Low-dimensional space

As explained in Graph Neural Networks in Action

A low-dimensional vector representation encodes an object as a vector with relatively few numerical dimensions.

A low-dimensional vector representation compresses a graph, node, or edge into relatively few numerical dimensions while retaining essential structural and relational information. This makes complex graph data easier to handle, visualize, and use with machine-learning models. section 2.4.1

Use a low-dimensional representation to move from graph structure to practical analysis:

• Represent a node, edge, or entire graph as a compact vector. section 2.4.1
• Interpret similarity through graph connectivity, hop counts, paths, or random-walk probabilities rather than only geometric distance. section 2.4.1
• Use the resulting vectors for two- or three-dimensional visualization, node classification, link prediction, or graph classification. section 2.4.1 section 1.2.3

Figure 2.11 Comparison of similarity concepts: using distance on a plane (left) and using steps along a graph (right)

Compare ordinary spatial distance with graph-based similarity measured by steps along connections. section 2.4.1

How Graph Neural Networks in Action covers it

How deep it goes

Defines embeddings and explains structural similarity through graph hops and random-walk probability, but does not detail embedding algorithms.

Examples

Uses two- or three-dimensional visualization, graph nodes, edges, whole graphs, Node2Vec, and DeepWalk as embedding examples.

What you can do after

Apply it via Node2Vec or DeepWalk to create vectors for visualization and graph prediction tasks.

How it connects

Connects low-dimensional representations to graph embeddings, node similarity, random walks, node classification, link prediction, and graph classification.

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Related concepts

• embeddingincludes · A numerical vector representation of an item or input that encodes features or relationships useful for computational tasks.
• principal component analysisproduced by · Principal component analysis is a linear dimensionality-reduction technique that transforms data into orthogonal components ordered by the variance they capture.
• UMAPproduced by · UMAP, or Uniform Manifold Approximation and Projection, is a nonlinear manifold-learning technique that embeds high-dimensional data in a lower-dimensional space while preserving important neighborhood and structural relationships.

This explainer was generated by Manning's knowledge graph from the text of Graph Neural Networks in Action. The section links point to the parts of the book it draws on.