python - tensorflow 中的矩阵行列式微分

Question

我对使用 TensorFlow 计算矩阵行列式的导数很感兴趣。我通过实验可以看出，TensorFlow 并没有实现通过行列式求微分的方法:

LookupError: No gradient defined for operation 'MatrixDeterminant' 
(op type: MatrixDeterminant)

进一步调查表明，实际上可以计算导数；参见例如 Jacobi's formula .我确定，为了实现这种通过行列式进行区分的方法，我需要使用函数装饰器，

@tf.RegisterGradient("MatrixDeterminant")
def _sub_grad(op, grad):
    ...

但是，我对 tensorflow 不够熟悉，无法理解这是如何实现的。有没有人对此事有任何见解？

这是我遇到此问题的示例:

x = tf.Variable(tf.ones(shape=[1]))
y = tf.Variable(tf.ones(shape=[1]))

A = tf.reshape(
    tf.pack([tf.sin(x), tf.zeros([1, ]), tf.zeros([1, ]), tf.cos(y)]), (2,2)
)
loss = tf.square(tf.matrix_determinant(A))


optimizer = tf.train.GradientDescentOptimizer(0.001)
train = optimizer.minimize(loss)

init = tf.initialize_all_variables()
sess = tf.Session()
sess.run(init)


for step in xrange(100):
    sess.run(train)
    print sess.run(x)

Answer 1

请查看“在 Python 中实现渐变”部分 here

具体来说，你可以如下实现

@ops.RegisterGradient("MatrixDeterminant")
def _MatrixDeterminantGrad(op, grad):
  """Gradient for MatrixDeterminant. Use formula from 2.2.4 from
  An extended collection of matrix derivative results for forward and reverse
  mode algorithmic differentiation by Mike Giles
  -- http://eprints.maths.ox.ac.uk/1079/1/NA-08-01.pdf
"""
  A = op.inputs[0]
  C = op.outputs[0]
  Ainv = tf.matrix_inverse(A)
  return grad*C*tf.transpose(Ainv)

然后一个简单的训练循环来检查它是否有效:

a0 = np.array([[1,2],[3,4]]).astype(np.float32)
a = tf.Variable(a0)
b = tf.square(tf.matrix_determinant(a))
init_op = tf.initialize_all_variables()
sess = tf.InteractiveSession()
init_op.run()

minimization_steps = 50
learning_rate = 0.001
optimizer = tf.train.GradientDescentOptimizer(learning_rate)
train_op = optimizer.minimize(b)

losses = []
for i in range(minimization_steps):
  train_op.run()
  losses.append(b.eval())

然后你可以想象随着时间的推移你的损失

import matplotlib.pyplot as plt

plt.ylabel("Determinant Squared")
plt.xlabel("Iterations")
plt.plot(losses)

应该看到这样的东西