Euclidean space

As explained in Knowledge Graphs and LLMs in Action

Euclidean space is a geometric space in which points have coordinate representations and distances follow the standard straight-line metric.

Euclidean space represents points as coordinate vectors, measures distance with straight lines, and follows the Pythagorean theorem. It is the default geometric setting for many machine-learning models and embeddings. section 11.1.1

Key properties include:

• Representing an item as coordinates such as (0.5, 0.3). section 11.1.1
• Calculating distance as the square root of the sum of squared coordinate differences. section 11.1.1
• Treating relationships as uniform, with space between points remaining constant regardless of position. section 11.1.1

Graph embeddings

An encoding function maps graph nodes into a multidimensional Euclidean embedding space. Training can arrange the vectors so that distances reflect relevant relationships in the original graph, although hierarchical structures may fit hyperbolic space better. section 11.1.1

Figure 11.2 Euclidean vs. non-Euclidean space [3]. Pixels in an image are all equally distant from each other and have eight neighbors (except those on the border). In a graph, the number of neighbors can vary enormously from one node to another.

Compare Euclidean regularity with graph structure: image pixels have predictable neighborhoods, while graph nodes can have widely varying numbers of neighbors. section 11.1.1

How Knowledge Graphs and LLMs in Action covers it

How deep it goes

Defines coordinates, straight-line distance, and spatial growth, then contrasts Euclidean geometry with hyperbolic alternatives.

Examples

Uses coordinate vectors, image pixels, graph nodes, and multidimensional embedding spaces as concrete examples.

What you can do after

Evaluate/understand when straight-line geometry is suitable for embeddings and when hierarchical data may require another space.

How it connects

Connects graph embeddings, graph neural networks, non-Euclidean space, and hyperbolic space.

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Related concepts

• embeddingused by · A numerical vector representation of an item or input that encodes features or relationships useful for computational tasks.
• projective spacecontrasts with · Projective space is a geometric space that contains Euclidean points and adds points at infinity, with points represented up to nonzero scale.
• hyperbolic spacealternative to · Hyperbolic space is a negatively curved geometric space whose available volume grows exponentially with distance from a center.
• real numberuses · A real number is a value corresponding to a point on the continuous number line, including rational and irrational numbers.
• coordinatehas part · A coordinate is a numerical value or ordered set of values specifying a position relative to a coordinate system.
• Euclidean distancemeasured by · Euclidean distance is the straight-line distance between two points in Euclidean space, calculated as the square root of the sum of squared differences between corresponding coordinates.

This explainer was generated by Manning's knowledge graph from the text of Knowledge Graphs and LLMs in Action. The section links point to the parts of the book it draws on.