Before revision:  We extend perturbation results for the singular subspace decomposition to derive the first-order perturbation expansion of the truncated operator about a matrix with rank no less than r. Observing that the first-order expansion can be greatly simplified when the matrix has exact rank r, we further show that the singular value truncation admits a simple second-order perturbation expansion about a rank-r matrix.
After revision:  We extend perturbation results for the singular subspace decomposition to derive the first-order perturbation expansion of the truncated operator about a matrix with rank greater than or equal to r. Observing that the first-order expansion can be greatly simplified when the matrix has exact rank r, we further show that the singular value truncation admits a simple second-order perturbation expansion about a rank-r matrix.