TLDR: Hierarchical Optimal Transport (HOT) is a novel framework for comparing how information is represented across different layers of AI models and regions of the brain. Unlike traditional methods, HOT allows for flexible, many-to-many layer correspondences, provides a single global alignment score, and naturally handles networks of varying depths. It consistently outperforms standard approaches in alignment quality and reveals clear hierarchical structures in large language models, vision models, and human visual cortex data, especially when augmented with rotation invariance (HOT+R).
Understanding how different neural networks or even different brains process information is a big challenge. Scientists often compare “representations” – how information is encoded at various stages (layers in AI models, regions in the brain). The standard way to do this is by matching layers one-to-one, which has limitations. It can be asymmetric, doesn’t give a single overall score, and struggles when networks have different numbers of layers. It also ignores the bigger picture of how information flows through the entire system.
To overcome these issues, researchers Shaan Shah and Meenakshi Khosla propose a new method called Hierarchical Optimal Transport (HOT). This framework offers a more unified and flexible way to compare how information is represented across different systems. Instead of rigid one-to-one matches, HOT allows for “soft” connections, meaning a part of one system can relate to multiple parts of another.
How Hierarchical Optimal Transport (HOT) Works
Imagine information as “mass” moving through layers. HOT works on two levels:
At a fine-grained level, it figures out how individual “neurons” (or units in AI models, voxels in brain scans) in one layer correspond to neurons in another.
At a higher level, it determines how entire layers in one network connect to layers in another. This is where the “soft” part comes in – a layer can distribute its “representational mass” across several layers in the target network. This is especially useful when comparing networks of different depths.
The goal is to find the most efficient way to “transport” this representational mass while ensuring that no information is lost or over-utilized. This process results in a single, overall alignment score for the entire network comparison.
Sometimes, two systems might represent the same information but in a “rotated” way. To account for this, an extension called HOT+R (Rotation-invariant HOT) was developed. This version can find correspondences even when the underlying features are geometrically similar but oriented differently.
Key Advantages of HOT
HOT brings several significant improvements to representational alignment:
Global Alignment: It considers the entire network structure, providing a more consistent and meaningful comparison.
Handles Depth Differences: It naturally aligns networks with different numbers of layers by allowing one layer to map to multiple others.
Better Alignment Scores: Empirically, HOT often achieves higher accuracy in reconstructing representations compared to older methods.
Reveals Hierarchy: Without being explicitly told to, HOT uncovers clear hierarchical patterns. For example, early layers in one model tend to align with early layers in another, and deeper layers maintain their relative positions.
Shows Feature Distribution: It can reveal how a single computation in a shallower network might be spread out across several layers in a deeper one.
Real-World Applications and Results
The researchers tested HOT across various domains:
Large Language Models (LLMs): When comparing different LLMs (like LLaMA and Qwen), HOT consistently showed higher alignment accuracy and revealed clear diagonal structures in the transport plans, indicating that early layers align with early layers, and deeper layers with deeper ones.
Human Visual Cortex: Applied to fMRI data from human visual cortex (V1-V4 regions), HOT recovered expected region-to-region correspondences across different subjects, which traditional methods missed.
Vision Models: For Vision Transformers (DINOv2, ViT-MAE), the rotation-augmented HOT+R variant significantly improved alignment scores and revealed clearer hierarchical mappings, highlighting the importance of accounting for rotational invariances in these models.
This innovative research is detailed in the paper, “Representational Alignment Across Model Layers and Brain Regions with Hierarchical Optimal Transport,” by Shaan Shah and Meenakshi Khosla.
Also Read:
- Enhancing Hierarchical Classification with Feature-Aware Contrastive Learning
- Decoding How AI Understands the World: A Multimodal Perspective
Looking Ahead
While powerful, HOT is computationally intensive. Future work aims to make it more efficient and explore even more complex alignments, such as tracking representational changes during learning. This framework is a significant step towards understanding the fundamental principles that govern intelligent systems, both biological and artificial.
