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Pardon my simplistic question, but when you mean rotation you’re essentially talking about diagonalization aren’t you?

So storing the diagonal as a matrix and the new bases is more compact?

I just today learned about Multi-Head Latent Attention, which is also sort of a way of compressing the KV cache. Can someone explain how this new development relates to MHLA?

If they didn't cite your paper that's bullshit.

But if they read your paper enough that they invited you to a talk, that probably means they were far enough along to independently inventing it they were going to do so anyway, and wanted to chat with someone who was also doing the thing they were already doing. Good ideas tend to reveal themselves to anyone who is aware of the problem.

Schmidhuber'd

Other people have answered here but the real answer is that deep neural networks don't learn isotropic distributions of activations.

What happens is that you get very spikey activations, there are so called "outlier" activations. A easy to read paper that tells you about this is SmoothQuant [0]. Another source from Anthropic and the Mechanistic Interperability people is calling these "privileged basis" [1].

Now based on the weight symmetries of a typical transformer, these actually don't need to exist. Weight symmetries means the ways you can change the weights without actually affecting the mathematical function, there are a broad class of these because the linear algebra has a lot of redundancies in it.

But the behaviour of the Adam optimizer is such that you do end up w/ these things because it sort of more quickly optimizes to produce them. This comes from the fact it is an elementwise dynamic learning rate (and probably partly to do with the epsilon).

[0] https://arxiv.org/pdf/2211.10438 [1] https://transformer-circuits.pub/2023/privileged-basis/index...

They are saying that models should be invariant to data's orientation - and only sensitive to the distance between vectors. This has a pretty significant effect on reducing the set of possible models, and may stabilize the optimization.

In simple terms, large ML models like LLMs often learn trivial rules such as "if the 21st decimal place of the 5th dimension in the embedding vector is 5 - then the image is of a cat." Learning such a memorization function is usually not what we are trying to do, and there are a variety of techniques to avoid these trivial solutions and "smooth" the optimization geometry.

The whole goal of quantisation is to put the data into 'bins' so that it can easily be 'packed' so that you can represent it using less bits (less information). You can think of it like rounding essentially (3.14159 -> 3). Now, sometimes within data, the distribution will be non-ideal for separating it out into bins (let's say that our rounding rules are simple -- we simply use a floor function so 2.45 maps to 2 and 6.4543 maps to 6 etc...) and our bins simply map to the floor -- if we had a set of numbers which look like this: [3.11, 4.43, 5.78, 12.33, 34.32], they would simply map to [3, 4, 5, 12, 34]. Now, we have one huge outlier in our data (34) so to create bins for those sets of numbers, we would need 6 bits of information (2 to the power of 6 = 64), but this is mostly due to the fact that we have one huge outlier (34.32). To get rid of this -- the algorithms applies a random rotation matrix which 'distorts' the original data so that it is more evenly distributed among the possible bins which are assigned to the data set. In linear algebra, a rotation matrix is an orthogonal matrix. When you multiply your vector by this matrix, you aren't changing the "amount" of data (the length of the vector remains the same), but you are recalculating every single number in that vector as a weighted sum of the originals. According to the Central Limit Theorem, when you sum up many random things, the result always starts looking like a bell curve. This is the magic TurboQuant relies on: they don't know what your data looks like, but they know that after the rotation, the data must look like a Beta Distribution and they use this fact to transform the original data into a more 'tightly packed' distribution which allows them to more efficiently pack (or quantise) the information. If most of the transformed data is huddled together into a predictable Bell curve shape, you can pack your bins tightly around that shape leading to much higher precision with fewer needed bits to store it. For example, after applying a rotation matrix, our original transform [3.11, 4.43, 5.78, 12.33, 34.32] might get mapped to something like [8.12, 8.65, 9.25, 10.53, 12.86] and we can crate bins which both are more accurate and need less bits in order to hold our original data set. To create the most optimal bins -- the Lloyd-Max algorithm is used. This algorithm is the gold standard for 1D quantisation. Its goal is to find the best places to put your "boundaries" (where you cut the data) and your "reconstruction values" (the number you store) to minimise the Mean Squared Error (MSE). After applying this, you have your 'rounded' values (or quantized data), but there is still an error value which is missing from our data set: and this is where the residual bit comes in. That bit doesn't represent the original data (or vector) - it simply represents our 'bias' after we apply the above algorithms. It's basically like a '1-bit note' which allows you to perfectly cancel out all the bias terms which our above quantisation algorithm produces to make the 'interactions' (or inner products) when we multiply our values together extremely accurate again even after transforming our original data. Does this make sense?

