text: import numpy as np
  from random import shuffle
  
  def softmax_loss_naive(W, X, y, reg):
    """
    Softmax loss function, naive implementation (with loops)
    Inputs:
    - W: C x D array of weights
    - X: D x N array of data. Data are D-dimensional columns
    - y: 1-dimensional array of length N with labels 0...K-1, for K classes
    - reg: (float) regularization strength
    Returns:
    a tuple of:
    - loss as single float
    - gradient with respect to weights W, an array of same size as W
    """
    # Initialize the loss and gradient to zero.
    loss = 0.0
    dW = np.zeros_like(W)
  
    #############################################################################
    # TODO: Compute the softmax loss and its gradient using explicit loops.     #
    # Store the loss in loss and the gradient in dW. If you are not careful     #
    # here, it is easy to run into numeric instability. Don't forget the        #
    # regularization!                                                           #
    #############################################################################
    num_train  = X.shape[1]
    num_classes = W.shape[0]
  
    fs = W.dot(X)
    for i in xrange(num_train):
      f = fs[:, i]
      # shift the f value to achieve numerial stability
      f -= np.max(f)
  
      # probability interpretation for each class
      p = np.exp(f) / np.sum(np.exp(f), axis=0)
  
      # p = np.exp(f[y[i]]) / np.sum( f, axis=0 )
      loss += - f[y[i]] + np.log( np.sum( np.exp(f), axis=0) )
  
      for j in xrange(num_classes):
        dW[j,:] +=  p[j] * X[:,i]
  
      dW[y[i],:] -= X[:, i]
      ##################################################
      # So the gradient in total is:
      # dW = (p[j] - 1) * X[:, i]     , if j == i
      #    =  p[j]      * X[:, j]     , otherwise
      #
      # just use chain rule to calculate gradient
      ##################################################
  
    # calculate average batch gradient
    dW /= num_train
    # calculate average batch loss
    loss /= num_train
  
    # regularization
    loss += 0.5 * reg * np.sum( W ** 2)
    dW += reg * W
  
    return loss, dW
  
  
  def softmax_loss_vectorized(W, X, y, reg):
    """
    Softmax loss function, vectorized version.
  
    Inputs and outputs are the same as softmax_loss_naive.
    """
    # Initialize the loss and gradient to zero.
    loss = 0.0
    dW = np.zeros_like(W)
  
    #############################################################################
    # TODO: Compute the softmax loss and its gradient using no explicit loops.  #
    # Store the loss in loss and the gradient in dW. If you are not careful     #
    # here, it is easy to run into numeric instability. Don't forget the        #
    # regularization!                                                           #
    #############################################################################
    num_train  = X.shape[1]
    num_classes = W.shape[0]  
  
    f = W.dot(X)
    f -= np.amax(f, axis=0)
  
    prob = np.exp(f) / np.sum(np.exp(f), axis=0)
    prob[y, range(num_train)] -= 1
    
    # Data loss
    loss = np.sum(- f[y, range(num_train)] + np.log( np.sum( np.exp(f), axis=0) )) / num_train
  
    # Weight sub-gradient
    dW = p.dot(X.T) / num_train
  
    # Regularization term
    loss += 0.5 * reg * np.sum( W ** 2)
    dW += reg * W
    return loss, dW
  