The universal weight subspace hypothesis
arxiv.org
arxiv.org
I visited one of the models they reference and huggingface says it has malware in it: https://huggingface.co/lucascruz/CheXpert-ViT-U-MultiClass
I see now that they did one experiment with trained from scratch models. They trained five Resnet-50s on five disjoint datasets of natural images, most quite small. And IIUC they were able to, without further training, combine them into one "universal" model that can be adapted to have only somewhat worse performance on any one of the five datasets (actually one of them is pretty bad) using only ~35 adaptation parameters. Which is kind of cool I guess but I also don't find it that surprising?
I don't expect that you'd get the same finding at large scale in LLMs trained from scratch on disjoint and dissimilar data with different optimizers etc. I would find that surprising. But it would be very expensive to do that experiment so I understand why they weren't able to.
The LLMs are finetuned on very disjoint data. I checked some are on Chinese and other are for Math. The pretrained model provides a good initialization. I'm convinced.
1) "pertaining"
2) architecture
2) the architecture and training methods matter. As a simple scenario to make things a bit easier to understand let's say we have two models with identical architectures and we'll use identical training methods (e.g. optimizer, learning rate, all that jazz) but learn on different data. Also to help so you can even reproduce this on your own let's train one on MNIST (numbers) and the other in FashionMNIST (clothing).
Do you expect these models to have similar latent spaces? You should! This is because despite the data being very different visually there are tons of implicit information that's shared (this is a big reason we do tuning in the first place!). One of the most obvious things you'll see is subnetworks that do edge detection (there's a famous paper showing this with convolutions but transformers do this too, just in a bit different way). The more similar the data (orders shouldn't matter too much with modern training methods but it definitely influences things) the more similar this will be too. So if we trained on LAION we should expect it to do really well on ImageNet because even if there aren't identical images (there are some) there are the same classes (even if labels are different)[0].
If you think a bit here you'll actually realize that some of this will happen even if you change architectures because some principles are the same. Where the architecture similarity and training similarity really help is that they bias features being learned at the same rate and in the same place. But this idea is also why you can distill between different architectures, not just by passing the final output but even using intermediate information.
To help, remember that these models converge. Accuracy jumps a lot in the beginning then slows. For example you might get 70% accuracy in a few epochs but need a few hundred to get to 90% (example numbers). So ask yourself "what's being learned first and why?" A lot will make more sense if you do this.
[0] I have a whole rant on the indirect of saying "zero shot" on ImageNet (or COCO) when trained in things like LAION or JFT. It's not zero shot because ImageNet is in distribution! We wouldn't say "we zero shotted the test set" smh
In any case, my impression is that this is not immediately more useful than a LoRA (and is probably not intended to be), but is maybe an avenue for further research.
The ResNet results hold from scratch because strict local constraints (e.g., 3x3 convolutions) force the emergence of fundamental signal-processing features (Gabor/Laplacian filters) regardless of the dataset. The architecture itself enforces the subspace.
The Transformer/ViT results rely on fine-tunes because of permutation symmetry. If you trained two ViTs from scratch, "Attention Head 4" in Model A might be functionally identical to "Head 7" in Model B, but mathematically orthogonal.
Because the authors' method (SVD) lacks a neuron-alignment step, scratch-trained ViTs would not look aligned. They had to use pre-trained models to ensure the weights shared a coordinate system. Effectively, I think that they proved that CNNs converge due to it's arch, but for Transformers, they mostly just confirmed that fine-tuning doesn't drift far from the parent model.
And this critique is likely not aimed at academics so much as the systems and incentives of academia. This is partially on the parties managing grants (caring much more about impact and visibility than actually moving science forwards, which means everyone is scrounging for or lying about low hanging fruit). It is partially on those who set (or rather maintain) the culture at academic institutions of gathering clout by getting 'impactful' publications. And those who manage journals also share blame, by trying to defend their moat, very much hamming up "high impact", and aggressively rent-seeking.
