torch.svd_lowrank

torch.svd_lowrank(A, q=6, niter=2, M=None)

Return the singular value decomposition (U, S, V) of a matrix, batches of matrices, or a sparse matrix \(A\) such that \(A \approx U \operatorname{diag}(S) V^{\text{H}}\). In case \(M\) is given, then SVD is computed for the matrix \(A - M\).

Note

The implementation is based on the Algorithm 5.1 from Halko et al., 2009.

Note

For an adequate approximation of a k-rank matrix \(A\), where k is not known in advance but could be estimated, the number of \(Q\) columns, q, can be choosen according to the following criteria: in general, \(k <= q <= min(2*k, m, n)\). For large low-rank matrices, take \(q = k + 5..10\). If k is relatively small compared to \(min(m, n)\), choosing \(q = k + 0..2\) may be sufficient.

Note

This is a randomized method. To obtain repeatable results, set the seed for the pseudorandom number generator

Note

In general, use the full-rank SVD implementation torch.linalg.svd() for dense matrices due to its 10x higher performance characteristics. The low-rank SVD will be useful for huge sparse matrices that torch.linalg.svd() cannot handle.

Args::

A (Tensor): the input tensor of size \((*, m, n)\)

q (int, optional): a slightly overestimated rank of A.

niter (int, optional): the number of subspace iterations to

conduct; niter must be a nonnegative integer, and defaults to 2

M (Tensor, optional): the input tensor’s mean of size

\((*, m, n)\), which will be broadcasted to the size of A in this function.

References::

• Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp, Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions, arXiv:0909.4061 [math.NA; math.PR], 2009 (available at arXiv).

Return type:

Tuple[Tensor, Tensor, Tensor]