Question

我正在尝试实施 ZCA美白，并发现了一些文章要做，但它们有点令人困惑..有人能为我发光吗？

感谢任何提示或帮助！

以下是我读过的文章：

http://courses.media.mit.edu/2010fall/mas622j/whiten.pdf http://bbabenko.tumblr.com/post/86756017649/learning-low-level-vision-feautres-in-10-lines-of

我尝试了几件事但其中大多数我都不明白，而且我已经锁定了一些步骤。现在我把它作为重新开始的基础：

dtype = np.float32
data = np.loadtxt("../inputData/train.csv", dtype=dtype, delimiter=',', skiprows=1)
img = ((data[1,1:]).reshape((28,28)).astype('uint8')*255)

Answer 1

这是一个用于生成ZCA白化矩阵的python函数：

def zca_whitening_matrix(X):
    """
    Function to compute ZCA whitening matrix (aka Mahalanobis whitening).
    INPUT:  X: [M x N] matrix.
        Rows: Variables
        Columns: Observations
    OUTPUT: ZCAMatrix: [M x M] matrix
    """
    # Covariance matrix [column-wise variables]: Sigma = (X-mu)' * (X-mu) / N
    sigma = np.cov(X, rowvar=True) # [M x M]
    # Singular Value Decomposition. X = U * np.diag(S) * V
    U,S,V = np.linalg.svd(sigma)
        # U: [M x M] eigenvectors of sigma.
        # S: [M x 1] eigenvalues of sigma.
        # V: [M x M] transpose of U
    # Whitening constant: prevents division by zero
    epsilon = 1e-5
    # ZCA Whitening matrix: U * Lambda * U'
    ZCAMatrix = np.dot(U, np.dot(np.diag(1.0/np.sqrt(S + epsilon)), U.T)) # [M x M]
    return ZCAMatrix

用法示例：

X = np.array([[0, 2, 2], [1, 1, 0], [2, 0, 1], [1, 3, 5], [10, 10, 10] ]) # Input: X [5 x 3] matrix
ZCAMatrix = zca_whitening_matrix(X) # get ZCAMatrix
ZCAMatrix # [5 x 5] matrix
xZCAMatrix = np.dot(ZCAMatrix, X) # project X onto the ZCAMatrix
xZCAMatrix # [5 x 3] matrix

希望它有所帮助！

有关Edgar Andrés Margffoy Tuay答案错误原因的详细信息：正如R.M comment Edgar Andrés Margffoy Tuay here中指出的那样美白功能包含一个小而重要的错误：应删除np.diag(S)。 Numpy返回S作为m x 1向量而不是m x m矩阵（对于其他svd实现（例如Matlab）是常见的）。因此，ZCAMatrix变量变为m x 1向量而不是m x m矩阵（当输入为m x n时）。（此外，Andfoy答案中的协方差矩阵仅在X预先居中时才有效，即均值为0。）

ZCA的其他参考：您可以在Python中看到斯坦福UFLDL ZCA美白练习this的完整答案。

Answer 2

您的数据是否存储在mxn矩阵中？其中m是数据的维数，n是案例的总数？如果情况并非如此，则应调整数据大小。例如，如果您的图像大小为28x28并且您只有一个图像，则应该具有1x784向量。你可以使用这个功能：

import numpy as np

def flatten_matrix(matrix):
    vector = matrix.flatten(1)
    vector = vector.reshape(1, len(vector))
    return vector

然后使用以下方法将ZCA Whitening应用于训练集：

def zca_whitening(inputs):
    sigma = np.dot(inputs, inputs.T)/inputs.shape[1] #Correlation matrix
    U,S,V = np.linalg.svd(sigma) #Singular Value Decomposition
    epsilon = 0.1                #Whitening constant, it prevents division by zero
    ZCAMatrix = np.dot(np.dot(U, np.diag(1.0/np.sqrt(np.diag(S) + epsilon))), U.T)                     #ZCA Whitening matrix
    return np.dot(ZCAMatrix, inputs)   #Data whitening

保存ZCAMatrix矩阵很重要，如果要在训练神经网络后进行预测，则应该将测试用例相乘。

最后，我邀请您参加http://ufldl.stanford.edu/wiki/index.php/UFLDL_Tutorial或http://ufldl.stanford.edu/tutorial/的斯坦福大学UFLDL教程。他们对MATLAB有很好的解释和一些编程练习，但几乎所有在MATLAB上找到的函数都在同名的Numpy上。我希望这可以提供一个见解。

