计算两个矩阵之间的欧式距离

在我们使用k-NN模型时，需要计算测试集中每一点到训练集中每一点的欧氏距离，即需要求得两矩阵之间的欧氏距离。在实现k-NN算法时通常有三种方案，分别是使用两层循环，使用一层循环和不使用循环。

使用两层循环

分别对训练集和测试集中的数据进行循环遍历，计算每两个点之间的欧式距离，然后赋值给dist矩阵。此算法没有经过任何优化。

num_test = X.shape[0]
    num_train = self.X_train.shape[0]
    dists = np.zeros((num_test, num_train)) 
    for i in xrange(num_test):
      for j in xrange(num_train):
        #####################################################################
        # TODO:                                                             #
        # Compute the l2 distance between the ith test point and the jth    #
        # training point, and store the result in dists[i, j]. You should   #
        # not use a loop over dimension.                                    #
        #####################################################################
        # pass
        dists[i][j] = np.sqrt(np.sum(np.square(X[i] - self.X_train[j])))
        #####################################################################
        #                       END OF YOUR CODE                            #
        #####################################################################
    return dists

使用一层循环

使用矩阵表示训练集的数据，计算测试集中每一点到训练集矩阵的距离，可以对算法优化为只使用一层循环。

def compute_distances_one_loop(self, X):
    """
    Compute the distance between each test point in X and each training point
    in self.X_train using a single loop over the test data.
    Input / Output: Same as compute_distances_two_loops
    """
    num_test = X.shape[0]
    num_train = self.X_train.shape[0]
    dists = np.zeros((num_test, num_train))
    for i in xrange(num_test):
      #######################################################################
      # TODO:                                                               #
      # Compute the l2 distance between the ith test point and all training #
      # points, and store the result in dists[i, :].                        #
      #######################################################################
      # pass
      dists[i] = np.sqrt(np.sum(np.square(self.X_train - X[i]), axis = 1))
      #######################################################################
      #                         END OF YOUR CODE                            #
      #######################################################################
    return dists

不使用循环

运算效率最高的算法是将训练集和测试集都使用矩阵表示，然后使用矩阵运算的方法替代之前的循环操作。但此操作需要我们对矩阵的运算规则非常熟悉。接下来着重记录如何计算两个矩阵之间的欧式距离。

记录测试集矩阵P的大小为M*D，训练集矩阵C的大小为N*D（测试集中共有M个点，每个点为D维特征向量。训练集中共有N个点，每个点为D维特征向量）
记 Pi P i 是P的第i行，记 Cj C j 是C的第j行
Pi=[Pi1Pi2⋯PiD] P i = [ P i 1 P i 2 ⋯ P i D ]   Cj=[Cj1Cj2⋯CjD] C j = [ C j 1 C j 2 ⋯ C j D ]

---

首先计算 Pi P i 和 Cj C j 之间的距离dist(i,j)
d(Pi,Cj)=(Pi1−Cj1)2+(Pi2−Cj2)2+⋯+(PiD−CjD)2−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√=(P2i1+P2i2+⋯+P2iD)+(C2j1+C2j2+⋯+C2jD)−2×(Pi1Cj1+Pi2Cj2+⋯+PiDCiD)−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√=∥Pi∥2+∥Cj∥2−2×PiCTj−−−−−−−−−−−−−−−−−−−−−√ d ( P i , C j ) = ( P i 1 − C j 1 ) 2 + ( P i 2 − C j 2 ) 2 + ⋯ + ( P i D − C j D ) 2 = ( P i 1 2 + P i 2 2 + ⋯ + P i D 2 ) + ( C j 1 2 + C j 2 2 + ⋯ + C j D 2 ) − 2 × ( P i 1 C j 1 + P i 2 C j 2 + ⋯ + P i D C i D ) = ‖ P i ‖ 2 + ‖ C j ‖ 2 − 2 × P i C j T

---

我们可以推广到距离矩阵的第i行的计算公式
dist[i]=(∥Pi∥2∥Pi∥2⋯∥Pi∥2)+(∥C1∥2∥C2∥2⋯∥CN∥2)−2×Pi(CT1CT2⋯CTN)−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√=(∥Pi∥2∥Pi∥2⋯∥Pi∥2)+(∥C1∥2∥C2∥2⋯∥CN∥2)−2×PiCT−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√ d i s t [ i ] = ( ‖ P i ‖ 2 ‖ P i ‖ 2 ⋯ ‖ P i ‖ 2 ) + ( ‖ C 1 ‖ 2 ‖ C 2 ‖ 2 ⋯ ‖ C N ‖ 2 ) − 2 × P i ( C 1 T C 2 T ⋯ C N T ) = ( ‖ P i ‖ 2 ‖ P i ‖ 2 ⋯ ‖ P i ‖ 2 ) + ( ‖ C 1 ‖ 2 ‖ C 2 ‖ 2 ⋯ ‖ C N ‖ 2 ) − 2 × P i C T

---

继续将公式推广为整个距离矩阵
dist=⎛⎝⎜⎜⎜⎜⎜⎜∥P1∥2∥P2∥2⋮∥PM∥2∥P1∥2∥P2∥2⋮∥PM∥2⋯⋯⋱⋯∥P1∥2∥P2∥2⋮∥PM∥2⎞⎠⎟⎟⎟⎟⎟⎟+⎛⎝⎜⎜⎜⎜⎜⎜∥C1∥2∥C1∥2⋮∥C1∥2∥C2∥2∥C2∥2⋮∥C2∥2⋯⋯⋱⋯∥CN∥2∥CN∥2⋮∥CN∥2⎞⎠⎟⎟⎟⎟⎟⎟−2×PCT−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−⎷ d i s t = ( ‖ P 1 ‖ 2 ‖ P 1 ‖ 2 ⋯ ‖ P 1 ‖ 2 ‖ P 2 ‖ 2 ‖ P 2 ‖ 2 ⋯ ‖ P 2 ‖ 2 ⋮ ⋮ ⋱ ⋮ ‖ P M ‖ 2 ‖ P M ‖ 2 ⋯ ‖ P M ‖ 2 ) + ( ‖ C 1 ‖ 2 ‖ C 2 ‖ 2 ⋯ ‖ C N ‖ 2 ‖ C 1 ‖ 2 ‖ C 2 ‖ 2 ⋯ ‖ C N ‖ 2 ⋮ ⋮ ⋱ ⋮ ‖ C 1 ‖ 2 ‖ C 2 ‖ 2 ⋯ ‖ C N ‖ 2 ) − 2 × P C T

表示为python代码：

def compute_distances_no_loops(self, X):
    """
    Compute the distance between each test point in X and each training point
    in self.X_train using no explicit loops.

    Input / Output: Same as compute_distances_two_loops
    """
    num_test = X.shape[0]
    num_train = self.X_train.shape[0]
    dists = np.zeros((num_test, num_train)) 
    #########################################################################
    # TODO:                                                                 #
    # Compute the l2 distance between all test points and all training      #
    # points without using any explicit loops, and store the result in      #
    # dists.                                                                #
    #                                                                       #
    # You should implement this function using only basic array operations; #
    # in particular you should not use functions from scipy.                #
    #                                                                       #
    # HINT: Try to formulate the l2 distance using matrix multiplication    #
    #       and two broadcast sums.                                         #
    #########################################################################
    # pass
    dists = np.sqrt(-2*np.dot(X, self.X_train.T) + np.sum(np.square(self.X_train), axis = 1) + np.transpose([np.sum(np.square(X), axis = 1)]))
    #########################################################################
    #                         END OF YOUR CODE                              #
    #########################################################################
    return dists