The Platonic Representation Hypothesis โ€” part 4/6

Cleaned-text code points [45878, 61502). Bibliography omitted here; source.md includes it.

Thus we have convergence to a representation of the statistics of X X , but what about Z Z ? Recall that our idealized world consists of bijective observation functions, which, over discrete random variables, preserve probabilities. So we have: P ๐–ผ๐—ˆ๐—ˆ๐—‹ โ€‹ ( x a , x b ) \displaystyle{P}_{\mathsf{coor}}(x_{a},x_{b}) = P ๐–ผ๐—ˆ๐—ˆ๐—‹ โ€‹ ( z a , z b ) \displaystyle={P}_{\mathsf{coor}}(z_{a},z_{b}) K ๐–ฏ๐–ฌ๐–จ โ€‹ ( x a , x b ) \displaystyle K_{\mathsf{PMI}}(x_{a},x_{b}) = K ๐–ฏ๐–ฌ๐–จ โ€‹ ( z a , z b ) , \displaystyle=K_{\mathsf{PMI}}(z_{a},z_{b}), where we use P ๐–ผ๐—ˆ๐—ˆ๐—‹ {P}_{\mathsf{coor}} and K ๐–ฏ๐–ฌ๐–จ K_{\mathsf{PMI}} in a modality-agnostic way to emphasize that different modalities share the same these quantities.

All these arguments hold not just for X X but also for Y Y (or any other bijective, discrete modality), implying: K ๐–ฏ๐–ฌ๐–จ โ€‹ ( z a , z b ) \displaystyle K_{\mathsf{PMI}}(z_{a},z_{b}) = โŸจ f X โ€‹ ( x a ) , f X โ€‹ ( x b ) โŸฉ โˆ’ c X \displaystyle=\langle f_{X}(x_{a}),f_{X}(x_{b})\rangle-c_{X} (7) = โŸจ f Y โ€‹ ( y a ) , f Y โ€‹ ( y b ) โŸฉ โˆ’ c Y . \displaystyle=\langle f_{Y}(y_{a}),f_{Y}(y_{b})\rangle-c_{Y}. (8) Therefore, for any modality in our idealized world, we observe representational convergence to the same kernel, which represents certain pairwise statistics of โ„™ โก ( ๐™ ) \mathbb{P}(\mathbf{Z}) .

This analysis suggests that certain representation learning algorithms may boil down to a simple rule: find an embedding in which similarity equals PMI . We note that this idea is consistent with prior works that have used PMI as a similarity measure for clustering in vision and language ( e.g. , Isola et al. (2014) ; Isola (2015) ; Isola et al. (2016) ; Chambers & Jurafsky (2008) ).

#### A study in color

We conduct a case study to verify that convergence does happen on real data. Abdou et al. (2021) discovered that color distances in learned language representations, when trained to predict cooccurrences in text ( Devlin et al., 2018 ) , closely mirror human perception of these distances, which we reproduce in Figure 8 with both contrastive and predictive models. Interestingly, they noted an increasing similarity as models scale larger and become better at modeling text cooccurrences. In Figure 8 , we also learn representations of color based on K ๐–ฏ๐–ฌ๐–จ K_{\mathsf{PMI}} from cooccurrences in images . Indeed, learning cooccurrence statistics in either domain recovers roughly the same perceptual representation. Details of this experiment are described in Appendix D .

We believe that our simple model encapsulates essential aspects of complex real-world systems, and offers a path toward understanding the representation that models are converging toโ€”a unified model that is proficient across various domains and modalities, grounded in the statistical properties of the underlying world. Section 6 further elaborates some limitations.

## 5 What are the implications of convergence?

#### Scaling is sufficient, but not necessarily efficient

Our arguments are roughly in line with the claim that โ€œscale is all you needโ€ to reach high levels of intelligence. We have argued that as resources are scaled (# parameters, # datapoints, # flops), representations are converging, regardless of other modeling choices and even data modality. Does this mean that scale is all that matters? Not quite: different methods can scale with different levels of efficiency ( Hestness et al., 2017 ; Kaplan et al., 2020 ) , and successful methods must still satisfy some general requirements ( e.g. , be a consistent estimator, model pairwise statistics of โ„™ โก ( ๐™ ) \mathbb{P}(\mathbf{Z}) ).

#### Training data can be shared across modalities

Suppose you have access to N N images and M M sentences, and want to learn the best representation. If there is indeed a modality-agnostic platonic representation, then both image and language data should help find it. The implication is that if you want to train the best vision model, you should train not just on N N images but also on M M sentences. This is already becoming common practice ( OpenAI, 2023 ; Radford et al., 2021 ) . Many vision models are finetuned from pre-trained LLMs. The other direction is less common, but also is implied by our hypothesis: if you want to build the best LLM, you should also train on image data . Indeed, OpenAI (2023) showed that training on images improved performance on text. In theory, there should be some conversion ratio: a pixel is worth a a words for training LLMs, and a word is worth b b pixels for training vision models.