Perhaps we need to revisit the concept and have a narrow abstract and a lay abstract, given how niche science has become.
The ViT models are never really trained from scratch - they are always finetuned as they require large amounts of data to converge nicely. The pretraining just provides a nice initialization. Why would one expect two ViT's finetuned on two different things - image and text classification end up in the same subspace as they show? I think this is groundbreaking.
I don't really agree with the drift far from the parent model idea. I think they drift pretty far in terms of their norms. Even the small LoRA adapters drift pretty far from the base model.
Given 500 fine tune datasets, we could expect the 500 drag directions to span a 500 dimensional space. After all, 500 random vectors in a high dimensional space are likely to be mutually orthogonal.
The paper shows, however, that the 500 drag directions live in a ~40 dimensional subspace.
Another way to say it is that you can compress fine tune weights into a vector of 40 floats.
Imagine if, one day, fine tunes on huggingface were not measured in gigabytes, megabytes, or even kilobytes. Suppose you started to see listings like 160 bytes. Would that be surprising?
I’m leaving out the detail that the basis direction vectors themselves would have to be on your machine and each basis direction is as big as the model itself. And I’m also taking for granted that the subspace dimension will not increase as the number of fine tune datasets increases.
I agree that the authors decision to use random models on hugging face is unfortunate. I’m hopeful that this paper will inspire follow up works that train large models from scratch.
They're using SVD to throw away almost all of the "new information" and apparently getting solid results anyhow. Which of course raises interesting questions if replicable. The code doesn't seem to have been released yet though.
> we selected five additional, previously unseen pretrained ViT models for which we had access to evaluation data. These models, considered out-of-domain relative to the initial set, had all their weights reconstructed by projecting onto the identified 16-dimensional universal subspace. We then assessed their classification accuracy and found no significant drop in performance
> we can replace these 500 ViT models with a single Universal Subspace model. Ignoring the task-variable first and last layer [...] we observe a requirement of 100 × less memory, and these savings are prone to increase as the number of trained models increases. We note that we are, to the best of our knowledge, the first work, to be able to merge 500 (and theoretically more) Vision Transformer into a single universal subspace model. This result implies that hundreds of ViTs can be represented using a single subspace model
So, they found an underlying commonality among the post-training structures in 50 LLaMA3-8B models, 177 GPT-2 models, and 8 Flan-T5 models; and, they demonstrated that the commonality could in every case be substituted for those in the original models with no loss of function; and noted that they seem to be the first to discover this.
For a tech analogy, imagine if you found a bzip2 dictionary that reduced the size of every file compressed by 99%, because that dictionary turns out to be uniformly helpful for all files. You would immediately open a pull request to bzip2 to have the dictionary built-in, because it would save everyone billions of CPU hours. [*]
[*] Except instead of 'bzip2 dictionary' (strings of bytes), they use the term 'weight subspace' (analogy not included here[**]) — and, 'file compression' hours becomes 'model training' hours. It's just an analogy.
[**] 'Hilbert subspaces' is just incorrect enough to be worth appending as a footnote[***].
[***] As a second footnote.
Could someone clarify what this means in practice? If there is a 'commonality' why would substituting it do anything? Like if there's some subset of weights X found in all these models, how would substituting X with X be useful?
I see how this could be useful in principle (and obviously it's very interesting), but not clear on how it works in practice. Could you e.g. train new models with that weight subset initialized to this universal set? And how 'universal' is it? Just for like like models of certain sizes and architectures, or in some way more durable than that?
No matter how large X is, one copy of X baked into the OS / into the silicon / into the GPU / into CUDA, is less than 50+177+8 copies of X baked into every single model. Would that permit future models to be shipped with #include <X.model> as line 1? How much space would that save us? Could X.model be baked into chip silicon so that we can just take it for granted as we would the mathlib constant "PI"? Can we hardware-accelerate the X.model component of these models more than we can a generic model, if X proves to be a 'mathematical' constant?