Answer 3

我可能有点迟到了讨论，但最近我发现这个帖子是因为我在TensorFlow中努力实现ZCA，因为我的PC处理器太差，处理大量数据的速度太慢。

如果有人有兴趣，我已经在TensorFlow中实现了gist我对ZCA的实施：

import tensorflow as tf

from keras.datasets import mnist

import numpy as np

tf.enable_eager_execution()

assert tf.executing_eagerly()


class ZCA(object):
    """
    Simple ZCA aka Mahalanobis transformation class made in TensorFlow.
    The code was largely ported from Keras ImageDataGenerator
    """

    def __init__(self, epsilon=1e-5, dtype='float64'):

        """epsilon is the normalization constant, dtype refers to the data type used in the computation.
         WARNING: the default precision is set to float64 as i have found that when computing the mean tensorflow'
         and numpy results can differ by a substantial amount.
         Usage: fit method computes the principal components and should be called first,
                compute method returns the actual transformed tensor
         NOTE : The input to both methods must be a 4D tensor.
        """

        assert dtype is 'float32' or 'float64', "precision must be float32 or float64"

        self.epsilon = epsilon
        self.dtype = dtype
        self.princ_comp = None
        self.mean = None

    def _featurewise_center(self, images_tensor):

        if self.mean is None:
            self.mean, _ = tf.nn.moments(images_tensor, axes=(0, 1, 2))
            broadcast_shape = [1, 1, 1]
            broadcast_shape[2] = images_tensor.shape[3]
            self.mean = tf.reshape(self.mean, broadcast_shape)

        norm_images = tf.subtract(images_tensor, self.mean)

        return norm_images

    def fit(self, images_tensor):

        assert images_tensor.shape[3], "The input should be a 4D tensor"

        if images_tensor.dtype is not self.dtype:  # numerical error for float32

            images_tensor = tf.cast(images_tensor, self.dtype)

        images_tensor = self._featurewise_center(images_tensor)

        flat = tf.reshape(images_tensor, (-1, np.prod(images_tensor.shape[1:].as_list())))
        sigma = tf.div(tf.matmul(tf.transpose(flat), flat), tf.cast(flat.shape[0], self.dtype))
        s, u, _ = tf.svd(sigma)
        s_inv = tf.div(tf.cast(1, self.dtype), (tf.sqrt(tf.add(s[tf.newaxis], self.epsilon))))
        self.princ_comp = tf.matmul(tf.multiply(u, s_inv), tf.transpose(u))

    def compute(self, images_tensor):

        assert images_tensor.shape[3], "The input should be a 4D tensor"

        assert self.princ_comp is not None, "Fit method should be called first"

        if images_tensor.dtype is not self.dtype:
            images_tensor = tf.cast(images_tensor, self.dtype)

        images_tensors = self._featurewise_center(images_tensor)

        flatx = tf.cast(tf.reshape(images_tensors, (-1, np.prod(images_tensors.shape[1:]))), self.dtype)
        whitex = tf.matmul(flatx, self.princ_comp)
        x = tf.reshape(whitex, images_tensors.shape)

        return x


def main():
    import matplotlib.pyplot as plt

    train_set, test_set = mnist.load_data()
    x_train, y_train = train_set

    zca1 = ZCA(epsilon=1e-5, dtype='float64')

    # input should be a 4D tensor

    x_train = x_train.reshape(*x_train.shape, 1)
    zca1.fit(x_train)
    x_train_transf = zca1.compute(x_train)

    # reshaping to 28*28 and casting to uint8 for plotting

    x_train_transf = tf.reshape(x_train_transf, x_train_transf.shape[0:3])


    fig, axes = plt.subplots(3, 3)

    for i, ax in enumerate(axes.flat):
        # Plot image.
        ax.imshow(x_train_transf[i],
                  cmap='binary'
                  )

        xlabel = "True: %d" % y_train[i]
        ax.set_xlabel(xlabel)
        ax.set_xticks([])
        ax.set_yticks([])

    plt.show()


if __name__ == '__main__':
main()