They are not doing random rotation, simplification here means they are aligning the outliers. If you threw a bunch of shapes on the ground they are picking up one that rolled away and putting it with the others.

>How can a boolean value preserve all of the relational and positional information between data points?

They aren't reducing entire vector to a bollean only each of its dimensions.

He even attempts to improve on the paper by replacing the random rotation operation which is O(d^2), by a Subsampled Randomized Hadamard Transform which can be computed in O(d*log d).

Hopefully Johnson–Lindenstrauss lemma applies in the same way for SRHTransformed vectors as they do for randomly rotated vectors and the independence of the distribution laws of the coordinates remains and therefore the quantization of each coordinates independently is still theoretically sound.

For some reason I thought the implementation would be way more complicated than that. I obviously lack the domain knowledge to tackle something like this, but it looks straight forward.

The pace of development in llama.cpp is really high, could see an implementation being merged in 4-6 weeks.

It has a lot clearer explanation of the method than Google's own post.

It is AI generated. Or was written by someone a bit far from the technical advances IMHO. The Johnson-Lindenstrauss Lemma is a very specific and powerful concept, when in the article the QLJ explanation is vacuous. A knowledgeable human would not have left the reader wanting for how that relates to the Lemma.

Yeah, and some parts of the article are just bizarre:

> Instead of looking at a memory vector using standard coordinates (i.e., X, Y, Z) that indicate the distance along each axis, PolarQuant converts the vector into polar coordinates using a Cartesian coordinate system. This is comparable to replacing "Go 3 blocks East, 4 blocks North" with "Go 5 blocks total at a 37-degree angle”

Why bother explaining this? Were they targeting the high school and middle school student reader base??

I think it is though-

“ TurboQuant, QJL, and PolarQuant are more than just practical engineering solutions; they’re fundamental algorithmic contributions backed by strong theoretical proofs. These methods don't just work well in real-world applications; they are provably efficient and operate near theoretical lower bounds.”

JPEG XL is mainly based on unique image-specific research, but you're right to say a lot of the techniques are compatible with videos in theory (the XYB color space comes to mind). AVIF is an AV1 OBU in an image-specific container, and required a lot of image-specific engineering to make AV1's tools useful for images; see libaom's tune "iq", and the same in SVT-AV1. The compression gains translated when engineering effort went into creating bespoke implementations, and the same may happen for LLMs if I had to guess.

If you think of it from the point of view of the universal approximation theorem, it's all efficiency optimisation. We know that it works if we do it incredibly inefficiently.

Every architecture improvement is essentially a way to achieve the capability of a single fully-connected hidden layer network n wide. With fewer parameters.

Given these architectures usually still contain fully connected layers, unless they've done something really wrong, they should still be able to do anything if you make the entire thing large enough.

That means a large enough [insert model architecture] will be able to approximate any function to arbitrary precision. As long as the efficiency gains with the architecture are retained as the scale increases they should be able to get there quicker.

Most breakthroughs that are published are for efficiency because most breakthroughs that are published are for open source.'

All the foundation model breakthroughs are hoarded by the labs doing the pretraining. That being said, RL reasoning training is the obvious and largest breakthrough for intelligence in recent years.

Efficiency gains can be used to make existing models more profitable, or to make new larger and more intelligent models.

> What are the most importsnt breakthroughs from the past two or three years for intelligence?

The most important one in that timeframe was clearly reasoning/RLVR (reinforcement learning with verifiable rewards), which was pioneered by OpenAI's Q* aka Strawberry aka o1.

https://mesuvash.github.io/blog/2026/turboquant-interactive/ has a little visualisation

1. Efficient recursive transform of kv embeddings into polar coordinates 2. Quantize resulting angles without the need for explicit normalization. This saves memory via key insight: angles follow a distribution and have analytical form.