Given a common X, theoretically, training for models could now start from X rather than from 0. The cost of developing X could be brutal; we've never known to measure it before. Thousands of dollars of GPU per complete training at minimum? Between Google, Meta, Apple, and ChatGPT, the world has probably spent a billion dollars recalculating X a million times. In theory, they probably would have spent another billion dollars over the next year calculating X from scratch. Perhaps now they won't have to?
We don't have a lot of "in practice" experience here yet, because this was first published 4 days ago, and so that's why I'm suggesting possible, plausible, ways this could help us in the future. Perhaps the authors are mistaken, or perhaps I'm mistaken, or perhaps we'll find that the human brain has X in it too. As someone who truly loathes today's "AI", and in an alternate timeline would have completed a dual-major CompSci/NeuralNet degree in ~2004, I'm extremely excited to have read this paper, and to consider what future discoveries and optimizations could result from it.
EDIT:
Imagine if you had to calculate 3.14159 from basic principles every single time you wanted to use pi in your program. Draw a circle to the buffer, measure it, divide it, increase the memory usage of your buffer and resolution of your circle if necessary to get a higher precision pi. Eventually you want pi to a billion digits, so every time your program starts, you calculate pi from scratch to a billion digits. Then, someday, someone realizes that we've all been independently calculating the exact same mathematical constant! Someone publishes Pi: An Encyclopedia (Volume 1 of ∞). It becomes inconceivably easier to render cones and spheres in computer graphics, suddenly! And then someone invents radians, because now that we can map 0..360° onto 0..τ, and no one predicted radians at all but it's incredibly obvious in hindsight.
We take for granted knowledge of things like Pi, but there was a time when we did not know it existed at all. And then for a long time it was 3. And then someone realized the underlying commonality of every circle and defined it plainly, and now we have Pi Day, and Tau Day, because not only do we know it exists, but we can argue about it. How cool is that! So if someone has discovered a new 'constant', then that's always a day of celebration in my book, because it means that we're about to see not only things we consider "possible, but difficult" to instead be "so easy that we celebrate their existence with a holiday", but also things that we could never have remotely dreamed of before we knew that X existed at all.
(In less tangible analogies, see also: postfix notation which was repeatedly invented for decades (by e.g. Dijkstra) as a programming advance, or the movie "Arrival" (2019) as a linguistic advance, or the BLIT Parrot (don't look!) as a biological advance. :)
1. As John Napier, who freely, generously, gifted his `Mirifici' for the benefit of all.
2. Here we go, patent trolls, have at it. OpenAI, et al burning midnight oil to grab as much real estate on this to erase any (even future?) debt stress, deprecating the AGI Philospher's Stone to first owning everything conceivable from a new miraculous `my precious' ring, not `open', closed.
Key point being: the parameters might be picked off a lower dimensional manifold (in weight space), but this doesn't imply that lower-rank activation space operators will be found. So translation to inference-time isn't clear.
Let's say you finetune a Mistral-7B. Now, there are hundreds of other fine-tuned Mistral-7B's, which means it's easy to find the universal subspace U of the weights of all these models combined. You can then decompose the weights of your specific model using U and a coefficient matrix C specific to your model. Then you can convert any operation of the type `out=Wh` to `out=U(C*x)` Both U and C are much smaller dimension that W and so the number of matrix operations as well as the memory required is drastically lower.
or is it just that 16 was arbitrarily chosen by them as close enough to the actual minimal number of dimensions necessary?
I also don’t understand what they write under figure 2, since resnet50 has 50 layers, not 31.
- Training costs: We might discover these universal subspaces without training thousands of models
- Storage requirements: Models could share common subspace representations
You can show for example that siamese encoders for time-series, with MSE loss on similarity, without a decoder, will converge to the the same latent space up to orthogonal transformations (as MSE is kinda like gaussian prior which doesn’t distinguish between different rotations).