我知道这不是原始问题的正确答案，但对于那些正在寻找ZCA的GPU实现但却找不到的人来说，它仍然有用。

Answer 4

虽然两个答案都引用UFLDL tutorial，但它们似乎都没有使用其中描述的步骤。

因此，我认为根据教程提供简单实施PCA / ZCA白化的答案可能并不是一个坏主意：

import numpy as np

# generate some random, 2D data
x = np.random.randn(1000, 2)
# and center it
x_c = x - np.mean(x, 0)

# compute the 2x2 covariance matrix
# (remember that covariance matrix is symmetric)
sigma = np.cov(x, rowvar=False)
# and extract eigenvalues and eigenvectors
# using the algorithm for symmetric matrices
l,u = np.linalg.eigh(sigma)
# NOTE that for symmetric matrices,
# eigenvalues and singular values are the same.
# u, l, _ = np.linalg.svd(sigma) should thus give equivalent results

# rotate the (centered) data to decorrelate it
x_rot = np.dot(x_c, u)
# check that the covariance is diagonal (indicating decorrelation)
np.allclose(np.cov(x_rot.T), np.diag(np.diag(np.cov(x_rot.T))))

# scale the data by eigenvalues to get unit variance
x_white = x_rot / np.sqrt(l)
# have the whitened data be closer to the original data
x_zca = np.dot(x_white, u.T)

我假设你可以自己把它包装成一个函数......

一种替代实现，有时可能更有效（并且应该从我从数学数学课程中记得的更好的稳定性），将是：

_,s,v = np.linalg.svd(x_c)
x_rot = np.dot(x_c, v.T)
x_white = x_rot / s * np.sqrt(len(x_c) - 1)
x_zca = np.dot(x_white, v)

对于背后的数学，我从交叉验证中引用this excellent answer。

Answer 5

这适用于48x48的数组：

def flatten_matrix(matrix):
    vector = matrix.flatten(order='F')
    vector = vector.reshape(1, len(vector))
    return vector

def zca_whitening(inputs): 
    sigma = np.dot(inputs, inputs.T)/inputs.shape[1] #Correlation matrix
    U,S,V = np.linalg.svd(sigma) #Singular Value Decomposition
    epsilon = 0.1                #Whitening constant, it prevents division by zero
    ZCAMatrix = np.dot(np.dot(U, np.diag(1.0/np.sqrt(np.diag(S) + epsilon))), U.T)  #ZCA Whitening matrix
    return np.dot(ZCAMatrix, inputs)   #Data whitening

def global_contrast_normalize(X, scale=1., subtract_mean=True, use_std=True,
                              sqrt_bias=10, min_divisor=1e-8):

    """
    __author__ = "David Warde-Farley"
    __copyright__ = "Copyright 2012, Universite de Montreal"
    __credits__ = ["David Warde-Farley"]
    __license__ = "3-clause BSD"
    __email__ = "wardefar@iro"
    __maintainer__ = "David Warde-Farley"
    .. [1] A. Coates, H. Lee and A. Ng. "An Analysis of Single-Layer
       Networks in Unsupervised Feature Learning". AISTATS 14, 2011.
       http://www.stanford.edu/~acoates/papers/coatesleeng_aistats_2011.pdf
    """
    assert X.ndim == 2, "X.ndim must be 2"
    scale = float(scale)
    assert scale >= min_divisor

    mean = X.mean(axis=1)
    if subtract_mean:
        X = X - mean[:, np.newaxis]  
    else:
        X = X.copy()
    if use_std:
        ddof = 1
        if X.shape[1] == 1:
            ddof = 0
        normalizers = np.sqrt(sqrt_bias + X.var(axis=1, ddof=ddof)) / scale
    else:
        normalizers = np.sqrt(sqrt_bias + (X ** 2).sum(axis=1)) / scale
    normalizers[normalizers < min_divisor] = 1.
    X /= normalizers[:, np.newaxis]  # Does not make a copy.
    return X

def ZeroCenter(data):
    data = data - np.mean(data,axis=0)
    return data

def Zerocenter_ZCA_whitening_Global_Contrast_Normalize(data):
    numpy_data = np.array(data).reshape(48,48)
    data2 = ZeroCenter(numpy_data)
    data3 = zca_whitening(flatten_matrix(data2)).reshape(48,48)
    data4 = global_contrast_normalize(data3)
    data5 = np.rot90(data4,3)
    return data5

例如来自此图片：

返回：

以下是代码：

https://gist.github.com/m-alcu/45f4a083cb5e388d2ed26ace4392ed66，需要将fer2013.csv文件放在同一目录中（https://www.kaggle.com/c/challenges-in-representation-learning-facial-expression-recognition-challenge/data）

如何实施ZCA美白？蟒蛇

5 个答案: