low_rank_toolbox.randomized.generalized_nystrom

low_rank_toolbox.randomized.generalized_nystrom(X, r, oversampling_params=(10, 15), epsilon=None, seed=1234, **extra_data)[source]

Generalized Nyström method

Reference:

“Fast and stable randomized low-rank matrix approximation” Nakatsukasa, 2019

Approximation with formula

X ~= X J (K^T X J)^{dagger} K^T X

Parameters:

  • X (LinearOperator) – Matrix to approximate (real-valued only, complex matrices not supported)

  • r (int) – Rank of approximation, must be positive

  • oversampling_params (tuple, optional) – Oversampling parameters (p1, p2) for the two sketch matrices (default: (10, 15))

  • epsilon (float, optional) – When given, perform stable GN with epsilon-truncation for SVD (default: None)

  • seed (int, optional) – Random seed for reproducibility (default: 1234)

  • **extra_data (dict) – Additional arguments passed to QuasiSVD constructor

Returns:

Near optimal best rank r approximation of X in QuasiSVD format

Return type:

QuasiSVD

Notes

This method returns a QuasiSVD (not SVD) because the middle matrix S is typically inverted, making it non-diagonal. Convert to SVD if needed:

result = generalized_nystrom(X, r).to_svd()