Similarly I would expect that transformers trained on the same loss function for predicting the next word, if the data is at all similar (like human language), would converge to approx the same space, up to some, likely linear, transformations. And to represent that same space probably weights are similar, too. Weights in general seem to occupy low-dimensional spaces.
All in all, I don’t think this is that surprising, and I think the theoretical angle should be (have been?) to find mathematical proofs like this paper https://openreview.net/forum?id=ONfWFluZBI
They also have a previous paper (”CEBRA”) published in Nature with similar results.
For CNNs, the 'Universal Subspace' is simply the strong inductive bias (locality) forcing filters into standard signal processing shapes (Laplacian/Gabor) regardless of the data. Since CNNs are just a constrained subset of operations, this convergence is not that surprising.
For Transformers, which lack these local constraints, the authors had to rely on fine-tuning (shared initialization) to find a subspace. This confirms that 'Universality' here is really just a mix of CNN geometric constraints and the stability of pre-training, rather than a discovered intrinsic property of learning.
Here's a very cool analogy from GPT 5.1 which hits the nail in the head in explaining the role of subspace in learning new tasks by analogy with 3d graphics.
Think of 3D character animation rigs:
• The mesh has millions of vertices (11M weights).
• Expressions are controlled via:
• “smile”
• “frown”
• “blink”
Each expression is just:
mesh += α_i \* basis_expression_i
Hundreds of coefficients modify millions of coordinates.Are there novel tasks? Inside the limits of physics, tasks are finite, and most of them are pointless. One can certainly entertain tasks that transcend physics, but that isn't necessary if one merely wants an immortal and indomitable electronic god.
What I don’t get is what is meant by a universal shared subspace, because there is some invariance regarding the specific values in weights and the directions of vectors in the model. For instance, if you were doing matrix multiplication with a weight tensor, you could swap two rows/columns (depending on the order of multiplication) and all that would do is swap two values in the resulting product, and whatever uses that output could undo the effects of the swap so the whole model has identical behavior, yet you’ve changed the direction of the principal components. There can’t be fully independently trained models that share the exact subspace directions for analogous weight tensors because of that.
As a really stupid example: the sets of integers less than 2, 8, 5, and 30 can all be embedded in the set of integers less than 50, but that doesn’t require that the set of integer is finite. You can always get a bigger one that embeds the smaller.
If all need just 16 dimensions if we ever make one that needs 17 we know we are making progress instead of running in circles.
Apparently it doesn't at least not in our models with our training applied to our tasks.
So if we expand one of those 3 things and notice that 17-th vector makes a difference then we are having progress.
It's known that large neural networks can even memorize random data. The number of random datasets is unfathomably large, and the weight space of neural networks trained on random data would probably not live in a low dimensional subspace.
It's only the interesting-to-human datasets, as far as I know, that drive the neural network weights to a low dimensional subspace.
it's interesting that this paper was discovered by JHU, not some groups from OAI/Google/Apple, considering that the latter probably have spent 1000x more resource on "rediscovering"
Similarly I would expect that transformers trained on the same loss function for predicting the next word, if the data is at all similar (like human language), would converge to approx the same space. And to represent that same space probably weights are similar, too. Weights in general seem to occupy low-dimensional spaces.
All in all, I don’t think this is that surprising, and I think the theoretical angle should be (have been?) to find mathematical proofs like this paper https://openreview.net/forum?id=ONfWFluZBI
But I always want Genetic Algorithms to show up in any discussion about neural networks...
I just stumbled upon a very nice description of the core of it, right here: https://www.youtube.com/watch?v=AyzOUbkUf3M&t=133s
Almost all talks by Geoffrey Hinton (left side on https://www.cs.toronto.edu/~hinton/) are in very approachable if you're passingly familiar with some ML.
On a basic level, it's kind of like if you had a calculation for aiming a cannon, and someone was giving you targets to shoot at 1 by 1, and each time you miss the target, they tell you how much you missed by and what direction. You could tweak your calculation each time, and it should get more accurate if you do it right.
