Computing Euclidean Distance Between Two Matrices: Methods and Code Implementation

Euclidean distance is a classical method for measuring the straight-line distance between two points in space, widely used in pattern recognition and machine learning for calculating similarity between samples. When computing distances between matrices, two typical scenarios commonly arise:

Element-wise Distance When two matrices have identical dimensions (both m×n), you can directly compute the sum of squared differences between corresponding elements, then take the square root. This approach effectively treats the matrices as flattened vectors and applies standard Euclidean distance calculation. It's particularly suitable for comparing feature maps with identical dimensions or assessing pixel-level differences in images. In Python/NumPy implementation, this can be achieved using: np.sqrt(np.sum((A - B)**2)).

Row/Column Vector Distance A more common requirement involves computing the Euclidean distance between each row of matrix A (m×k) and each row of matrix B (n×k), resulting in an m×n distance matrix. This is essential for algorithms like K-Nearest Neighbors (KNN), where distances between test samples and training samples need calculation. Efficient implementation can avoid explicit loops through broadcasting mechanisms: first expand matrix dimensions using np.newaxis, then apply the formula sqrt(sum((A[:, np.newaxis, :] - B[np.newaxis, :, :])**2, axis=2)). This vectorized approach leverages matrix operations for optimal performance.

Extended Considerations For large-scale data scenarios, dimensionality reduction techniques like PCA can be incorporated to reduce computational complexity. Other distance metrics (Manhattan distance, cosine similarity) between matrices can be derived using similar computational frameworks. In deep learning applications, matrix Euclidean distance can be utilized to design custom loss functions for specific learning objectives, particularly in similarity-based learning tasks.