Embeddings & Vector Search — Prepflix AI Notes

What Are Embeddings? CRITICAL

An embedding is a vector of floating-point numbers (typically 384–3072 dimensions) that represents the semantic meaning of a piece of text.

The key property: semantically similar text maps to vectors that are close together in the vector space.

embedding("king") - embedding("man") + embedding("woman") ≈ embedding("queen") embedding("dog") ≈ embedding("puppy") embedding("dog") ≠ embedding("database")

          Intuition: imagine a 768-dimensional space where every concept has a location. Similar concepts cluster together. Embeddings are the coordinates.
        

What Embeddings Enable

🔍Semantic search: find results by meaning, not keywords
📚RAG retrieval: find the most relevant document chunks for a query
🎯Recommendations: find items similar to what a user liked
🔄Deduplication: find near-duplicate content at scale
🏷️Classification: cluster by similarity without labeled data

Embedding Models — Which to Use

Model	Dimensions	Cost	Best For
OpenAI text-embedding-3-small	1536	$0.02 / 1M tokens	Good balance of cost and quality. Default choice for most applications.
OpenAI text-embedding-3-large	3072	$0.13 / 1M tokens	Highest quality. Use when retrieval accuracy matters most.
Google text-embedding-004	768	Free (with Gemini API)	Excellent free option. Required for Firestore native vector search.
sentence-transformers/all-MiniLM-L6-v2	384	Free (local)	Fastest, smallest, runs on CPU. For prototyping or cost-sensitive setups.
BAAI/bge-large-en-v1.5	1024	Free (local)	Best open-source model. Competitive with OpenAI on MTEB benchmarks.

Interview answer for "which embedding model would you use?": "For a Firebase/GCP project I'd use text-embedding-004 — free and native to Firestore vector search. For a general production system, text-embedding-3-small for cost efficiency, or large if we have a precision budget. For local/offline: BAAI/bge-large."

Similarity Metrics — When to Use What CRITICAL

Cosine Similarity

cos(A,B) = dot(A,B) / (|A|×|B|)

Measures the angle between vectors. Range: −1 (opposite) to 1 (identical direction).

Ignores magnitude — only direction matters.

✓ Default for text embeddings. Most APIs return normalized vectors → cosine sim = dot product.

Dot Product

dot(A,B) = Σ aᵢ × bᵢ

For normalized vectors: dot product = cosine similarity. Faster to compute (no division).

Affected by magnitude — if vectors aren't normalized, higher-magnitude vectors dominate.

✓ Use when vectors are pre-normalized (most embedding APIs do this by default).

Euclidean Distance

d(A,B) = √(Σ(aᵢ-bᵢ)²)

Straight-line distance in vector space. Range: 0 (identical) to ∞.

Affected by vector magnitude — less reliable for text embeddings.

Less common for text. Better for image embeddings or when absolute distance matters.

# Python: cosine similarity (NumPy) import numpy as np def cosine_sim(a, b): a, b = np.array(a), np.array(b) return np.dot(a, b) / (np.linalg.norm(a) * np.linalg.norm(b)) # For normalized vectors (most APIs normalize by default): def cosine_sim_normalized(a, b): return np.dot(a, b) # same result, faster

Why Brute Force Fails at Scale & How HNSW Solves It CRITICAL

The Problem: Brute-Force KNN

You have 1 million documents, each embedded as a 768-dim vector. A user asks a question. You need the top-5 most similar documents.

Brute force: compare query to every one of 1M vectors.
Cost: O(n × d) = O(1M × 768) = 768M multiply-adds per query.
At 1000 queries/second: 768 billion operations/second — impractical.

Approximate Nearest Neighbor (ANN)

Idea: trade a tiny bit of accuracy (miss ~2-5% of true nearest neighbors) for a massive speedup.

ANN algorithms return results that are "approximately" nearest neighbors — fast enough for production, accurate enough to be useful.

HNSW — How It Works

Hierarchical Navigable Small World graph:

Build a multi-layer graph of vectors at insertion time
Top layers: sparse graph with long-range connections (fast navigation, like a highway)
Bottom layers: dense graph with local connections (precise search, like city streets)
Query: start at the top layer, greedily navigate toward the query vector, zoom in at each lower layer

Brute force: O(n × d) HNSW query: O(log n × d) At n=1M: 20 steps instead of 1M

HNSW is the default index in Qdrant, Weaviate, Chroma, and FAISS. Pinecone uses a proprietary variant. Firestore uses Google's internal ANN algorithm.

Vector Database Decision Framework CRITICAL

Database	Type	Best For	Key Characteristic
Pinecone	Managed SaaS	Getting started fast, no infra to manage	Fully managed, easiest to set up, closed source. Good free tier.
Qdrant	Open source (Rust)	Self-hosted production systems	Very fast, excellent metadata filtering, Docker or cloud. Best OSS option.
Weaviate	Open source	Hybrid search (keyword + vector) needed	BM25 + vector hybrid built-in. Good for production with mixed search needs.
Chroma	Open source (Python)	Prototyping, local development, learning	Simplest API, in-memory or persistent, zero infra. Not for high-scale production.
pgvector	PostgreSQL extension	Already using PostgreSQL, don't want another service	Add vector column to existing Postgres table. Limited scale vs dedicated vector DBs.
Firestore Vector	Managed (Google)	Firebase/GCP projects with Gemini embeddings	Native KNN search in Firestore (launched late 2024). Zero extra infrastructure. Best for Firebase apps.

How to Answer in an Interview

"We already use Firebase"

→ Firestore native vector search. Zero extra infrastructure, uses Gemini embeddings natively.

"We already use PostgreSQL"

→ pgvector. Add vector column to existing tables. No new service to operate.

"We want managed, no ops"

→ Pinecone. Best developer experience, production-ready immediately.

"We want open source in production"

→ Qdrant. Best performance, excellent filtering, active community.

Hybrid Search: BM25 + Vector HIGH

Why Neither Alone Is Enough

Pure vector searchGreat for semantic understanding ("what does this mean?"). Misses exact product names, codes, IDs.

Pure keyword (BM25)Great for exact terms and proper nouns. Misses synonyms and conceptual matches.

Hybrid (both)Better than either alone. Handles semantic + exact match in one query.

Real-world Example

Query: "iPhone 15 Pro Max camera"

Keyword search: finds "iPhone 15 Pro Max" exactly ✓
Vector search: understands "camera" in photography context ✓
Hybrid: gets the product right AND understands the intent ✓✓

BM25 Formula (know conceptually)

BM25(q,d) = Σ IDF(qᵢ) × f(qᵢ,d)×(k+1) / (f(qᵢ,d) + k×(1-b+b×|d|/avgdl)) k=1.5, b=0.75 (standard defaults) f = term frequency in document IDF = log((N-n+0.5)/(n+0.5)) N = total docs, n = docs containing term

          Don't memorize BPE formula for interviews. Know the concept: BM25 scores documents by term frequency (how often the query term appears) weighted by inverse document frequency (rare terms get higher weight). It's the gold standard keyword retrieval algorithm.
        

Weaviate has hybrid search built-in. For other vector DBs, run BM25 and vector search separately, then combine scores using Reciprocal Rank Fusion (RRF).