If Information Retrieval is the process of finding relevant facts, then Embeddings are the language that makes that process possible for a machine. While humans perceive language through syntax and definitions, machine learning models require a mathematical representation to understand the relationship between different pieces of data.

Representation

At its most basic level, an embedding is a numerical representation of an object, such as a word, a sentence, or an entire document. This object is transformed into a fixed-length array of numbers called a vector. These are not simple binary values. They are continuous floating-point numbers that act as coordinates in a high-dimensional space.

When we say this space is high-dimensional, we mean it may have hundreds or even thousands of axes. For example, modern embedding models (for example, the OpenAI embedders) frequently generate vectors with 1,536 dimensions. Each dimension conceptually represents a specific feature or trait of the data, such as its topic, tone, or relationship to other concepts.

Geometric Properties

The power of embeddings lies in their ability to preserve semantic relationships. In a well-trained embedding space, items with similar meanings are positioned closer to one another than items that are unrelated.

This spatial arrangement allows machines to perform semantic arithmetic. A classic example in natural language processing is the relationship between gender and royalty. The vector for "King" minus the vector for "Man" plus the vector for "Woman" will result in a coordinate very close to the vector for "Queen". Because the model has mapped royalty and gender as distinct dimensions in its mathematical world, it can navigate these concepts without needing a dictionary.

Similarity

Once data is converted into vectors, the task of finding relevant information becomes a geometry problem. We use distance metrics to quantify how similar two embeddings are.

Cosine Similarity: This is the most common metric for text embeddings. Instead of measuring the raw distance between two points, it measures the angle between two vectors. If two vectors point in exactly the same direction, their similarity score is 1. If they are perpendicular, it is 0.

This is particularly useful for text because it focuses on the orientation of the meaning rather than the length of the document.

Euclidean Distance (L2): This measures the straight-line distance between two points in the vector space. While intuitive, it can be sensitive to the magnitude of the vectors, meaning it might struggle if your documents vary greatly in length.

Purpose

Without embeddings, an AI agent's information retrieval would be limited to exact keyword matching. This approach is brittle and easily confused by synonyms or varied phrasing. By using embeddings, a developer can ensure that if a user asks about "reducing household energy costs", the system can retrieve documents about "insulation" or "solar panels" even if the original query did not use those specific terms.

This capability is the engine behind semantic search and RAG, and is what allows the system to bridge the gap between human messy, natural language queries and the structured or unstructured data stored in your apps and databases.