Low-rank approximation — where it appears
Named by 37 essays across 8 fields — each of them below, with the objects they name alongside it.
A block nobody can call sparse
A 96 × 96 block of a kernel matrix has ninety-six nonzero singular values and five that matter. It has no zero entries, it is not described by fewer numbers than it contains, and neither of the two ways this collection already knows to make a large matrix affordable applies to it.
A bound that holds with probability
Every other guarantee in this collection 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.
A rank that is a number of digits
Ask a kernel block for two digits and it costs two columns; ask for fourteen and it costs nine. The curve is a straight line at 0.55 columns a decade, and the bound the geometry gives is a straight line too — at 3.32, which is the same shape and six times the price.
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.
The size the rank does not notice
Sample a kernel block at 32, 64, 128 and 256 points a side and it needs five columns, five, five and five. Sample the touching block next to it at the same four sizes and it needs nine, eleven, twelve and thirteen. Same kernel, same accuracy, one number and a logarithm.
A nearest point that is not there
Eckart and Young guarantee that a matrix has a best rank-k approximation and that the truncated SVD is it. For three indices the guarantee is false in the strongest available way — there are tensors whose distance to the rank-two set is zero and which no rank-two tensor equals.
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.
A rank that is not a property of the tensor
The same eight real numbers have rank three over the reals and rank two over the complexes, and a random 2 × 2 × 2 tensor has rank two with probability exactly π/4. Neither sentence has an analogue for matrices, where the rank is one number and a random matrix has the largest one.
A decomposition made only of SVDs
Everything the definition of tensor rank loses comes back if the SVD's algorithm is carried across instead of its definition — take the leading left singular subspace of every unfolding and project onto all of them. It exists, it costs d matrix decompositions, and its error is within √d of the best there is.
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.
The orthogonality that cannot be diagonal
A matrix decomposition hands over orthonormal factors and a diagonal middle at once. For three indices the two come apart, and there is no arrangement that has both — so the question stops being which decomposition to use and becomes which of the two properties the computation needs.
The format that does not notice the dimension
A Tucker core is r^d numbers, so the format that repaired the definition still cannot go past five indices. Cutting between the indices rather than across them gives d − 1 ranks instead of d, storage linear in the number of indices, and a family whose ranks are two everywhere by an addition formula.
Why a Gramian can be truncated at all
Every method in this field rests on one fact nobody states the reason for — the eigenvalues of a Gramian fall off a cliff. The equation defining it has a rank-one right-hand side and no low-rank structure anywhere — and the answer's decay is a rational approximation problem with a closed-form rate.
Where to put the poles of a rational function
Three times in one field the same question arrives from different directions — ADI shifts, rational approximation of a square root, the decay of a Gramian — and it has one answer. Cluster them geometrically towards wherever the function is difficult, and the alternative that looks reasonable costs orders.
An iteration that walks out of the set
Every sweep of alternating least squares is the exact minimiser of its own subproblem, so the objective can only fall. What it cannot do is converge, when the target's nearest rank-r point is not in the rank-r set — and a plateau at a small residual looks identical to slow convergence unless the size of the terms is plotted beside it.
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.
The zero you are allowed to write
A deflation criterion sets a subdiagonal entry to zero because it is small. A drop tolerance discards an entry of a factor because it is small. A truncation discards a singular value because it is small. Three fields, three vocabularies, no shared arithmetic — and plotted as work saved against error accepted, one curve.
A factorisation that is unique for once
A rank-r factorisation of a matrix is never unique — AB is (AM)(M⁻¹B) for any invertible M, so no factor means anything on its own. For three indices a checkable condition on the factors' k-ranks makes the decomposition unique up to permuting and scaling the terms, and it holds generically.
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.
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.
The cliff behind the count
The fill's rank is an integer between three and six across every separator two dense half-eliminations can afford, and this field has already recorded that a handful of such integers cannot carry a law. The singular values underneath are real numbers. They say the cliff's first step is 23.0 at a separator of eleven, 19.1 at fifteen and 16.2 at twenty-three — and that a control with no differential operator behind it gives 14,672.
A compression of 10¹⁴ that still does not fit
A Tucker core of a twenty-index array at rank four is 1.1·10¹² numbers against the tensor's 1.05·10²⁶ — a compression by a factor of 9.5·10¹³ that is still nearly nine terabytes. The ratio is not the verdict. The verdict is a ceiling, and the ceiling is a number of indices.
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.
