Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Low-rank approximation — where it appears

3 essays name this object, across 2 fields. What follows is each of them, and the objects they name alongside it.

Also named here as singular value decomposition — the same set of essays touches all of them, so they are one junction rather than several.

04812162010⁻¹10⁻⁰.⁵1target rank k‖A − A_k‖₂published boundrandomisedσ_{k+1}, optimalhow far apart the three areworst seed spread1.6bound / median at k = 125.9median / optimum at k = 121.960×60, 6 seeds, oversampling p = 5band is best to worst

A bound that holds with probability

Every other guarantee on this site is deterministic. The randomised low-rank approximation offers one that holds with a probability, the seed changes the answer, and the honest figure is a band rather than a line.

randomised · randomised 12345678910⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1rank k of the approximation‖A − Aₖ‖measured, 2-normσₖ₊₁, from theorymeasured, Frobeniusthe first two agreeto 4.3·10⁻⁹worst |‖A−Aₖ‖₂ − σₖ₊₁| / σₖ₊₁4.3·10⁻⁹worst Frobenius discrepancy4.3·10⁻⁹κ = 10⁹; 30 random rank-3 matrices, none closerthe error is σₖ₊₁

The best approximation there is

The error of the best rank-k approximation is not bounded by the next singular value. It is equal to it. That is an unusually sharp theorem, and it makes the theorem itself usable as an independent check on the computation.

spectra · svd 04812162010⁻¹10⁻⁰.⁵1target rank k‖A − A_k‖₂published boundrandomisedσ_{k+1}, optimalhow far apart the three areworst seed spread1.1bound / median at k = 1211median / optimum at k = 12160×60, 6 seeds, oversampling p = 5band is best to worst

Randomisation does not create structure

On a matrix whose singular values are all equal, a rank-ten randomised approximation has error 1.0 — and so does the optimal deterministic one. Neither achieved anything, and only one of them is usually sold with the implication that it might.

randomised · randomised

Named alongside it

The objects these essays reach for when they reach for this one.

Singular value decompositionSingular valuesEckart–YoungRandomised SVDFrobenius normGram matrixJacobi's eigenvalue methodOrthogonal projectionOrthogonalityOversamplingProbabilistic boundsResidual

All concepts