Backpropagation is based on a mathematical solution for how exactly you make those tweaks, taking advantage of some calculus. If you're comfortable with calculus you can probs understand it. If not, you might have some background knowledge to pick up first.
For example, evolving program tapes is not something you can back propagate. Having a symbolic, procedural representation of something as effective as ChatGPT currently is would be a holy grail in many contexts.
Then I had a multi-layer network - I don't remember how many layers.
Then I was using a simple Genetic Algorithm to try to set the weights.
Essentially, it was like breeding up a winner for the snake game - but you always know where all of the food is, and the ant always started in the same square. I was trying to maximize the score for how many food items the ant would eventually find.
In retrospect, it was pretty stupid. Too much of it was hard-coded, and I didn't have near enough middle layers to do anything really interesting. And I was essentially coming up with a way to not have to do back-propagation.
At the time, I convinced myself I was selecting for instinctive knowledge...
And I was very excited by research that said that, rather than having one pool of 10,000 ants...
It was better to have 10 islands of 1,000 ants, and to occasionally let genetic information travel from one island to another island. The research claimed the overall system would converge faster.
I thought that was super cool, and made me excited that easy parallelism would be rewarded.
I daydream about all of that, still.
¹ https://www.rogeralsing.com/2008/12/07/genetic-programming-e...
Something I've been interested in recently is - I wonder if it'd be possible to encode a known-good model - some massive pretrained thing - and use that as a starting point for further mutations.
Like some other comments in this thread have suggested, it would mean we can distill the weight patterns of things like attention, convolution, etc. and not have to discover them by mutation - so - making use of the many phd-hours it took to develop those patterns, and using them as a springboard. If papers like this are to be believed, more advanced mechanisms may be able to be discovered.
What does this mean? Probably not nothing, but probably not “the cosmos is the mind of god.” It probably means that we live in a universe that tends to produce repeating nested patterns at different scales.
But maybe that’s part of what makes it possible to evolve or engineer brains that can understand it. If it had no regularity there’d be no common structural motifs.
Beyond the practical implications of this (i.e. reduced training and inference costs), I'm curious if this has any consequences for "philosophy of the mind"-type of stuff. That is, does this sentence from the abstract, "we identify universal subspaces capturing majority variance in just a few principal directions", imply that all of these various models, across vastly different domains, share a large set of common "plumbing", if you will? Am I understanding that correctly? It just sounds like it could have huge relevance to how various "thinking" (and I know, I know, those scare quotes are doing a lot of work) systems compose their knowledge.
This is a little outside my area, but I think the relevant part of that abstract is "Gradient-based optimization follows horizontal lifts across low-dimensional subspaces in the Grassmannian Gr(r, p), where r p is the rank of the Hessian at the optimum"
I think this question is super interesting though: why can massively overparametrised models can still generalise?
E.g
https://youtu.be/Qp0rCU49lMs?si=UXbSBD3Xxpy9e3uY
https://thoughtforms.life/symposium-on-the-platonic-space/
e.g see this paper on Universal Embeddings https://arxiv.org/html/2505.12540v2
"The Platonic Representation Hypothesis [17] conjectures that all image models of sufficient size have the same latent representation. We propose a stronger, constructive version of this hypothesis for text models: the universal latent structure of text representations can be learned and, furthermore, harnessed to translate representations from one space to another without any paired data or encoders.
In this work, we show that the Strong Platonic Representation Hypothesis holds in practice. Given unpaired examples of embeddings from two models with different architectures and training data, our method learns a latent representation in which the embeddings are almost identical"
Also from the OP's Paper we see this on statement:
"Why do these universal subspaces emerge? While the precise mechanisms driving this phenomenon remain an open area of investigation, several theoretical factors likely contribute to the emergence of these shared structures.
First, neural networks are known to exhibit a spectral bias toward low frequency functions, creating a polynomial decay in eigenvalues that concentrates learning dynamics into a small number of dominant directions (Belfer et al., 2024; Bietti et al., 2019).
