## What is a Vector Database?

A vector database is a specialized database technology designed to efficiently manage and query high-dimensional vector representations of data. Unlike traditional relational databases that are optimized for structured and tabular data, vector databases are built to handle unstructured or semi-structured data such as text, images, and audio.

One of the key features of a vector database is its ability to perform similarity-based searches. When a query is made to the database, it returns the vectors that are closest to the query vector based on a chosen similarity metric such as cosine similarity or Euclidean distance. This approach enables quick and efficient retrieval of similar data points even in large and complex datasets.

Vector databases find applications in a wide range of domains, including information retrieval, recommendation systems, image recognition, and natural language processing. As the need to extract insights from unstructured data continues to grow, vector databases are becoming an increasingly important tool in the modern data stack.

## Semantic Search

Semantic search is an advanced information retrieval technique aimed at improving the accuracy and relevance of search results by understanding the contextual meaning and intent behind a user's query. Unlike traditional search algorithms that primarily rely on keyword matching, semantic search engines strive to decode the underlying semantics in both the query and the content of the document.

At the core of semantic search lies the representation of documents and queries as high-dimensional vectors that capture their semantic content. This process, known as vectorization or embedding, transforms unstructured text into a numerical representation that can be manipulated and compared mathematically.

When a user submits a query, the search engine converts it into a vector representation and then calculates its similarity with the vectors of each indexed document using measures such as cosine similarity. Documents considered to be semantically closest to the query are returned as the top results.

By focusing on the underlying meaning rather than just keyword occurrence, semantic search methods enable more intuitive and context-aware queries. Users can express their information needs in natural language and still expect relevant results even when synonyms, related concepts, or complex sentence structures are used.

As the amount of digital information continues to grow exponentially, semantic search techniques are becoming increasingly important for unlocking the value of unstructured data and providing insights that might otherwise be difficult to access through traditional search strategies.

#### Cosine Similarity

In a high-dimensional vector space, cosine similarity can be used to measure the similarity between two vectors based on the angle between them. The closer the angle is to 0°, the more similar the vectors are. Cosine similarity is a key principle in information retrieval, where it is used to find documents that are semantically similar to a given search query.

$\cos(\theta) = \frac{\vec{A} \cdot \vec{B}}{||\vec{A}|| \cdot ||\vec{B}||}$ Let $\vec{A} = (3, 4)$ and $\vec{B} = (4, 3)$ be two vectors. The cosine similarity between them can be calculated as: $\cos(\theta) = \frac{3 \cdot 4 + 4 \cdot 3}{\sqrt{3^2 + 4^2} \cdot \sqrt{4^2 + 3^2}} \approx 0.96$ The high cosine value indicates that the vectors are very similar in direction.Used in: Semantic search, document classification, recommendation systems.

#### Dot Product

The dot product is a fundamental operation in linear algebra that takes two vectors of the same dimensions and returns a scalar. It is often used as a measure of similarity between two vectors because it encapsulates both the angles between the vectors and their lengths. In high-dimensional spaces, the dot product can be effectively used to find vectors that are closest to a given query vector.

$\vec{A} \cdot \vec{B} = \sum_{i=1}^{n} A_i B_i$ Given two vectors $\vec{A} = (2, 3, 1)$ and $\vec{B} = (1, 2, 3)$, their dot product can be calculated as: $\vec{A} \cdot \vec{B} = 2 \cdot 1 + 3 \cdot 2 + 1 \cdot 3 = 11$ The resulting scalar quantifies the overlap or similarity between the two vectors.Used in: Machine learning, pattern recognition, recommendation systems.

#### Euclidean Distance

Euclidean distance is a metric that measures the straight-line distance between two points in a multi-dimensional space. It is one of the most intuitive and commonly used distance measures in various domains such as data analysis, pattern recognition, and machine learning. The smaller the Euclidean distance between two vectors, the more similar they are considered to be.

$d(\vec{p}, \vec{q}) = \sqrt{\sum_{i=1}^{n} (q_i - p_i)^2}$ Let $\vec{p} = (1, 2, 3)$ and $\vec{q} = (4, 5, 6)$ be two points in a three-dimensional space. The Euclidean distance between them can be calculated as: $d(\vec{p}, \vec{q}) = \sqrt{(4 - 1)^2 + (5 - 2)^2 + (6 - 3)^2} = \sqrt{27} = 5.2$ The distance 5.2 indicates the degree of similarity (or lack thereof) between the two points.Used in: Clustering analysis, anomaly detection, nearest neighbor classification.

#### Jaccard Similarity

Jaccard similarity is a statistical measure of similarity between two sets. It is calculated by dividing the size of the intersection of the sets by the size of the union of the sets. Jaccard similarity is particularly useful for comparing the similarity between two datasets with binary or categorical features, such as the presence or absence of specific elements.

$J(A, B) = \frac{|A \cap B|}{|A \cup B|}$ Consider two sets $A = \{a, b, c, d\}$ and $B = \{c, d, e, f\}$. The Jaccard similarity between them can be calculated as: $J(A, B) = \frac{|\{c, d\}|}{|\{a, b, c, d, e, f\}|} = \frac{2}{6} = 0.33$ The Jaccard similarity of 0.33 indicates that one-third of the elements in the two sets are common.Used in: Text comparison, market analysis, recommendation systems.