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tf.linalg.svd

Computes the singular value decompositions of one or more matrices.

Computes the SVD of each inner matrix in tensor such that tensor[..., :, :] = u[..., :, :] * diag(s[..., :, :]) * transpose(conj(v[..., :, :]))

# a is a tensor.
# s is a tensor of singular values.
# u is a tensor of left singular vectors.
# v is a tensor of right singular vectors.
s, u, v = svd(a)
s = svd(a, compute_uv=False)

tensor Tensor of shape [..., M, N]. Let P be the minimum of M and N.
full_matrices If true, compute full-sized u and v. If false (the default), compute only the leading P singular vectors. Ignored if compute_uv is False.
compute_uv If True then left and right singular vectors will be computed and returned in u and v, respectively. Otherwise, only the singular values will be computed, which can be significantly faster.
name string, optional name of the operation.

s Singular values. Shape is [..., P]. The values are sorted in reverse order of magnitude, so s[..., 0] is the largest value, s[..., 1] is the second largest, etc.
u Left singular vectors. If full_matrices is False (default) then shape is [..., M, P]; if full_matrices is True then shape is [..., M, M]. Not returned if compute_uv is False.
v Right singular vectors. If full_matrices is False (default) then shape is [..., N, P]. If full_matrices is True then shape is [..., N, N]. Not returned if compute_uv is False.

Numpy Compatibility

Mostly equivalent to numpy.linalg.svd, except that

  • The order of output arguments here is s, u, v when compute_uv is True, as opposed to u, s, v for numpy.linalg.svd.
  • full_matrices is False by default as opposed to True for numpy.linalg.svd.
  • tf.linalg.svd uses the standard definition of the SVD \(A = U \Sigma V^H\), such that the left singular vectors of a are the columns of u, while the right singular vectors of a are the columns of v. On the other hand, numpy.linalg.svd returns the adjoint \(V^H\) as the third output argument.
import tensorflow as tf
import numpy as np
s, u, v = tf.linalg.svd(a)
tf_a_approx = tf.matmul(u, tf.matmul(tf.linalg.diag(s), v, adjoint_b=True))
u, s, v_adj = np.linalg.svd(a, full_matrices=False)
np_a_approx = np.dot(u, np.dot(np.diag(s), v_adj))
# tf_a_approx and np_a_approx should be numerically close.