Inside a model every token becomes a vector, an embedding. Training places tokens used in similar ways close together, so meaning becomes direction.

The usual way to compare two vectors is cosine similarity: the cosine of the angle between them. It is 1 for the same direction, 0 for unrelated, -1 for opposite, and it ignores length.

import torch
import torch.nn.functional as F

a = torch.tensor([1.0, 2.0])
b = torch.tensor([2.0, 4.0])
print(F.cosine_similarity(a, b, dim=0))
# tensor(1.): same direction

Search engines, recommendation and the retrieval in chat assistants all start here: turn things into vectors, then find the nearest ones.

Write cosine(a, b) returning the cosine similarity of two 1-D tensors as a float. Write nearest(query, matrix, k) returning the indices of the k rows of matrix most similar to query, best first, as a list of ints.

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