from using llms regularly. They let you operate at a higher conceptual level, and they paper over enough of the actual details that your conceptual model might not actually be correct.
I'm a mechanical engineer by training, and have similar vibes with the similarities I see between llm training and metallurgy. I could probably put together a formal concept for these vibes at this point, but is there actually a "there" there? I have no idea. And it would take me years to actually dive in and learn everything to gain the deep understanding that would be required to know if I'm just experiencing my own brand of AI psychosis or not.
> that will instantly 10x or 100x current capabilities.
In the 1920s we had legions of very smart, highly trained (arguably better trained in mathematics) basically chucking relays and vacuum tubes together with reckless abandon to build the most valuable and complicated systems mankind had ever come up with (telephony, radio, radar, etc). They had no idea how they worked and only ad-hoc rules of thumb to construct them.
It took the insight of a handful of these people both in and outside of industry to formalize the theory of operation of most of what people were already building and then use that theory to establish formal design practices.
The people before these theories were realized were exceptionally smart and good at what they did, it's just they didn't have better design tools to reason about the things they were building.
And once they had those tools they didn't 10x or 100x overnight.
I feel like this is an inverted interpretation? Transmission tech uses those methods because the math shows the desired properties.
Linear algebra is used everywhere, orthogonalization, SVD, eigenvalues etc are valuable because the resulting properties are very useful in many places.
Yea, I could have used a better word choice. I was thinking about the domains here in the generalized sense such as signal processing and wireless communication being applicable to the domain of artificial intelligence. In reality, you are correct that it's all tied together under of domain of applied maths or computer science.
I suspect with "orthogonalization" they mean to find vectors that form an orthogonal bases (same subspace) for the vectors in the source matrix.
I wonder what would be the result if they used a matrix that is orthogonal and closest to the source matrix.
Usually one uses the Frobenius norm (root of the sum of all squared matrix entries). Maybe, one could even try another norm that gives a sparser matrix.
The Newton-Schulz iteration they use approximates setting all singular values of the matrix to 1. That computes the nearest orthogonal matrix under the Frobenius norm.
3D graphics and kinematics people dodge the need for periodic orthonormalization by using quaternions. When they need a rotation matrix, they create it on demand rather than having to maintain it incrementally.
I wonder if there's a similar shortcut representation that we will eventually realize we should be using for ML. I suppose if there is one, it won't have native GPU support, so no one will bother looking for it.
You can take the output of the matrix LSTM, which is going to be matrix for each token, and compute the SVD. To get better storage, we want U and V to be the same for all tokens, so that we can operate on the diagonal S matrix. But LSTM is likely highly nonlinear, U and V will be vastly different for different tokens.
I don’t know AI, but, weight matrices aren’t square in general, right? My first guess for something like this would be to take the SVD instead, since you can always do that, but I’m sure that’s been tried already.
I wouldn't say that making the matrix diagonal in some basis is some further step.
If we have an singular value decomposition, M=USV^*, the columns of U are linearly independent they are a basis for the space M maps things into, and the columns of V are linearly independent then it's a basis for the space it maps things from, and [M]_{BB'} = S.
Now I’m wondering what is the eigenspace of an LLM? If I take a set of LLM’s with the same number of parameters, then what are the eigenvectors? Do they have different personalities?
Neural networks are non-linear, so I think you wouldn’t be able to compute typical eigenvalues. You could compute the eigenvalues and/or singular of the individual weight matrices (I’m sure this has been studied). SVDs are very conventional for making low-rank approximations, so it must have been studied.
The concept of nonlinear eigenvalues exists, but it is a bit more exotic.
Here is a pytorch optimizer that can maintain a matrix as orthogonal throughout optimization:
https://github.com/adrianjav/pogo — POGO: A Proximal One-step Geometric Orthoptimizer
https://arxiv.org/abs/2602.14656 — An Embarrassingly Simple Way to Optimize Orthogonal Matrices at Scale; Adrián Javaloy, Antonio Vergari
That's useful, but wouldn't help with this particular experiment because they orthogonalize activations, not weights
I can't help but think of orthogonal frequency-division multiplexing and it's use in encoding data on multiple carrier frequencies, and it makes me wonder what other parallels we will discover between digital transmission technology for cross-domain stuff like this.
Not even cross-domain. (Nor cross-co-domain.)
Trigonometric polynomials are also polynomials. And linear spaces are all "the same". That is what the definition is for. Even the transpose-mapping is linear.
