Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Recompression — where it appears

Truncating a low-rank representation back to a smaller rank after an operation has grown it. It is unavoidable in any format that is not closed under its own arithmetic, and what it costs is measured rather than assumed.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

0816243211.11.21.31.4terms added, each followed by a truncationerror ⁄ best rank-k erroroptimalterms with nothing in commona subspace that driftsthe rounding nobody should have feareddrifting, worst excess1independent, worst excess1a linear bound would say32energy discarded, first1.8·10⁻⁷energy discarded, last0.03thirty-two roundingsand four per cent

The rounding that was not the problem

A rank-k block plus a rank-k block is a rank-2k block, exactly, so every arithmetic in this format truncates after every addition. A Cholesky performed inside it does ninety-eight of those and its residual is 1.14·10⁻⁹ against a representation error of 1.40·10⁻⁹ — the roundings cost nothing measurable.

hierarchy · Recompression 10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²ε the blocks were compressed at‖A − LLᵀ‖ ⁄ ‖A‖the factorisationthe knob, on a factorisationtruncations34residual at 10⁻⁴2.2·10⁻⁵residual at 10⁻¹⁰9.9·10⁻¹²slope1worst ratio to the representation1.8ten decadesand a slope of one

A knob calibrated in residuals

A formatted Cholesky has two numbers in it and only one of them is an accuracy. Across twelve trees — three sizes by four leaf sizes — the leaf moves the truncation count from 0 to 258 and moves the ranks of the blocks not at all, while the residual follows the tolerance at slopes between 1.022 and 1.046 and sits at about a tenth of it throughout.

hierarchy · Recompression 02040608010010⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸truncations performed by the factorisationrelative errorthe representation's own error‖A − LLᵀ‖ ⁄ ‖A‖no decomposition without its residualresidual, 0 truncations2.1·10⁻¹⁰residual, 98 truncations1.1·10⁻⁹representation, deepest1.4·10⁻⁹residual ⁄ representation0.81levels, deepest5a hundred approximate stepsand an exact-looking factorisation

The count that is not the budget

A Cholesky performed inside a low-rank format truncates 0, 2, 10, 34 and 98 times as the leaf falls from 128 to 8, and those five integers are the same at every accuracy from 10⁻¹² to 10⁻². Across all ten decades the factorisation's residual stays below the representation's own error at a ratio between 0.81 and 1.00 — with two entries that read 1.83 and 1.78, and neither of them is accumulation.

hierarchy · Recompression 081624324010⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²conjugate gradient steprelative residualabove, on their own scale: rank asked 5, rank kept 3solid: the budgeted residual · light: no budgetbudget 3rank asked for5rank kept3residual, budgeted3.8·10⁻⁴residual, unbudgeted1.6·10⁻¹⁴numbers stored150the step asks for moreat every step

An iterate that must be made smaller

Applying a Kronecker-sum operator to a low-rank iterate multiplies its ranks by d and adding two of them adds their ranks, so a solver in a compressed format cannot keep what it produces. Every step is followed by a truncation — and whether that truncation is a floor on the residual depends on the right-hand side rather than on the truncation.

iterative · Low-rank iteration 012345678910⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹rank budgetresidual the run stalls atb with no structure at allb constant: the answer is a traintwo ladders, one truncationstructured, rank 10.12structured, rank 45.5·10⁻⁶structured, rank 85.9·10⁻¹⁴unstructured, rank 10.97unstructured, rank 80.47the floor is not the truncation'sit is the answer's

A run that is over at step five

A conjugate gradient whose every iterate is cut to a rank budget reaches the floor that budget allows at step 5, 36, 42 or 59, and then does nothing for the rest of the run. Four times the iterations move the floor by a factor of 1.8, and past the answer's own rank they move it the wrong way.

iterative · Low-rank iteration

Named alongside it

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

Low-rank approximationResidualCholesky factorisationFormatted arithmeticHierarchical matrixTruncated SVDAdmissibilityConjugate gradientsEckart–YoungError accumulationKronecker sumKrylov subspace

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