Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Subspace embedding — where it appears

A random map preserving the lengths of every vector in a subspace, whose required size depends on the subspace's dimension and not on the vectors' length.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

10²10².³10².⁵⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁶10⁻¹10⁻⁰.⁵1rows in the sketchworst relative distortiondimension 64dimension 2565 seeds per point, band is best to worstthe dimension does not appear

The dimension does not appear

A random projection preserves the lengths of a set of vectors to within a distortion that depends on how many vectors there are and not on how many coordinates each one has. That is the fact the whole field rests on, and it is genuinely surprising.

randomised · sketching 10²10⁴10⁶10⁸10¹⁰110²10⁴10⁶10⁸10¹⁰condition number of the matrixcondition number seen by the iterationκ(A), unchangedκ(AR⁻¹)a bound with no κ(A) in itκ(AR⁻¹), every κ(A)2.2κ(SU), the other route2.2κ(A), across the sweep10⁸κ(AR⁻¹), across the sweep1the sketch never sees the spectrumand the spectrum cancels out of the answer

The sketch that is not the answer

Sketch-and-solve throws away the original problem and keeps the small one's answer, which is why its answer moves with the seed. Use the same sketch as a preconditioner instead and the condition number the iteration sees is the same number at every κ from a hundred to ten billion — identically the same, to nine digits, because the spectrum cancels out of it.

randomised · sketching

Named alongside it

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

SketchingCondition numberDimension reductionJohnson–LindenstraussKrylov subspaceLsqrMatrix-freeOrthogonal projectionOrthogonalityPreconditioningProbabilistic boundsRandom projection

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