Language models are injective and hence invertible

As I read it, what they did there was a sanity-check by trusting the birthday paradox. Kind of: "If you get orthogonal vectors due to mere chance once, that's okay, but you try it billions of times and still get orthogonal vectors every time, mere chance seems a very unlikely explanation."

The nature of high-dimensional spaces kind of intuitively supports the argument for invertability though, no? In the sense that:

> I would expect the chance of two inputs to map to the same output under these constraints to be astronomically small.

I envy your intuition about high-dimensional spaces, as I have none (other than "here lies dragons"). (I think your intuition is broadly correct, seeing as billions of collision tests feels quite inadequate given the size of the space.)

> Just intuitively, in such a high dimensional space, two random vectors are basically orthogonal.

What's the intuition here? Law of large numbers?

And how is orthogonality related to distance? Expansion of |a-b|^2 = |a|^2 + |b|^2 - 2<a,b> = 2 - 2<a,b> which is roughly 2 if the unit vectors are basically orthogonal?

> Since the outputs are normalized, that corresponds to a ridiculously tiny patch on the surface of the unit sphere. Since the outputs are normalized, that corresponds to a ridiculously tiny patch on the surface of the unit sphere.

I also have no intuition regarding the surface of the unit sphere in high-dimensional vector spaces. I believe it vanishes. I suppose this patch also vanishes in terms of area. But what's the relative rate of those terms going to zero?

I think that the latent space that GPT-2 uses has 768 dimensions (i.e. embedding vectors have that many components).

> Just intuitively, in such a high dimensional space, two random vectors are basically orthogonal.

Which, incidentally, is the main reason why deep learning and LLM are effective in the first place.

A vector of a few thousands dimensions would be woefully inadequate to represent all of human knowledge, if not for the fact that it works as the projection of a much higher, potentially infinite-dimensional vector representing all possible knowledge. The smaller-sized one works in practice as a projection, precisely because any two such vectors are almost always orthogonal.

> If I am understanding this paper correctly, they are claiming that the model weights can be inverted in order to produce the original input text.

No, that is not the claim at all. They are instead claiming that given an LLM output that is a summary of chapter 18 of Mary Shelley's Frankenstein, you can tell that the input prompt that led to this output was "give me a summary of chapter 18 of Mary Shelley's Frankenstein". Of course, this relies on the exact wording: for this to be true, it means that if you had asked "give me a summary of chapter 18 of Frankenstein by Mary Shelley", you would necessarily receive a (slightly) different result.

Importantly, this needs to be understood as a claim about an LLM run with temperature = 0. Obviously, if the infra introduces randomness, this result no longer perfectly holds (but there may still be a way to recover it by running a more complex statistical analysis of the results, of course).

Edit: their claim may be something more complex, after reading the paper. I'm not sure that their result applies to the final output, or it's restricted to knowing the internal state at some pre-output layer.

There is a clarification tweet from the authors:

- we cannot extract training data from the model using our method

- LLMs are not injective w.r.t. the output text, that function is definitely non-injective and collisions occur all the time

- for the same reasons, LLMs are not invertible from the output text

https://x.com/GladiaLab/status/1983812121713418606

Clarification [0] by the authors. In short: no, you can't.

[0] https://x.com/GladiaLab/status/1983812121713418606

The input isn't the training data, the input is the prompt.

> they are compressing the data beyond the known limits, or they are abstracting the data into more efficient forms.

I would argue that this is two ways of saying the same thing.

Compression is literally equivalent to understanding.

For sure! Measuring parameters given data is central to statistics. It’s a way to concentrate information for practical use. Sufficient statistics are very interesting, bc once computed, they provably contain as much information as the data (lossless). Love statistics, it’s so cool!

> That's why an LLM (I tested this on Grok) can give you a summary of chapter 18 of Mary Shelley's Frankenstein, but cannot reproduce a paragraph from the same text verbatim.

