Mathematical Foundations of Vector Embeddings in AI Systems

Artificial intelligence utilities rely heavily on transforming unstructured textual content into numerical vector representations. Understanding vector embeddings requires exploring linear algebra and high-dimensional space geometry.

1. Vector Spaces & Dimensions An embedding maps a token or document into a continuous $N$-dimensional vector space $\mathbb{R}^N$. Nearby vectors in this mathematical space represent semantic similarity.

2. Cosine Similarity Formula To measure the semantic distance between two vectors $\mathbf{A}$ and $\mathbf{B}$, we compute their cosine similarity:

By ranking cosine similarity scores, retrieval-augmented generation (RAG) engines extract relevant context with high precision.