#### Ease of translation and adaptation across modalities

When two representations are aligned, transitioning from one to the other should be a simple function thatโ€™s easily obtained. Our hypothesis could explain the phenomenon that conditional generation is easier than unconditional ( Mirza & Osindero, 2014 ; Liu et al., 2020 ; Sauer et al., 2022 ) , as the data we condition on may have the same platonic structure as the data we are generating. In line with this, recent work has found that representation-conditioning is even easier ( Li et al., 2023 ) . Similarly, representational convergence could act as a bridge that lets us find mappings between domains even without paired data; this may underlie the success of unpaired translation in vision ( Zhu et al., 2017 ; Shi et al., 2024 ; Xie et al., 2022 ) and language ( Tran et al., 2017 ; Lample et al., 2018 ) . We emphasize that this doesnโ€™t mean that models trained on a single modality ( e.g. , language) can immediately process raw data from another ( e.g. , vision). What makes them adaptable to the new modalities is that they share a common modality-agnostic representation, and can readily process representations of new modalities. Furthermore, this implies that language models would achieve some notion of grounding in the visual domain even in the absence of cross-modal data 5 5 5 In 1688, William Molyneux asked if a person born blind, upon gaining sight, could distinguish shapes by vision alone ( Locke, 1690 ) . Our arguments suggest they could not do so immediately, but after some visual experience, they could easily map shapes to their prior touch-based representations. Empirical data supports this, showing that congenitally blind children given sight can quickly learn these abilities ( Held et al., 2011 ) . . The primary advantage of cross-modal data could then simply be sample efficiency.

#### Scaling may reduce hallucination and bias

A prominent shortcoming of current LLMs is their propensity to hallucinate, or output false statements. If models are indeed converging toward an accurate model of reality, and scale powers this convergence, then we may expect hallucinations to decrease with scale. Of course, our hypothesis is conditioned on the training data for future models constituting a sufficiently lossless and diverse set of measurements. This may not come to pass, but it is an implication of our hypothesis worth pointing out. A similar argument can be made about certain kinds of bias. It has been shown that large models can exacerbate existing biases present in their training data ( Hall et al., 2022 ) . Our hypothesis implies that, while this may be true, we should expect larger models to amplify bias less . This does not mean bias will be removed, rather that the modelโ€™s biases will more accurately reflect the dataโ€™s biases, rather than exacerbating them.

## 6 Counterexamples and limitations

#### Different modalities may contain different information

One immediate objection to our hypothesis is: what about the information that is unique to a given modality? Can language really describe the ineffable experience of watching a total solar eclipse? Or, how could an image convey the a concept like โ€œI believe in the freedom of speech,โ€ which is easy to write in English? Two different models cannot converge to the same representation if they have access to fundamentally different information.

More precisely, our mathematical argument in Section 4 only strictly holds for bijective projections of ๐™ \mathbf{Z} , so that the information in all the projections is equivalent to the information in the underlying world. This will not hold true for either lossy or stochastic observation functions. Nonetheless, similar arguments have been made theoretically and empirically that cooccurrence relations are learned by practical contrastive ( Wang & Isola, 2020 ; Zimmermann et al., 2021 ) and predictive learners ( Papyan et al., 2020 ; Roeder et al., 2021 ) . Lu et al. (2021) and Mirchandani et al. (2023) also showed that models trained to autoregressively generate text also capture statistical relations in many other modalities, including symbolic reasoning, vision, protein folding, and robotics.

A more nuanced version of our hypothesis will need to be developed to handle the case of non-bijective observations and abstract concepts. A starting point could be: different models will converge to the same representation when the input signals are sufficiently high information and the models are sufficiently high capacity ; when they are not, the lower-information representation will only align with the higher-information one up to a level capped by the mutual information between the input signals and by the capacity of each model. This cap might or might not be practically important. Popular representations like CLIP are explicitly optimized to only capture the shared information between vision and language, yet are highly successful on many pure vision tasks. We perform a preliminary test of the effect of information level in Figure 9 (detailed in Appendix E ), and find that the more descriptive (higher information) a caption is, the better its LLM representation aligns with the visual representation of the corresponding image.

#### Not all representations are presently converging

Our argument has mainly focused on two modalities: vision and language. While we do expect other modalities will follow similar trends, we have yet to see the same level of convergence across all domains. For example, in robotics there is not yet a standardized approach to representing world states in the same way as there is for representing images and text. One limitation lies in the hardware used in robotics, which is often expensive and slow. This creates a bottleneck in the quantity and diversity of training data.

#### Sociological bias in producing AI models

Researcher bias and collective preferences within the AI community have shaped the trajectory of model development. There is often an explicit or implicit goal of designing AI systems that mimic human reasoning and performance, and this could lead to convergence toward human-like representations even if other kinds of intelligence are in fact possible. Additionally, the โ€œhardware lotteryโ€ ( Hooker, 2021 ) suggests that the success of AI models can also depend on the compatibility of their design with available computational architectures, further contributing to convergent trends.

