Python欧式距离矩阵算法

根据欧式距离矩阵(Euclidean Distance Matrix, EDM) 的定义，它是一个n $\times$ n的矩阵，表示欧式空间中一组n个点的空间距离，每个元素为点之间的距离平方。假设 $\{x_i \in \mathbb{R}^d : i \in \{1,...,n\}\}$

$=\begin{pmatrix} | & | & & |\\ x_1 & x_2 &... & x_n\\ | & | & & |\\ \end{pmatrix}_{d\times n}$

距离矩阵， $D_{ij} = ||x_i-x_j||^2_2$
$x_i - x_j)^T(x_i - x_j)$
$x_i^Tx_i + x_jx_j^T-2x_i^Tx_j$
$=\begin{pmatrix} 0 &d_{12}^2 & d_{13}^2 & \dots & d_{1n}^2\\ d_{21}^2&0 & d_{23}^2 & \dots & d_{2n}^2\\ d_{31}^2 & d_{32}^2 & 0&\dots & d_{3n}^2\\ \vdots&\vdots&\vdots &\ddots & \vdots\\ d_{n1}^2 & d_{n2}^2 & d_{1n}^2 & \dots& 0\\ \end{pmatrix}$
欧式距离矩阵，edm(X)定义为
$\overset{\mathrm{def}}{=} diag(G) + diag(G)^T - 2G$
其中，格拉姆矩阵(Gram matrix) $G=X^TX$ , $d i a g (G)$ 为含有对角线元素的列向量。

在计算两个矩阵 $X_{d\times m}$ 和 $Y_{d\times n}$ 之间的欧式距离矩阵时，可从上式推导出两个矩阵的欧式距离矩阵公式:
$G_X = X^TX$
$G_Y = Y^TY$
$edm(X) = diag(G_X) + diag(G_Y)^T - 2X^TY$
$=\begin{pmatrix} x_{11}\\ x_{22}\\ \vdots\\ x_{mm}\\ \end{pmatrix}_{m\times1} + \begin{pmatrix} y_{11}\\ y_{22}\\ \vdots\\ y_{nn}\\ \end{pmatrix}^T_{n\times1} - 2X^TY$

这里的 $G_X$ 和 $G_Y$ 的对角线元素会经过numpy数组的广播机制(broadcasting)，分别广播成 $n\times n$ 和 $m\times m$ 的矩阵，进行相加。EDM矩阵的大小应为 $m\times n$ 。

例子：
$\begin{pmatrix} 1&2&3\\ 2&3&4\\ 0&1&2\\ \end{pmatrix}^T_{3\times 3}; Y = \begin{pmatrix} 1&2&3\\ 4&3&2\\ \end{pmatrix}^T_{2\times 3}$

Python代码:

import numpy as np


def euclidean_dist(x, y=None):
    """
    Compute all pairwise distances between vectors in X and Y matrices.
    :param x: numpy array, with size of (d, m)
    :param y: numpy array, with size of (d, n)
    :return: EDM:   numpy array, with size of (m,n). 
                    Each entry in EDM_{i,j} represents the distance between row i in x and row j in y.
    """
    if y is None:
        y = x

    # calculate Gram matrices
    G_x = np.matmul(x.T, x)
    G_y = np.matmul(y.T, y)

    # convert diagonal matrix into column vector
    diag_Gx = np.reshape(np.diag(G_x), (-1, 1))
    diag_Gy = np.reshape(np.diag(G_y), (-1, 1))

    # Compute Euclidean distance matrix
    EDM = diag_Gx + diag_Gy.T - 2*np.matmul(x.T, y)  # broadcasting
    
    print('This is matrix EDM: ')
    print(EDM)
    return EDM


x = np.array([[1.0, 2.0, 3.0], [2.0, 3.0, 4.0], [0.0, 1.0, 2.0]]).T
y = np.array([[1.0, 2.0, 3.0], [4.0, 3.0, 2.0]]).T
euclidean_dist(x,y)

输出： $\begin{pmatrix} 0&11\\ 3&8\\ 3&20\\ \end{pmatrix}_{3\times 2}$

备注：欧式距离矩阵中的元素为距离的平方，若需要距离，可用np.sqrt(EDM)开方。

参考文献:

https://www.dabblingbadger.com/blog/2020/2/27/implementing-euclidean-distance-matrix-calculations-from-scratch-in-python
Dokmanic, Ivan, et al. “Euclidean distance matrices: essential theory, algorithms, and applications.” IEEE Signal Processing Magazine 32.6 (2015): 12-30.
Albanie, Samuel. “Euclidean Distance Matrix Trick.” Retrieved from Visual Geometry Group, University of Oxford (2019).
https://ccrma.stanford.edu/~dattorro/EDM.pdf

Python欧式距离矩阵算法

Python欧式距离矩阵算法

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