Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Sketching — where it appears

Replacing a large matrix by its product with a small random one, which preserves lengths well enough to solve with and costs one pass. What it buys is fewer passes over the data for the same quality of answer, which is a statement about cost; on a matrix with no structure to find it buys nothing.

Named by 13 essays across 3 fields — 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 04812162010⁻¹10⁻⁰.⁵1target rank k‖A − Aₖ‖₂published boundrandomisedσₖ₊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 0481216202428321rank keptrelative errormedianthe answer movesspread at rank 81.8spread at rank 241best median error0.14widest where the method is worstand the bound does not say so

An answer that changes with the seed

A randomised rank-k solve is a truncation computed in a random subspace, and it reaches the same floor as the deterministic ones. What it does not do is return the same answer twice — a factor of 1.84 across four seeds at rank 8, and 1.02 at the rank where the method is best.

combination · Randomised 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 012345610⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹round‖AΩ‖ ⁄ ‖A‖ at that roundredrawn each roundhalf redrawnkept: the sketch is now the zero matrixthe randomness is spentkept, round 12.6kept, round 29·10⁻¹⁵redrawn, round 23.1kept, final error0.28redrawn, final error0.11a random matrix used twiceis not random the second time

The sketch that is spent

Every other object a sequence carries has a shelf life. A random sketch has one use. Deflate what its first round found and apply it again, and it returns the zero matrix — 9.0·10⁻¹⁵ where the first round saw 2.62 — because the input has been made orthogonal to the very draw the guarantee is over.

randomised · Sketching 12345678910¹10³10⁵10⁷number of indicesrandom numbers drawna dense Gaussian sketchdashes: the tensor's own entriesa Khatri–Rao sketcha random matrix nobody can afforddense at d = 81.5·10⁸structured4032the tensor's entries1.7·10⁷dense ⁄ structured3.7·10⁴crossing at d2the sketch outgrows its tensorand the structured one does not

Sketching what is never unfolded

A range finder multiplies its matrix by a few random vectors. For a mode-k unfolding those vectors have nᵈ⁻¹ entries, so the random object is the size of the tensor divided by n — and by six indices it is larger than the tensor it is sketching.

randomised · Sketching 2468101210⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²log₁₀ τ‖Bx − d‖ / ‖d‖sketched with the objectiveweighted, not sketchedkept out of the sketchone power instead of twokept out — feasibility1.7·10⁻¹⁶sketched at τ = 10⁸1.7·10⁻⁸unsketched at τ = 10⁸1.7·10⁻¹⁶objective ÷ optimum1.2a sketch preserves a normand a constraint is not one

The half of a problem a sketch may touch

A sketch guarantees that a norm is preserved to within a factor. An equality constraint is a statement that a quantity is zero, and no multiplicative guarantee says anything about zero. Sketch a constrained problem written as a weighted one and the constraint is not destroyed — it is demoted, from a violation of 1/τ² to one of ε/τ, exactly half the exponent.

randomised · Sketching 111315171921232510⁻¹110¹sketch width l, for rank 10‖(I − QQᵀ)A‖ ÷ σ₁₁GaussianHadamardone nonzero a rowthree nonzeros a rowcoherent, geometric 0.8, width 20, twenty seedsGaussian: median 0.41, worst0.78Hadamard: median 0.37, worst0.94one nonzero a row: median 3.3, worst7.1three nonzeros a row: median 0.41, worst0.99solid: median · dashed: worst of twentybelow one: better than the best rank-10 error

A sketch that finds the columns it can see

A sparse sketch with one nonzero in each row is twenty times cheaper to apply than a Gaussian one, and on a matrix whose important directions are spread across its columns it finds the same range: a median error of 0.45 against 0.43. Put the same ten directions into ten particular columns and it is eight times worse — 3.33 against 0.41, with a worst draw of 7.1 — because two important columns hashed to one bucket are one direction. Three nonzeros a row repair it at a sixth of the Gaussian's cost, and a randomised Hadamard transform never had the problem.

randomised · Randomised 00.10.20.30.40.50.60.70.80.9110⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹decay rate of the spectrumrelative spread of the rank-k errorthe seedthe machinetwo kinds of variationseeds drawn8partitionings7seed spread0.31machine spread9.8·10⁻¹⁶ratio3.1·10¹⁴one of these is recordedand it is the large one

