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Learning Objectives

By the end of this section, you will be able to:
  • Understand how vector embeddings represent text in numerical space
  • Explain similarity metrics like cosine similarity
  • Visualize how embeddings enable semantic search
  • Compare different types of similarity calculations

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Vector Space Concepts

Vector Embeddings & Semantic Search

Vector embeddings convert text into numerical representations that capture semantic meaning. This allows us to find similar content based on meaning rather than just keyword matching.

Traditional Search

Matches exact keywords and phrases. Limited understanding of context and meaning.

Semantic Search

Understands meaning and context. Can find relevant content even with different wording.

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What Are Embeddings?

Embeddings are numerical representations of text that capture semantic meaning. Think of them as coordinates in a high-dimensional space where similar concepts are positioned close together.

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How Embeddings Work

1

Text Input

Raw text is processed and tokenized into smaller units
2

Neural Processing

A neural network processes the tokens through multiple layers
3

Vector Output

The final layer produces a fixed-length vector (e.g., 1536 dimensions)
4

Semantic Representation

The vector captures the semantic meaning of the original text

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Embedding Properties

Fixed Dimensions

All embeddings have the same length (e.g., 1536 for OpenAI’s text-embedding-ada-002)

Semantic Similarity

Similar concepts have similar vector representations

Mathematical Operations

Vectors can be compared, added, subtracted, and averaged

Language Agnostic

Works across different languages and domains

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Similarity Metrics

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Cosine Similarity

The most common metric for comparing embeddings is cosine similarity, which measures the angle between two vectors:
  • Mathematical Formula
  • Implementation
  cosine_similarity = (A · B) / (||A|| × ||B||)

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Similarity Examples

High Similarity (0.9+): “machine learning” and “artificial intelligence”
Medium Similarity (0.5-0.8): “python programming” and “software development”
Low Similarity (0.0-0.3): “machine learning” and “cooking recipes”

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Other Similarity Metrics

Euclidean Distance

Measures straight-line distance between vectors. Lower values = more similar.

Dot Product

Simple multiplication of corresponding vector elements. Higher values = more similar.

Manhattan Distance

Sum of absolute differences. Less sensitive to outliers than Euclidean.

Jaccard Similarity

Measures overlap between sets. Useful for comparing document features.
Try This: Imagine you have these embeddings:
  • “machine learning” → [0.1, 0.8, 0.3, …]
  • “artificial intelligence” → [0.2, 0.7, 0.4, …]
  • “cooking recipes” → [0.9, 0.1, 0.8, …]
Which pair would have the highest cosine similarity?
Advanced Exercise: Build a simple similarity calculator:
  1. Create a function that takes two text inputs
  2. Generate embeddings for both texts
  3. Calculate and display the cosine similarity
  4. Provide interpretation of the similarity score
  5. Test with various text pairs to understand patterns

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Interactive Learning

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Hands-On Embedding Analysis

  • Exercise 1: Embedding Comparison
  • Exercise 2: Similarity Patterns
// Compare embeddings for different text types
const texts = [
  "Machine learning algorithms",
  "AI and deep learning",
  "Cooking recipes and ingredients",
  "Programming and software development",
  "Machine learning algorithms for data analysis"
];

// Generate embeddings and compare similarities
for (let i = 0; i < texts.length; i++) {
for (let j = i + 1; j < texts.length; j++) {
const similarity = cosineSimilarity(
getEmbedding(texts[i]),
getEmbedding(texts[j])
);
console.log(`${texts[i]} vs ${texts[j]}: ${similarity.toFixed(3)}`);
}
}

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Self-Assessment Quiz

What is the primary purpose of embeddings in RAG systems?
  • A) To compress text data
  • B) To represent text as numerical vectors for similarity comparison
  • C) To encrypt sensitive information
  • D) To translate text between languages
Which similarity metric is most commonly used for comparing embeddings?
  • A) Euclidean distance
  • B) Cosine similarity
  • C) Manhattan distance
  • D) Hamming distance
Which property allows embeddings to capture semantic meaning?
  • A) Fixed dimensions
  • B) Semantic similarity
  • C) Mathematical operations
  • D) Language agnostic

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Reflection Questions

Why is cosine similarity preferred over Euclidean distance for embeddings?
  • Think about the properties of high-dimensional vectors
  • Consider what cosine similarity actually measures
How does the concept of vector space help us understand semantic relationships?
  • Think about how similar concepts are positioned in space
  • Consider how this enables mathematical operations on meaning

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Next Steps

You’ve now explored the fundamental concepts of vector embeddings and similarity metrics! In the next section, we’ll learn about chunking strategies and performance considerations:
  • Document chunking techniques for effective embedding
  • Performance optimization strategies
  • Storage and retrieval considerations
Key Takeaway: Embeddings convert text into numerical vectors that capture semantic meaning, enabling powerful similarity-based search and retrieval in RAG systems.
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