Second, modern architectures impose strong inductive biases that constrain the solution space: convolutional structures inherently favor local, Gabor-like patterns (Krizhevsky et al., 2012; Guth et al., 2024), while attention mechanisms prioritize recurring relational circuits (Olah et al., 2020; Chughtai et al., 2023).
Third, the ubiquity of gradient-based optimization – governed by kernels that are largely invariant to task specifics in the infinite-width limit (Jacot et al., 2018) – inherently prefers smooth solutions, channeling diverse learning trajectories toward shared geometric manifolds (Garipov et al., 2018).
If these hypotheses hold, the universal subspace likely captures fundamental computational patterns that transcend specific tasks, potentially explaining the efficacy of transfer learning and why diverse problems often benefit from similar architectural modifications."
https://arxiv.org/abs/2007.00810
Without properly reading the linked article, if thats all this is, not a particularly new result. Nevertheless this direction of proofs is imo at the core of understanding neural nets.
> We analyze over 1,100 deep neural networks—including 500 Mistral-7B LoRAs and 500 Vision Transformers. We provide the first large-scale empirical evidence that networks systematically converge to shared, low-dimensional spectral subspaces, regardless of initialization, task, or domain.
I instantly thought of muon optimizer which provides high-rank gradient updates and Kimi-k2 which is trained using muon, and see no related references.
The 'universal' in the title is not that universal.
Isn't it obvious?
It isn’t obvious that these parameters are universal across all models.
The surprising thing is inter-modality shared variation. I wouldn't have bet against it but I also wouldn't have guessed it.
I would like to see model interpretability work into whether these subspace vectors can be interpreted as low level or high level abstractions. Are they picking up low level "edge detectors" that are somehow invariant to modality (if so, why?) or are they picking up higher level concepts like distance vs. closeness?
The "human" part of that matters. This is all human-made data, collected from human technology, which was created to assist human thinking and experience.
So I wonder if this isn't so much about universals or Platonic ideals. More that we're starting to see the outlines of the shapes that define - perhaps constrict - our own minds.
Basically, what if we're not actually "training" the model, but rather the model was randomly initialized and the learning algorithm is just selecting the vectors that happen to point into the right direction? A left multiplication of the form D*A with a diagonal matrix is equivalent to multiplying each row in A with the corresponding diagonal element. Low values mean the vector in question was a lottery blank and unnecessary. High values means that this turns out to be correct vector, yay!
But this trivial explanation doesn't work for the full SVD, because you now have a right multiplication U*D. This means each column gets multiplied against the corresponding diagonal element. Both the column in U and row vector in V^T have to perfectly coincide to make the "selection" theory work, which is unlikely to be true for small models, which happen to work just fine.*
Not a technical person just trying to put it in other words.
Now imagine you discover that all 500 are really just the same 11 base ingredients plus something extra.
What they've done here is use SVD, (which is normally used for image compression and noise reduction), to find that "base recipe". Now we can reproduce those other recipes by only recording the one igredient that differs.
More interestingly it might tell us something new about smoothies in general to know that they all share a common base. Maybe we can even build a simpler base using this info.
At least in theory. The code hasn't actually been released yet.
And that: "Defining 'novel' as 'not something that you've said before even though your using all the same words, concepts, linguistic tools, etc., doesn't actually make it 'novel'"
Point being, yeah duh, what's the difference between what any of these models are doing anyway? It would be far more surprising if they discovered a *different* or highly-unique subspace for each one!
Someone gives you a magic lamp and the genie comes out and says "what do you wish for"?
That's still the question. The question was never "why do all the genies seem to be able to give you whatever you want?"
Why else would they put so much money into something if not to try and get more out of it?
Capitalists' morals are driven by their social position. To them this is right becauae its rewarding. To us its an akin abomination we create that destroys us
But the problem isnt inherently tech. Its how society is structured around it that allows it to be used against us.
- I know what I do not know.
-- I do not know AI.