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Vector Search

Vector search finds semantically similar text by comparing query embeddings with document embeddings in high-dimensional vector space. It uses metrics like Cosine Similarity or Inner Product, allowing search engines to match queries based on semantic meaning rather than exact keywords.

Complexity Profile​

CaseComplexity
Best CaseO(log N) - HNSW index
Average CaseO(log N)
Worst CaseO(N) - Flat scan
Space ComplexityO(N * D)

Code Implementation​

import numpy as np

def cosine_similarity(v1, v2):
    dot_product = np.dot(v1, v2)
    norm_v1 = np.linalg.norm(v1)
    norm_v2 = np.linalg.norm(v2)
    return dot_product / (norm_v1 * norm_v2)

# Flat search scanning candidate pool
def flat_vector_search(query_vec, candidate_matrix, top_k=5):
    # candidate_matrix is of shape (N, D)
    similarities = np.dot(candidate_matrix, query_vec) / (
        np.linalg.norm(candidate_matrix, axis=1) * np.linalg.norm(query_vec)
    )
    return np.argsort(similarities)[-top_k:][::-1]

Real-World Applications​

  • Retrieval-Augmented Generation (RAG) contexts locator.
  • Recommendation systems (recommending similar products or tracks).
  • Image similarity and reverse visual matching engines.