Euclidean and Manhattan Distance in Machine Learning

Distance metrics are fundamental in various fields, including machine learning, statistics, and data analysis. Two commonly used distance measures are Euclidean and Manhattan distances. Both have unique characteristics and applications, influencing how machine learning algorithms perform and interpret data.

Euclidean Distance

Euclidean distance, also known as L2 norm, is the most common distance metric. It measures the straight-line distance between two points in Euclidean space. Mathematically, for two points in an n-dimensional space, the Euclidean distance is given by:

This distance metric is intuitive and geometrically meaningful, representing the “as-the-crow-flies” distance.

Manhattan Distance

Manhattan distance, also known as L1 norm or taxicab distance, measures the distance between two points along axes at right angles. It is the sum of the absolute differences of their Cartesian coordinates. For the same points, the Manhattan distance is calculated as:

This metric is reminiscent of navigating a city grid, where you can only move horizontally or vertically.

Differences Between Euclidean and Manhattan Distance

1. Calculation:

• Euclidean distance involves squaring the differences between coordinates, summing them, and taking the square root.
• Manhattan distance sums the absolute differences without squaring or taking a root.

2. Path:

• Euclidean distance measures the shortest path (straight line) between two points.
• Manhattan distance measures the path along grid lines (axis-aligned paths).

3. Impact of Dimensions:

• In higher dimensions, Euclidean distance tends to give a larger value due to the squaring of differences.
• Manhattan distance increases linearly with the number of dimensions.

4. Sensitivity to Changes:

• Euclidean distance is more sensitive to larger differences in individual dimensions due to squaring.
• Manhattan distance treats differences in each dimension equally.

Applications in Machine Learning

1. Nearest Neighbor Algorithms:

• Euclidean Distance: Often used in k-nearest neighbors (k-NN) when features are continuous and the dataset is relatively low-dimensional.
• Manhattan Distance: Preferred when dealing with high-dimensional spaces or when features are not continuous.

2. Clustering:

• Euclidean Distance: Common in algorithms like k-means clustering, where centroid calculations benefit from the Euclidean metric.
• Manhattan Distance: Suitable for clustering algorithms like k-medians, particularly in high-dimensional spaces.

3. Regularization:

• L2 Regularization (Ridge Regression): Uses the Euclidean norm to penalize large coefficients, encouraging smaller, more distributed weights.
• L1 Regularization (Lasso Regression): Uses the Manhattan norm to create sparsity, resulting in models with fewer non-zero coefficients.

4. Optimization Problems:

• Euclidean distance is often used in gradient-based optimization due to its smooth and differentiable nature.
• Manhattan distance can be used in linear programming and other combinatorial optimization problems where the solution space is grid-like.

Which Distance Metric is Used Most Widely?

The choice between Euclidean and Manhattan distance depends on the specific application and data characteristics.

• Euclidean Distance: More widely used in general due to its intuitive geometric interpretation and effectiveness in low-dimensional, continuous feature spaces. It is the default in many machine learning algorithms and applications involving physical spaces or continuous data.
• Manhattan Distance: Preferred in high-dimensional spaces or when dealing with categorical data, sparse data, or grid-like structures. It can be more robust to outliers in certain contexts.

In practice, the selection of the distance metric should be guided by the nature of the problem, the data characteristics, and the specific algorithmic requirements.

Both Euclidean and Manhattan distances play crucial roles in machine learning and data analysis. Understanding their differences and applications allows practitioners to make informed decisions about which metric to use for their specific tasks. While Euclidean distance is more commonly used, especially in low-dimensional spaces with continuous features, Manhattan distance provides valuable advantages in high-dimensional or structured spaces. The choice of distance metric can significantly impact the performance and interpretability of machine learning models.