Bracketing an error nobody can measure
The error of a low-rank Gramian factor is the one quantity a caller cannot compute, because computing it needs the Gramian the factor exists to avoid forming. Two numbers that can be computed sit either side of it — a rational factor known before the run, and a residual known after — and they stay a factor of four apart across a fourfold change of size.
The digit that costs more than the tensor
Ask a three-index reciprocal tensor on six points a side for seven digits and its train is 288 numbers against 216 entries. The break-even rank is n − 1 at all four grids measured, and a train that reaches it fits with exactly n numbers to spare.
The offset that moved the slope
The accuracy is supposed to lift a storage curve and leave its growth alone. Measured at seven tolerances, the strong partition adds 10.9 numbers per unknown per doubling at two digits and 36.5 at twelve — the growth rate more than triples, so ten decades of accuracy cost 33 per cent more storage at 64 unknowns and 118 per cent at 512.
The partition that does not move
The storage curve's growth rate more than triples between two digits and twelve, and the earlier measurement said so without saying what moves it. There are two candidates and the measurement separates them decisively: the partition is identical at every tolerance — 250 blocks, 156 of them low-rank, 94 dense — and the mean rank rises by exactly one per two decades of accuracy. The slope is 5.11 numbers per unknown per doubling per unit of rank, with a worst residual of 0.03 over six tolerances.
Two knobs on one number
The tolerance raises every block's rank and leaves the partition alone. The leaf size does the opposite — it changes how many blocks there are and does not move a single rank. Both act on the same storage, and only one of them has a best setting: at eight digits the least storage is at a leaf of 16 and the largest leaf tried costs 59 per cent more. The best leaf rises with the accuracy, from 4 at two digits to 16 at twelve, and the influence runs only that way.
The rank a certificate charges
A randomised range finder can choose its own rank: grow the basis a column at a time and stop when ten fresh probes all come back short. With the published safety factor it never stopped early in any draw measured, and on a matrix whose singular values fall by 0.8 a step it stopped at rank 30 for a tolerance the best rank-11 approximation already meets. The nineteen extra columns are three separate prices — four for building the basis from random vectors, five because a probe reads more than the spectral norm, and ten for the constant — and the spectrum decides which of them dominates.
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.
A second objective that is the first one doubled
Counting the operations a hierarchical product does was supposed to give the leaf size a second optimum, somewhere other than where the storage puts it. The count is 160,128, 139,904, 135,936, 155,136 and 216,064 against stored totals of 80,064, 69,952, 67,968, 77,568 and 108,032 — twice each, at every leaf, exactly. The two objectives cannot disagree, and what separates the leaf that stores least from the leaf that runs fastest is a charge of about 139 operations for reaching a block at all.
A tolerance is priced by the problem
Three tolerances from three fields sit on one pair of axes and agree to within a factor of 5.74. That factor is the ratio of the two curves that cannot move. Change the only problem in the comparison and the third curve's fitted slope swings from 0.188 to 0.040 while the printed spread does not shift by a digit.
A geometry setting that is a second accuracy
The tolerance raises every rank and moves no block; the leaf moves every block and no rank. The constant that decides how far apart two clusters must be before their block may be compressed moves both — 592 blocks at mean rank 3.25 at one end and 94 at mean rank 8.94 at the other — and the storage falls the whole way while the error against the dense matrix rises by a factor of 3.5, at a tolerance that never changed. It has no best setting, and it is not a third knob on the storage.
A prediction that arrives a decade late
A rougher kernel raises every rank, a higher rank raises the side at which compressing a block stops paying, so the leaf that stores least should rise. Across four kernels whose largest rank runs 5, 5, 9 and 10, it is 16 on three of them and a tie between 8 and 16 on the fourth. The prediction is right in sign and out by a decade of tolerance, and the reason is a ceiling: an oscillation multiplies the rank by exactly two — 3 and 6, 4 and 8, 5 and 10, 6 and 12, 7 and 14 — at any frequency, because a cosine of a difference is a rank-two function.
A good curve and a bad verdict
The diagonal of a column-pivoted R is famous for the one matrix it is wrong about. On that matrix it is right about thirty-nine of its forty entries — every |rₖₖ| within a factor of six of the σₖ it stands for — and wrong by 4·10⁶ at the fortieth, which is the only one a rank verdict ever reads.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Numerical rankEckart–YoungTruncated SVDToleranceHierarchical matrixTruncationAdmissibilityOff-diagonal rankSingular valuesUnfoldingResidualSpectral decay