Meridian: A New Paradigm for Visual-Language Representation by Moving CLIP to Hyperbolic Manifolds

The Meridian project breaks through the limitations of traditional Euclidean space representation by mapping CLIP's multimodal features to hyperbolic manifolds (Lorentz manifolds), providing a more natural geometric expression for hierarchical semantic structures. Built on CLIP, this project explores a new paradigm for multimodal representation.

CLIP, as a cornerstone model for visual-language representation learning, is based on the Euclidean space assumption. However, natural language and world knowledge are inherently hierarchical (e.g., animal → mammal → dog → golden retriever). Expressing such nested relationships in Euclidean space requires complex encoding, while hyperbolic geometry is naturally suited for tree-like structures and hierarchical relationships.

Meridian retains CLIP's ViT-B/16 visual encoder and text encoder as backbones, mapping the extracted features to Lorentz manifolds (the standard model of hyperbolic space). Lorentz manifolds have three key properties: 1. Volume grows exponentially with radius, enabling efficient embedding of tree-like structures; 2. The distance from a point to the origin corresponds to semantic hierarchy (farther points are more specific); 3. Preserves topological structure, avoiding semantic entanglement.

Meridian's design balances practicality and innovation: 1. Computational efficiency: Leverages CLIP's pre-trained weights, only needing to learn feature mapping, reducing training costs; 2. Compatibility: Compatible with the CLIP ecosystem (pre-trained weights, fine-tuning strategies, etc.); 3. Interpretability: Analyzes the model's semantic organization through the geometric properties of hyperbolic space (distance, angle).

The application prospects of hyperbolic representation learning include: 1. Fine-grained classification: Hierarchical categories are naturally distributed (coarse-grained near the center, fine-grained in the outer layer); 2. Zero-shot reasoning: Improves compositional reasoning ability (e.g., "red dog"); 3. Knowledge graph embedding: Combines with hyperbolic knowledge graphs to achieve tri-modal unification; 4. Long-tail distribution learning: Exponential capacity accommodates sample imbalance.

Meridian faces the following challenges: 1. Optimization difficulty: Gradient descent in hyperbolic space is more complex, requiring special optimizers and incurring higher computational overhead; 2. Visualization difficulties: Human intuition is based on Euclidean space, and tools for hyperbolic space are scarce; 3. Evaluation benchmarks: Existing benchmarks are designed for Euclidean models, so new protocols are needed to reflect the advantages of hyperbolic models.

Meridian represents an exploration of 'returning to first principles': instead of pursuing larger models or more data, it rethinks multimodal representation from the geometric foundation. The combination of hyperbolic geometry and deep learning is still in its early stages, and Meridian provides an implementation reference, reminding us that representation spaces do not have to be limited to Euclidean—richer geometric structures may lead to stronger semantic expression.