The variation that comes with a seed

A randomised low-rank approximation's error moves by 31% between draws and by 10⁻¹⁵ between partitionings of one draw. In the one field whose answer already comes as a band, the machine is inside the width of the line — and it is still there.

machine · Run-to-run variation median ratioone nonzero, t = 03.5one nonzero, t = 0.011.210⁻⁴10⁻³10⁻²10⁻¹11mixture t (t = 0 at the left edge)error ÷ σ₁₁one nonzero a rowheavy columns reservedthree nonzeros a rowGaussianthe left edge is exact coherencethe failure fades over four decades of mixing

The leverage that did not move

A one-nonzero sketch fails on a matrix whose leading directions sit on ten particular columns, and coherence — the largest column leverage — is the statistic that names the failure. Turn the directions away from their columns by a hundredth of a radian and the sketch's median error falls from 3.47 to 1.23 times σ₁₁ while the coherence stays at 6.40 to three figures. Giving the heaviest columns buckets of their own repairs the rest, but only when it reserves more buckets than the rank: ten reserved leave 1.16, sixteen reach 0.36, below the Gaussian's 0.41.

randomised · Randomised fraction left after a decademedian, t = 10⁻³ → 10⁻²0.57median, t = 10⁻² → 10⁻¹0.6210⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1110¹mixing angle terror ÷ σ₁₁median of 80single drawseach draw drops once, somewherethe median's slope is where they drop

A fade made of drops

A one-nonzero sketch's median error fell by about forty per cent for every factor of ten in how far a coherent matrix had been mixed toward an incoherent one, and nothing explained the rate. Followed one draw at a time, no draw fades at that rate. Each holds its coherent error — as large as the singular value of the direction its hash lost — and then drops, within one to three decades of mixing, never faster than one decade of error per decade of mixing. The median's steady slope is where the drops happen to fall. Change the spectrum and they fall elsewhere: at a decay of 0.9 there is no slope, only a cliff.

randomised · Randomised all Gaussian30incoherentcost 1.0030coherentcost 1.00all sparse30incoherentcost 0.0410812coherentcost 0.08sparse, checked30incoherentcost 0.291218coherentcost 0.33sparse, promoted30incoherentcost 0.2930coherentcost 0.54green: certified, error under ε · red: certified, over · grey: refuseddense probes keep it honest

The probe that refuses becomes the column

An adaptive range finder grows its basis from probes and stops when ten fresh ones come back short. Sparse probes cost a twentieth as much and fail on a matrix whose important directions sit on particular columns. Used for both jobs on such a matrix they certify falsely on eight draws of thirty, with the error ten times the tolerance. Let Gaussian probes do the certifying and the sparse basis is never certified falsely — but on eighteen draws of thirty it is never certified at all. The repair is to let a certifying probe that sees eight times more, per unit of its own length, than the next sparse candidate become the column instead. On the incoherent matrix that never happens and the rule costs 29 per cent of the all-Gaussian one; on the coherent matrix it happens a median ten times — about once for each direction the sparse probes cannot find — and every draw certifies correctly at 54 per cent.

randomised · Randomised 00.10.20.30.40.50.6mixture angle tprobe cost over the all-Gaussian rule's00.050.10.20.30.50.81.2π/2303030313132303030583836333232303031promoteddense certificate, no promotionsparse certificatenumbers: median rank; the unpromoted rule refuses at t = 0promotion buys rank, not safety

Promotion buys rank, not safety

A range finder that grows its basis from cheap sparse probes and certifies it with dense ones was made safe on a coherent matrix by promoting a refusing dense probe into the basis — ten promotions for ten coherent directions, none for none. On matrices mixed between the two, the promotions fade as predicted: ten at the coherent end, nine at a mixture angle of 0.1, none from 0.8. But the danger they guard against does not fade with them. It is gone at an angle of 0.05, where the rule that never promotes already certifies every draw correctly. What promotion buys there is rank — a basis of 30 columns instead of 38 — for four fifths more probe work, and the threshold that decides it is a price, not a safety margin.

randomised · Randomised

Named alongside it

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

Randomised SVDRandom projectionLow-rank approximationProbabilistic boundsRange finderLeverageOversamplingSpectral decaySubspace-embeddingCertificateCondition numberFlop count

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