What is a low-dimensional representation?
A low-dimensional representation is a compact numeric description of something that is large or messy in its raw form. A photograph contains millions of pixel values. A customer’s purchase history may span thousands of product columns, most of them empty. A low-dimensional representation reduces each of these to a few dozen or a few hundred numbers, usually called a vector, a latent representation, or an embedding.
The approach works because real data occupies only a small part of the space its raw format allows. A few underlying factors, such as a customer’s trade, size, and buying season, explain most of what the raw columns record. Principal component analysis, the encoder half of an autoencoder, and trained embedding models all produce representations of this kind. Similar items receive similar vectors, so the distance between two vectors measures how alike the items are, which is the basis of vector search.
Low-dimensional representation in practice
An industrial supply distributor sells tens of thousands of products to several thousand business customers. Matrix factorization, a standard technique in recommendation systems, compresses the customer-by-product purchase table into about 40 numbers per customer and 40 per product. Products bought by the same kinds of customers land close together even when their descriptions share no words, such as a cutting fluid and the carbide drill bits used with it. Inside sales representatives use the vectors to suggest add-on items. The individual numbers have no meaning a person would recognize, so explanations come from showing which items sit nearby.