Approximate Fiber Product: A Preliminary Algebraic-Geometric Perspective on Multimodal Embedding Alignment

Multimodal tasks, such as image-text retrieval and generation, require embedding data from diverse modalities into a shared representation space. Aligning embeddings from heterogeneous sources while preserving shared and modality-specific information is a fundamental challenge. This paper provides an initial attempt to integrate algebraic geometry into multimodal representation learning, offering a foundational perspective for further exploration. We model image and text data as polynomials over discrete rings, $\mathbb{Z}_{256}[x]$ and $\mathbb{Z}_{|V|}[x]$, respectively, enabling the use of algebraic tools like fiber products to analyze alignment properties. To accommodate real-world variability, we extend the classical fiber product to an approximate fiber product with a tolerance parameter $\epsilon$, balancing precision and noise tolerance. We study its dependence on $\epsilon$, revealing asymptotic behavior, robustness to perturbations, and sensitivity to embedding dimensionality. Additionally, we propose a decomposition of the shared embedding space into orthogonal subspaces, $Z = Z_s \oplus Z_I \oplus Z_T$, where $Z_s$ captures shared semantics, and $Z_I$, $Z_T$ encode modality-specific features. This decomposition is geometrically interpreted via manifolds and fiber bundles, offering insights into embedding structure and optimization. This framework establishes a principled foundation for analyzing multimodal alignment, uncovering connections between robustness, dimensionality allocation, and algebraic structure. It lays the groundwork for further research on embedding spaces in multimodal learning using algebraic geometry.