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Randomized strong rank-revealing QR for column subset selection and low-rank matrix approximation
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abstract
We discuss a randomized strong rank-revealing QR factorization that effectively reveals the spectrum of a matrix $\textbf{M}$. This factorization can be used to address problems such as selecting a subset of the columns of $\textbf{M}$, computing its low-rank approximation, estimating its rank, or approximating its null space. Given a random sketching matrix $\pmb{\Omega}$ that satisfies the $\epsilon$-embedding property for a subspace within the range of $\textbf{M}$, the factorization relies on selecting columns that allow to reveal the spectrum via a deterministic strong rank-revealing QR factorization of $\textbf{M}^{sk} = \pmb{\Omega}\textbf{M}$, the sketch of $\textbf{M}$. We show that this selection leads to a factorization with strong rank-revealing properties, making it suitable for approximating the singular values of $\textbf{M}$.
Forward citations
Cited by 4 Pith papers
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Computing Strong Rank-Revealing Factorizations for Matrices with Orthonormal Rows
Bischof-Stewart pivoting on orthonormal-row matrices provably yields strong rank-revealing QR factorizations, and a randomized variant attains the same column-selection bounds with large practical speedups.
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Accelerating the Canonical Polyadic Alternating Least Squares Optimization via a Randomized Interpolative Decomposition
Randomized QR pivots of the target tensor supply a fixed leverage-score-like sampling for CPD-ALS, reducing tensor re-sampling and storage overhead.
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Randomized biorthogonalization through a two-sided Gram-Schmidt process
Randomized two-sided Gram-Schmidt builds bases Q and P with (ΩQ)^T ΩP = I, at about half the cost and often with better conditioning than the deterministic version.
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Efficient QR-based Column Subset Selection through Randomized Sparse Embeddings
SE-QRCS selects k representative columns of a wide matrix by applying strong rank-revealing QR to a sparse sketch and to the small induced column set, with spectral bounds that shrink the dependence on n.
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