Compute approximate nearly optimal truncated Singular Value Decomposition of an input matrix a.

Decomposition is truncated to nb_components.

Increasing nb_oversamples or nb_iter increases the accuracy of the approximation

Input:

• A, a matrix of shape M, N
• nb_components: rank/dimension of the approximation i.e. number of singular values and vectors to extract Must be lower than min(M, N) Default to 2 for 2D visualization
• nb_oversamples: Additional number of random projections in the sampling matrix Recommended range 2 .. 10
• nb_power_iter: Number of power iterations Power iterations enforce rapid decay of singular values and allow the algorithm to sample dominant singular values and suppress irrelevant information. Useful for noisy problems.

Returns: with K = nb_components

• U: Unitary matrix of shape M, K with rank-K approximation of left singular vectors as columns
• S: Rank-k approximation of singular values diagonal of length K in decreasing order
• Vh: Unitary matrix of shape K, N with rank-K approximation of right singular vectors as rows

This is an approximate solution of the equation: A = U S V.h

• with S being a diagonal matrix of singular values
• with V being the right singular vectors and V.h being the hermitian conjugate of V for real matrices, this is equivalent to V.t (transpose)

⚠️: Input must not contain NaN

Exception:

• This can throw if the algorithm did not converge.

References:

• A Randomized Algorithm for Principal Component Analysis Rockhlin et al, 2009 https://epubs.siam.org/doi/10.1137/080736417
• Finding structure with randomness, Probabilistic algorithms for constructing approximate matrix decomposition Halko et al, 2011 http://users.cms.caltech.edu/~jtropp/papers/HMT11-Finding-Structure-SIREV.pdf
• A randomized algorithm for the decomposition of matrices Martinsson et al, 2011 https://www.sciencedirect.com/science/article/pii/S1063520310000242
• Randomized Algorithms for Low-Rank Matrix Factorizations: Sharp performance bounds Witten et al, 2013 https://arxiv.org/abs/1308.5697
• Subspace Iteration Randomization and Singular Value Problems Gu, 2014 https://epubs.siam.org/doi/10.1137/130938700
• An implementation of a randomized algorithm for principal component analysis Szlam et al, 2014 https://arxiv.org/abs/1412.3510
• Randomized methods for matrix computations Martinsson, 2016 https://arxiv.org/abs/1607.01649
• RSVDPACK: An implementation of randomized algorithms for computing the singular value, interpolative, and CUR decompositions of matrices on multi-core and GPU architectures Voronin, 2016 https://arxiv.org/abs/1502.05366
• Randomized Matrix Decompositions using R Erichson, 2016 https://arxiv.org/abs/1608.02148