#### Special-purpose intelligences might not converge

Different intelligent systems can be designed to accomplish different tasks. For instance: A bioinformatics systems might predict protein structure; an autonomous vehicle might follow lanes on highways. Itโ€™s possible that not much is shared between these two narrow tasks. Our argument only holds for intelligences that are optimized to perform well on many tasks. We have argued that a representation of reality is a structure that is useful across many tasks, but for any special purpose there may be shortcuts, or even effective representations detached from reality. Such shortcuts may be more efficient and necessary for continued improvements in specific domains. This will become more relevant if continued scaling comes up against boundary conditions around resources like energy and compute.

#### How do we measure alignment?

We focused on one particular alignment measure, mutual nearest-neighbor, in our experiments, and cited experiments using several others. However, there is active debate on the merits and deficiencies of all these ways of measuring alignment ( Bansal et al., 2021 ; Sucholutsky et al., 2023 ) . We discuss our choice and show results for other alignment metrics in Appendix A .

#### Lots left to explain

We have shown results where different models arrive at similar but not the same representations. For example, in Figure 3 , alignment clearly increases but only reaches a score of 0.16 0.16 , according to our mutual nearest-neighbor metric. The maximum theoretical value for this metric is 1 1 . Is a score of 0.16 0.16 indicative of strong alignment with the remaining gap being โ€œnoiseโ€ or does it signify poor alignment with major differences left to explain? We leave this as an open question.

## Acknowledgements

We thank Lindsey & Brown for sharing their data for our experiments shown in Figure 8 . We thank the anonymous reviewers for helpful feedback, and for providing the counterexample on how to visually convey โ€œI believe in the freedom of speech.โ€ Thanks for Yonglong Tian, Dilip Krishnan, Anna Decker, Yoon Kim, Jyo Pari, Ani Nrusimha, Dave Epstein, Victor Butoi, and Seungwook Han for helpful discussions and suggestions. We thank Mingzhong Sun for catching a typo. This work was supported by a Packard Fellowship and a Sloan Research Fellowship to P.I., by the MIT-IBM Watson AI Lab, by ONR MURI grant N00014-22-1-2740, by the Center for Brains, Minds, and Machines, the MIT Quest for Intelligence, NSF STC award CCF-1231216, the DARPA Knowledge Management at Scale and Speed (KMASS) program, and the DARPA Machine Common Sense (MCS) program.

## Appendix A Mutual k k -Nearest Neighbor Alignment Metric

For two models with representations f f , g g the mutual k k -nearest neighbor metric measures the average overlap of their respective nearest neighbor sets. In this section, we refer to this metric as m NN m_{\texttt{NN}} , which we will formally define below.

For cross-modal domains, define ( x i , y i ) โˆˆ ๐’ณ (x_{i},y_{i})\in\mathcal{X} as a sample from the data distribution ๐’ณ \mathcal{X} ( e.g. image-caption dataset). For the single domain alignment measurements, the samples are equivalent x i = y i x_{i}=y_{i} ( e.g. , images for vision, and text for language). Let { x i , y i } i = 1 b \{x_{i},y_{i}\}_{i=1}^{b} be the corresponding mini-batch sampled from this data distribution. Then given two model representations f f and g g the corresponding features are: ฯ• i = f โก ( x i ) \phi_{i}=f(x_{i}) and ฯˆ i = g โก ( y i ) \psi_{i}=g(y_{i}) , where the collection of these features are denoted as ฮฆ = { ฯ• 1 , โ€ฆ , ฯ• b } \Phi=\{\phi_{1},\dots,\phi_{b}\} and ฮจ = { ฯˆ 1 , โ€ฆ , ฯˆ b } \Psi=\{\psi_{1},\dots,\psi_{b}\} . Then for each feature pair ( ฯ• i , ฯˆ i ) (\phi_{i},\psi_{i}) , we compute the respective nearest neighbor sets ๐’ฎ โก ( ฯ• i ) \mathcal{S}(\phi_{i}) and ๐’ฎ โก ( ฯˆ i ) \mathcal{S}(\psi_{i}) . d ๐—„๐—‡๐—‡ โ€‹ ( ฯ• i , ฮฆ โˆ– ฯ• i ) = ๐’ฎ โก ( ฯ• i ) \displaystyle d_{\mathsf{knn}}(\phi_{i},\Phi\setminus\phi_{i})=\mathcal{S}(\phi_{i}) (9) d ๐—„๐—‡๐—‡ โ€‹ ( ฯˆ i , ฮจ โˆ– ฯˆ i ) = ๐’ฎ โก ( ฯˆ i ) \displaystyle d_{\mathsf{knn}}(\psi_{i},\Psi\setminus\psi_{i})=\mathcal{S}(\psi_{i}) (10) where d knn d_{\texttt{knn}} returns the set of indices of its k k -nearest neighbors. Then we measure its average intersection via m NN โ€‹ ( ฯ• i , ฯˆ i ) = 1 k โ€‹ | ๐’ฎ โก ( ฯ• i ) โˆฉ ๐’ฎ โก ( ฯˆ i ) | \displaystyle m_{\texttt{NN}}(\phi_{i},\psi_{i})=\frac{1}{k}\lvert\mathcal{S}(\phi_{i})\cap\mathcal{S}(\psi_{i})\rvert (11) where | โ‹… | \lvert{}\cdot{}\rvert is the size of the intersection.

#### The choice to use mutual nearest-neighbors