Abstract
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 t