Euclidean Distance (L2 Norm) Explanation

Euclidean distance, also known as L2 norm, is a widely used metric in vector databases to measure the distance between two points in Euclidean space. It is a simple and intuitive metric that calculates the straight-line distance between two points.

Mathematical Formula

Given two vectors, A and B, the Euclidean distance between them can be calculated using the following formula:

euclidean_distance(A, B) = sqrt(sum((A[i] - B[i])^2 for i in range(len(A))))

where:

• A[i] and B[i] are the elements of vectors A and B at index i, respectively.
• len(A) is the length of vector A.

Geometric Interpretation

Euclidean distance can be visualized as the length of the hypotenuse of a right triangle formed by the vectors A and B. The legs of the triangle represent the differences between the corresponding elements of the vectors.

Applications of Euclidean Distance

• Numerical Data: Euclidean distance is well-suited for comparing numerical data, such as measurements, sensor readings, or financial data.
• Image and Video Analysis: Euclidean distance can be used to compare images or videos based on their pixel values.
• Recommendation Systems: Euclidean distance can be used to find items that are similar to a user’s preferences based on numerical features.
• Clustering: Euclidean distance is commonly used in clustering algorithms to group similar data points together.

Advantages and Disadvantages

• Advantages:
  • Simple and intuitive to understand.
  • Widely used and well-supported.
  • Suitable for numerical data.
• Disadvantages:
  • Sensitive to outliers: Large differences between corresponding elements can significantly increase the distance.
  • May not be appropriate for categorical or text data.

Euclidean distance is a fundamental metric in vector databases, offering a simple and effective way to measure the similarity between two points in Euclidean space. Its applications extend to various domains, including numerical data analysis, image and video processing, recommendation systems, and clustering. By understanding the principles of Euclidean distance, you can make informed decisions about its use in your vector database projects.