gan_metrics.py 文件源码

def trace_sqrt_product(sigma, sigma_v):
    """Find the trace of the positive sqrt of product of covariance matrices.
    '_symmetric_matrix_square_root' only works for symmetric matrices, so we
    cannot just take _symmetric_matrix_square_root(sigma * sigma_v).
    ('sigma' and 'sigma_v' are symmetric, but their product is not necessarily).
    Let sigma = A A so A = sqrt(sigma), and sigma_v = B B.
    We want to find trace(sqrt(sigma sigma_v)) = trace(sqrt(A A B B))
    Note the following properties:
    (i) forall M1, M2: eigenvalues(M1 M2) = eigenvalues(M2 M1)
       => eigenvalues(A A B B) = eigenvalues (A B B A)
    (ii) if M1 = sqrt(M2), then eigenvalues(M1) = sqrt(eigenvalues(M2))
       => eigenvalues(sqrt(sigma sigma_v)) = sqrt(eigenvalues(A B B A))
    (iii) forall M: trace(M) = sum(eigenvalues(M))
       => trace(sqrt(sigma sigma_v)) = sum(eigenvalues(sqrt(sigma sigma_v)))
                                     = sum(sqrt(eigenvalues(A B B A)))
                                     = sum(eigenvalues(sqrt(A B B A)))
                                     = trace(sqrt(A B B A))
                                     = trace(sqrt(A sigma_v A))
    A = sqrt(sigma). Both sigma and A sigma_v A are symmetric, so we **can**
    use the _symmetric_matrix_square_root function to find the roots of these
    matrices.
    Args:
      sigma: a square, symmetric, real, positive semi-definite covariance matrix
      sigma_v: same as sigma
    Returns:
      The trace of the positive square root of sigma*sigma_v
    """

    # Note sqrt_sigma is called "A" in the proof above
    sqrt_sigma = _symmetric_matrix_square_root(sigma)

    # This is sqrt(A sigma_v A) above
    sqrt_a_sigmav_a = tf.matmul(
        sqrt_sigma, tf.matmul(sigma_v, sqrt_sigma))

    return tf.trace(_symmetric_matrix_square_root(sqrt_a_sigmav_a))