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 A A such that A ≈ U diag ( S ) VH A \approx U \operatorname{diag}(S) V^{\text{H}} A≈Udiag(S)VH
Note
The relation of (U, S, V) to PCA is as follows:
A A A is a data matrix with m samples and n features
the V V V columns represent the principal directions
S ∗ ∗ 2 / ( m − 1 ) S ** 2 / (m - 1) S∗∗2/(m−1) contains the eigenvalues of AT A / ( m − 1 ) A^T A / (m - 1) ATA/(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 U U is m x q matrix
S S S is q-vector
V V V is n x q matrix
Note
To obtain repeatable results, reset the seed for the pseudorandom number generator
A (Tensor) – the input tensor of size ( ∗ , m , n ) (*, m, n) (∗,m,n)
q (int, optional) – a slightly overestimated rank of A A 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.
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>`_).
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