That overview is frustratingly high-level. I know what a vector is, a bit, and yet that compression description is crazy uninformative. And that PolarQuant visualization is.. Very abstract.

The way I understand it, it's a way of compressing vectors by switching from their per-component representation to polar coordinates representation, where the nearby vectors are clumped together to a single line, allowing to describe them by different lengths

This seems to be an inference-time optimization and they are putting AI on every search result page. That seems like plenty of incentive to optimize.

I can assure you OpenAI and Anthropic pay for hardware. They don’t receive it for free.

These are very different media types with very different goals.

Agreed. The practical implications are often more interesting than the math anyway — smaller models running locally means you can afford to run multiple models in parallel for cross-validation, which changes how you approach tasks like code analysis or bug detection.

KV cache compression, so how much memory the model needs to use for extending its context. Does not affect the weight size.

Is the number of bits per coordinate. So, 1 bit is 2x2 grid. 3 bit is a 64 cell grid (2^3 x 2^3). Here you have a demo.

https://mesuvash.github.io/blog/2026/turboquant-interactive/

The explanation is terrible, but it's clear that it's not actually lossless.

If in short, for many inference tasks the bottleneck is memory bandwidth. Suppose you have a machine with a memory bandwidth of 256 GB/s, and let's say you want to do inference for 4B model (model with 4 billion parameters). If you will load the model in BF16 format (16 bits), each forward pass (i.e. each token generated) will require roughly ~8 GB of memory bandwidth. So, 256/8 = 32 t/s, and that's the generation speed you will be strictly capped at even if your processing power is measured in exaFLOPS. But let's say now that you have decided to instead quantize the model and then run the quantized version. Suppose you have made a Q4_K_M version (4 bits + some weights will take more). Now each of your forward passes will take roughly 2-3 GB (rough approximations, reality is different) of memory bandwith (actually, it will be around 2 GB), and even in the worst case 256/3 = 85.3, while 256/2 = 128 t/s. Quants can reduce quality of the model and lower it's performance, but in most modern quantization methods those losses are usually negligible (although, of course, they're still present). So, as you can see, it can be concluded that quantization "widens" (it's not removing it fully) memory bottleneck while still preserving (not always though) acceptable quality.

(Sorry for my terrible English, it's not my native language)

So let’s start with a really simple decoder transformer with a single layer and single attention head, and train it to predict the next token in a sequence of text. To predict the next token you need a few things: a query for the very last token in the sequence, and a key and value for every prior token. You take your query and compute a dot product with every prior key (two large vectors in, scaler attention score out). That scaler attention score first goes through softmax, and then becomes the weight you use to compute a weighted average of your values, new value goes through the mlp, mlp output is projected into the logits from which you sample your next token (that’s the general idea at least skipped a few steps).

The last query in the sequence will be new for every new token you predict, but the set of prior keys and values stay the same, ie keys and values are reusable. The key value cache gets bigger and bigger for each new token you add to the sequence, and that’s where compression comes in. You have to store the keys and values in vram, and you’d like to keep the size down by not storing the raw uncompressed tensors. To make this work well your compression needs two things: it needs to be fast so that you can compress and decompress on the fly, and it needs to play well with softmax attention. Prior attempts at compression usually suck at one or the other, either the speed to decompress is too slow and your token/s takes a hit, or you lose important precision and the model output quality suffers. The claim in the paper is that they’ve made progress on both.

r is a single value per vector. You don't have to quantize it, you can keep it and quantize the billion+ other coordinates of the vector.

Nah, those are completely different beasts. DeepSeek's MLA solves the KV cache issue via low-rank projection - they literally squeeze the matrix through a latent vector at train time. TurboQuant is just Post-Training Quantization where they mathematically compress existing weights and activations using polar coordinates

Efficient execution on the GPU appears to have been one of the specific aims of the authors. Table 2 of their paper shows real world performance that would appear at a glance to be compatible with inference.

Apparently MLX confirmed it - https://x.com/prince_canuma/status/2036611007523512397

Classic academic move. If the authors show accuracy-vs-space charts but hide end-to-end latency, it usually means their code is slower in practice than vanilla fp16 without any compression. Polar coordinates are absolute poison for parallel GPU compute