torch.pca_lowrank

torch.pca_lowrank(A, q=None, center=True, niter=2)

Performs linear Principal Component Analysis (PCA) on a low-rank matrix, batches of such matrices, or sparse matrix.

This function returns a namedtuple (U, S, V) which is the nearly optimal approximation of a singular value decomposition of a centered matrix \(A\) such that \(A \approx U \operatorname{diag}(S) V^{\text{H}}\)

Note

The relation of (U, S, V) to PCA is as follows:

• \(A\) is a data matrix with m samples and n features

• the \(V\) columns represent the principal directions

• \(S ** 2 / (m - 1)\) contains the eigenvalues of \(A^T A / (m - 1)\) which is the covariance of A when center=True is provided.

• matmul(A, V[:, :k]) projects data to the first k principal components

Note

Different from the standard SVD, the size of returned matrices depend on the specified rank and q values as follows:

• \(U\) is m x q matrix

• \(S\) is q-vector

• \(V\) is n x q matrix

Note

To obtain repeatable results, reset the seed for the pseudorandom number generator

Parameters:

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

• q (int, optional) – a slightly overestimated rank of \(A\). By default, q = min(6, m, n).

• center (bool, optional) – if True, center the input tensor, otherwise, assume that the input is centered.

• niter (int, optional) – the number of subspace iterations to conduct; niter must be a nonnegative integer, and defaults to 2.

Return type:

Tuple[Tensor, Tensor, Tensor]

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 <http://arxiv.org/abs/0909.4061>`_).