Unfortunately, the reality is more boring. https://www.litcharts.com/lit/frankenstein/chapter-18 https://www.cliffsnotes.com/literature/frankenstein/chapter-... https://www.sparknotes.com/lit/frankenstein/sparklets/ https://www.sparknotes.com/lit/frankenstein/section9/ https://www.enotes.com/topics/frankenstein/chapter-summaries... https://www.bookey.app/freebook/frankenstein/chapter-18/summ... https://tcanotes.com/drama-frankenstein-ch-18-20-summary-ana... https://quizlet.com/content/novel-frankenstein-chapter-18 https://www.studypool.com/studyGuides/Frankenstein/Chapter_S... https://study.com/academy/lesson/frankenstein-chapter-18-sum... https://ivypanda.com/essays/frankenstein-by-mary-shelley-ana... https://www.shmoop.com/study-guides/frankenstein/chapter-18-... https://carlyisfrankenstein.weebly.com/chapters-18-19.html https://www.markedbyteachers.com/study-guides/frankenstein/c... https://www.studymode.com/essays/Frankenstein-Summary-Chapte... https://novelguide.com/frankenstein/summaries/chap17-18 https://www.ipl.org/essay/Frankenstein-Summary-Chapter-18-90...

I have not known an LLM to be able to summarise a book found in its training data, unless it had many summaries to plagiarise (in which case, actually having the book is unnecessary). I have no reason to believe the training process should result in "abstracting the data into more efficient forms". "Throwing away most of the training data" is an uncharitable interpretation (what they're doing is more sophisticated than that) but, I believe, a correct one.

Spoiler but for those who do not want to open the paper, the contribution statement is:

"Equal contribution; author order settled via Mario Kart."

If only more conflicts in life would be settled via Mario Kart.

Yeh it would be fun if we could reverse engineer the prompts from auto generated blog posts. But this is not quite the case.

Still, it is technically correct. The model produces a next-token likelihood distribution, then you apply a sampling strategy to produce a sequence of tokens.

> - This impacts privacy, deletion, and compliance

Surely that's a stretch... Typically, the only thing that leaves a transformer is its output text, which cannot be used to recover the input.

This mental model is also in direct contradiction to the whole purpose of the embedding, which is that the embedding describes the original text in a more interpretable form. If a piece of content in the original can be used for search, comparison etc., p much by definition it has to be stored in the embedding.

Similarly, this result can be rephrased as "Language Models process text." If the LLM wasn't invertible with regards to a piece of input text, it couldn't attend to this text either.

> There is an easy fix to that - a random rotation; preserves all distances.

Is that like homomorphic encryption, in a sense, where you can calculate the encryption of a function on the plaintext, without ever seeing the input or calculated function of plaintext.

I don't think vector databases are intended to be secure, encrypted forms of data storage in the first place.

No, it's about the distribution being injective, not a single sampled response. So you need a lot of outputs of the same prompt, and know the LLM, and then you should in theory be able to reconstruct the original prompt.

No, it says nothing about LLM output being invertible.

I think their claims are limited to the "theoretical" LLM, not to the way we typically use one.

The LLM itself has a fixed size input and a fixed size, deterministic output. The input is the initial value for each neuron in the input layer. The LLM output is the vector of final outputs of each neuron in the output layer. For most normal interactions, these vectors are almost entirely 0s.

Of course, when we say LLM, we typically consider infrastructure that abstracts these things for us. Especially we typically use infra that takes the LLM outputs as probabilities, and thus typically produces different results even for the exact same input - but that's just a choice in how to interpret these values, the values themselves are identical. Similarly on the input side, the max input is typically called a "context window". You can feed more input into the LLM infra than the context window, but that's not actual input to the model itself - the LLM infra will simply pick a part of your input and feed that part into the model weights.

Well that is not how I reed it, but: Every final state has an unique prompt. You could have several final states have the same unique prompt.

That's not what you get out of LLMs.

LLMs produce a distribution from which to sample the next token. Then there's a loop that samples the next token and feeds it back to to the model until it samples a EndOfSequence token.

In your example the two distributions might be {"OK": 0.997, EOS: 0.003} vs {"OK": 0.998, EOS: 0.002} and what I think the authors claim is that they can invert that distribution to find which input caused it.

I don't know how they go beyond one iteration, as they surely can't deterministically invert the sampling.

Edit: reading the paper, I'm no longer sure about my statement below. The algorithm they introduce claims to do this: "We now show how this property can be used in practice to reconstruct the exact input prompt given hidden states at some layer [emp. mine]". It's not clear to me from the paper if this layer can also be the final output layer, or if it must be a hidden layer.

They claim that they can reverse the LLM (get prompt from LLM response) by only knowing the output layer values, the intermediate layers remain hidden. So, Their claim is that indeed you shouldn't be able to do that (note that this claim applies to the numerical model outputs, not necessarily to the output a chat interface would show you, which goes through some randomization).