Geometry of learned representations
Embeddings map discrete objects (words, images, users) into a continuous vector space where geometry encodes meaning. In word embeddings trained on large corpora, semantically related words end up near each other. 'Dog' and 'cat' are close; 'dog' and 'stone' are far apart. Vector arithmetic works: if you subtract 'man' from 'king' and add 'woman', you get close to 'queen'. This emerges naturally because the model is trained to predict words in context, forcing nearby words to have similar representations. The space is learned; nothing explicitly encodes that 'dog' and 'cat' should be close. Yet the training objective (predict the next word based on context) induces this structure.
Using embedding space for search and clustering
Once a learned embedding space exists, applications follow naturally. Similar items cluster together spatially: cosine similarity between embedding vectors measures semantic similarity. Search becomes nearest neighbor: to find documents like a query, embed the query and find nearest documents. Clustering algorithms (K-means, hierarchical clustering) operate on embeddings to find groups of similar items. Dimensionality reduction (t-SNE, UMAP) projects embeddings into 2D for visualization. The success of these applications depends entirely on embedding quality: if the learned space captures the right notions of similarity, everything works. Modern embedding models are fine-tuned on specific tasks (semantic search, question-answering, image-text matching) rather than relying on generic word embeddings, improving performance significantly.