Query Expansion

Query expansion is an algorithmic technique in information retrieval designed to improve retrieval performance by augmenting an initial user query with additional, semantically related terms, synonyms, or contextual vectors. By transforming sparse or ambiguous user inputs into richer, expanded query representations via pseudo-relevance feedback, knowledge graphs, or dense embedding projections, systems bridge the vocabulary mismatch problem between user intent and document representation.

import numpy as np
from sklearn.feature_extraction.text import TfidfVectorizer
from sklearn.metrics.pairwise import cosine_similarity

# Initialize corpus and vectorizer
corpus = [
    "machine learning algorithms optimize data pipelines",
    "deep learning neural networks process natural language",
    "search engine retrieval uses vector space models"
]
vectorizer = TfidfVectorizer(stop_words='english')
X = vectorizer.fit_transform(corpus)

def expand_query_rocchio(query, corpus, vectorizer, X, top_k=2, alpha=1.0, beta=0.75):
    """
    Expands a query using the Rocchio pseudo-relevance feedback formula.
    Q_new = alpha * Q_orig + (beta / |D_rel|) * sum(D_rel)
    """
    q_vec = vectorizer.transform([query])
    similarities = cosine_similarity(q_vec, X).flatten()
    top_indices = similarities.argsort()[::-1][:top_k]
    
    # Compute centroid of top relevant documents
    rel_docs_matrix = X[top_indices]
    centroid = np.asarray(rel_docs_matrix.mean(axis=0))
    
    # Apply Rocchio weights
    expanded_vector = alpha * q_vec.toarray() + beta * centroid
    
    # Map back to terms
    feature_names = vectorizer.get_feature_names_out()
    top_term_indices = expanded_vector.argsort()[0][::-1]
    
    expanded_terms = [feature_names[i] for i in top_term_indices[:5]]
    return " ".join(expanded_terms)

def test_query_expansion():
    query = "machine learning"
    expanded = expand_query_rocchio(query, corpus, vectorizer, X)
    assert len(expanded.split()) > 0, "Expansion failed to yield terms"
    print(f"Original: {query} | Expanded: {expanded}")

test_query_expansion()

def harden_query(expanded_query, max_terms=8):
    tokens = expanded_query.split()
    # Deduplicate and clamp length to mitigate performance degradation
    unique_tokens = list(dict.fromkeys(tokens))
    return " ",join(unique_tokens[:max_terms])

Mathematical Formulation

Let Q be the original query represented as a dense or sparse vector in a vocabulary space V of size V. In Pseudo-Relevance Feedback (PRF), the system executes the initial query Q_0, retrieves the top M documents, and selects the top K documents as the pseudo-relevant set D_rel. The expanded query Q_exp is calculated via the Rocchio algorithm: Q_exp = alpha * Q_0 + (beta / |D_rel|) * sum(d) for d in D_rel – (gamma / |D_non_rel|) * sum(d_prime) for d_prime in D_non_rel. Here, alpha, beta, and sesquilinear scaling parameters control the influence of the original query, positive feedback, and negative feedback respectively.

Algorithmic Complexity

The time complexity is dominated by document scoring and centroid calculation. For a vocabulary size V, corpus size N, and top-k retrieved documents M, vector dot-products require O(V * N) operations. Extracting and averaging the top-k document vectors requires O(M * V) time. Space complexity is O(V * N + V^2) to maintain the inverted index or document-term matrix alongside semantic similarity caches.

Production Deployment & Failure Modes

A primary production failure mode is semantic drift, where expanding a polysemous term (e.g., 'Apple' the company vs. 'apple' the fruit) introduces irrelevant terms that degrade precision. Mitigations include corpus-specific stop lists, thresholding expansion term weights against a minimum cosine similarity cutoff, and leveraging transformer-based dense embeddings (e.g., ColBERT, SPLADE) to perform sparse-dense hybrid expansions securely.

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