def zca_white(x):
  """ perform zca whitening 
  
      inputs:
        x: numpy array of images

      outputs:
        y: numpy array of whitened images        
  """

  # *** put code for zca whitening here ***

  return y

def sample(x, patch_size=16, num_patches=10):
    ''' randomly sample patches from x 
    
        inputs:
            x: numpy array of images
            patch_size: patch dims for patch_size x patch_size images 
                        (default 16)
            num_patches: number of patches to sample (default 10)

        outputs:
            y: numpy array of patches
    '''

    return y

def ica(x, **args):
    ''' perform independent component analysis (ICA)
    
        inputs:
            x: numpy array of images
            args:
                lr: learning rate (default 1e-3)
                nsteps: maximum iterations (default 1000)
                k: number of latent variables (defualt 20)
        
        returns:
            L: numpy array of loss function value for all iterations
            W: numpy array of ICA basis vectors
    '''

    # default parameters
    if not len(args):
        args['lr'] = 1
        args['nsteps'] = 200
        args['k'] = 64

    lr=args['lr']
    nsteps = args['nsteps']
    k = args['k']
    
    # ***initialize variables here***

    '''training loop using graident descent'''
    for step in range(nsteps):
        # ***insert gradient descent code here***
        ''' use backtracking line search '''
        
        # print loss
        print('step: {} / {}, L: {}'.format(step, nsteps, L[step]))
        
    return L, W