I have this strange sensation that I can't put into words that somehow we are on the brink of unveiling an entirely new paradigm of AIs or perhaps even of combining AI with classical algorithms in a way to rapidly iterate between each other (and sensor data) that will instantly 10x or 100x current capabilities.
Anyone else feel this?
I think part of it is the feeling of false understanding that comes from using llms regularly. They let you operate at a higher conceptual level, and they paper over enough of the actual details that your conceptual model might not actually be correct.
I'm a mechanical engineer by training, and have similar vibes with the similarities I see between llm training and metallurgy. I could probably put together a formal concept for these vibes at this point, but is there actually a "there" there? I have no idea. And it would take me years to actually dive in and learn everything to gain the deep understanding that would be required to know if I'm just experiencing my own brand of AI psychosis or not.
It's a brave new world, that's for sure.
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> that will instantly 10x or 100x current capabilities.
In the 1920s we had legions of very smart, highly trained (arguably better trained in mathematics) basically chucking relays and vacuum tubes together with reckless abandon to build the most valuable and complicated systems mankind had ever come up with (telephony, radio, radar, etc). They had no idea how they worked and only ad-hoc rules of thumb to construct them.
It took the insight of a handful of these people both in and outside of industry to formalize the theory of operation of most of what people were already building and then use that theory to establish formal design practices.
The people before these theories were realized were exceptionally smart and good at what they did, it's just they didn't have better design tools to reason about the things they were building.
And once they had those tools they didn't 10x or 100x overnight.
no. we're approach a sigmoid. AI is bloated carcass and we're tweaking out the size of the models and speed they'll run on smaller hardware.
I think to feel what you're feeling, you've bought into "all we need is more context". I think evolution demonstrates that's not really true.
8 replies →
I feel like this is an inverted interpretation? Transmission tech uses those methods because the math shows the desired properties.
Linear algebra is used everywhere, orthogonalization, SVD, eigenvalues etc are valuable because the resulting properties are very useful in many places.
Yea, I could have used a better word choice. I was thinking about the domains here in the generalized sense such as signal processing and wireless communication being applicable to the domain of artificial intelligence. In reality, you are correct that it's all tied together under of domain of applied maths or computer science.
I suspect with "orthogonalization" they mean to find vectors that form an orthogonal bases (same subspace) for the vectors in the source matrix.
I wonder what would be the result if they used a matrix that is orthogonal and closest to the source matrix. Usually one uses the Frobenius norm (root of the sum of all squared matrix entries). Maybe, one could even try another norm that gives a sparser matrix.
The Newton-Schulz iteration they use approximates setting all singular values of the matrix to 1. That computes the nearest orthogonal matrix under the Frobenius norm.
Interesting, thanks!
3D graphics and kinematics people dodge the need for periodic orthonormalization by using quaternions. When they need a rotation matrix, they create it on demand rather than having to maintain it incrementally.
I wonder if there's a similar shortcut representation that we will eventually realize we should be using for ML. I suppose if there is one, it won't have native GPU support, so no one will bother looking for it.
If it can be made orthogonal, can you go a step further and diagonalize it? The storage and performance improvement from that would be huge.
You can take the output of the matrix LSTM, which is going to be matrix for each token, and compute the SVD. To get better storage, we want U and V to be the same for all tokens, so that we can operate on the diagonal S matrix. But LSTM is likely highly nonlinear, U and V will be vastly different for different tokens.
I don’t know AI, but, weight matrices aren’t square in general, right? My first guess for something like this would be to take the SVD instead, since you can always do that, but I’m sure that’s been tried already.
But orthogonal matrices are square.
1 reply →
I wouldn't say that making the matrix diagonal in some basis is some further step.
If we have an singular value decomposition, M=USV^*, the columns of U are linearly independent they are a basis for the space M maps things into, and the columns of V are linearly independent then it's a basis for the space it maps things from, and [M]_{BB'} = S.
Now I’m wondering what is the eigenspace of an LLM? If I take a set of LLM’s with the same number of parameters, then what are the eigenvectors? Do they have different personalities?
Neural networks are non-linear, so I think you wouldn’t be able to compute typical eigenvalues. You could compute the eigenvalues and/or singular of the individual weight matrices (I’m sure this has been studied). SVDs are very conventional for making low-rank approximations, so it must have been studied.
The concept of nonlinear eigenvalues exists, but it is a bit more exotic.
I saw a presentation about this in 2022.
Someone found a way to get "something like" a tri-diagonal matrix that was equivalent to the LLM they were studying in 2022.
Apologies for being informal and hand-wavey. Been a long time and I probably forgot a few important points.
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