mars.tensor.linalg.svd#

mars.tensor.linalg.svd(a, method='tsqr')[source]#

Singular Value Decomposition.

When a is a 2D tensor, it is factorized as u @ np.diag(s) @ vh = (u * s) @ vh, where u and vh are 2D unitary tensors and s is a 1D tensor of a’s singular values. When a is higher-dimensional, SVD is applied in stacked mode as explained below.

Parameters

  • a ((..., M, N) array_like) – A real or complex tensor with a.ndim >= 2.

  • method ({'tsqr'}, optional) –

    method to calculate qr factorization, tsqr as default

    TSQR is presented in:

    A. Benson, D. Gleich, and J. Demmel. Direct QR factorizations for tall-and-skinny matrices in MapReduce architectures. IEEE International Conference on Big Data, 2013. http://arxiv.org/abs/1301.1071

Returns

  • u ({ (…, M, M), (…, M, K) } tensor) – Unitary tensor(s). The first a.ndim - 2 dimensions have the same size as those of the input a. The size of the last two dimensions depends on the value of full_matrices. Only returned when compute_uv is True.

  • s ((…, K) tensor) – Vector(s) with the singular values, within each vector sorted in descending order. The first a.ndim - 2 dimensions have the same size as those of the input a.

  • vh ({ (…, N, N), (…, K, N) } tensor) – Unitary tensor(s). The first a.ndim - 2 dimensions have the same size as those of the input a. The size of the last two dimensions depends on the value of full_matrices. Only returned when compute_uv is True.

Raises

LinAlgError – If SVD computation does not converge.

Notes

SVD is usually described for the factorization of a 2D matrix \(A\). The higher-dimensional case will be discussed below. In the 2D case, SVD is written as \(A = U S V^H\), where \(A = a\), \(U= u\), \(S= \mathtt{np.diag}(s)\) and \(V^H = vh\). The 1D tensor s contains the singular values of a and u and vh are unitary. The rows of vh are the eigenvectors of \(A^H A\) and the columns of u are the eigenvectors of \(A A^H\). In both cases the corresponding (possibly non-zero) eigenvalues are given by s**2.

If a has more than two dimensions, then broadcasting rules apply, as explained in Linear algebra on several matrices at once. This means that SVD is working in “stacked” mode: it iterates over all indices of the first a.ndim - 2 dimensions and for each combination SVD is applied to the last two indices. The matrix a can be reconstructed from the decomposition with either (u * s[..., None, :]) @ vh or u @ (s[..., None] * vh). (The @ operator can be replaced by the function mt.matmul for python versions below 3.5.)

Examples

>>> import mars.tensor as mt
>>> a = mt.random.randn(9, 6) + 1j*mt.random.randn(9, 6)
>>> b = mt.random.randn(2, 7, 8, 3) + 1j*mt.random.randn(2, 7, 8, 3)

Reconstruction based on reduced SVD, 2D case:

>>> u, s, vh = mt.linalg.svd(a)
>>> u.shape, s.shape, vh.shape
((9, 6), (6,), (6, 6))
>>> np.allclose(a, np.dot(u * s, vh))
True
>>> smat = np.diag(s)
>>> np.allclose(a, np.dot(u, np.dot(smat, vh)))
True