Truncated SVD — where it appears
Named by 24 essays across 5 fields — each of them below, with the objects they name alongside it.
When the answer is a choice
A backward-stable least-squares solve of this problem returns an answer whose relative error is 5.5·10⁸. Nothing went wrong. The singular values decay exponentially with no gap anywhere in them, the data does not determine the answer, and something outside the data has to choose — which is the computation rather than a preliminary to 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.
Four knobs and one floor
A truncation, a Tikhonov parameter, a step count and a randomised rank, on one problem with an answer that is known. Their best errors are 0.1445, 0.1406, 0.1426 and 0.1449 — a spread of 3% across four methods that share no arithmetic.
Where the answer stops being in the data
The Picard condition finds the index where a noisy right-hand side stops carrying signal, from the data alone, with no knowledge of the answer. It lands at 32 where the truncation that actually minimises the error is 28 — and at 45 where the best is 38. It overshoots at every stop from 10% noise to 0.0001%, and it overshoots for a reason. The best truncation walks up the spectrum in a straight line, six or seven indices a decade; the crossing climbs in jumps of 11, 0, 8, 5 and 1.
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.
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.
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.
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.
A second blur, narrower than the first
A regularised answer is not the truth with the noise taken out. It is the truth seen through a second blur, V F Vᵀ, which depends on the operator and λ and on nothing that was measured. At the best λ for 0.1% noise its rows are 2.82 points wide against the instrument's 5.89, they dip to −0.075 on either side, and their width times the number of components kept stays between 1.10n and 1.27n across seven decades of λ. Two spikes four points apart come back as two; three apart, as one.
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.
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 grid was the first filter
A continuous deconvolution discretised on n points and solved with no regularisation at all is not unregularised. Its error against the continuous signal is least at 24, 26 and 34 points for noise of 1%, 0.1% and 0.01% per sample — beside best truncations of 24, 28 and 32 components on a 64-point grid — and within 4 to 16 per cent of their error. The grid's own filter factors sum to n exactly and fall through a half at k = n. Choosing the grid was choosing a truncation, before anybody chose a λ.
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 method that cannot use a smooth answer
On the collection's own signal four regularisers reach the same floor to a few per cent. Score them instead against answers of increasing smoothness and one stops improving. Across six decades of noise Tikhonov's error falls with a fitted slope of 0.70 whether the answer is twice or four times as smooth, while truncation's rises to 0.94 — and at 10⁻⁶ noise Tikhonov's best is 38 times truncation's.
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.
A better discretisation is a weaker filter
A coarse grid's error has two sources — how well the discrete operator approximates the integral, and how well the grid's function represents the answer — and the grid essay could not separate them. Changed one at a time they separate: integrating the kernel against the hat functions takes a fifth off the 12-point error, reading the answer as a cubic spline takes 15 per cent more, and both roughly double the condition number on every grid. At 0.1% noise the spline discretisation's unregularised solve on 26 points reaches the best truncation of a 64-point grid to 0.3%. At 1% it is worse than the crude grid.
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.
Where the grid hands over to λ
An unregularised solve on a coarse grid comes within a tenth of the best Tikhonov answer on a fine one, and the pair of a grid and a λ was left unmeasured. Measured, the two do not trade. On every grid up to the best unregularised one no λ helps at all. On every grid of 40 points and more the best λ is the same to within a quarter of a decade — 3.2·10⁻² at 1% noise per sample, 10⁻³ at 0.01% — and the 96-point grid with it beats the best coarse grid by 7, 9 and 13 per cent. The grids between the two, given their own λ, land between them.
One arc, and what each filter pays to be on it
Conjugate gradients and Tikhonov stop at the same effective dimension, and that could have meant two curves crossing once or one curve. It is one curve over a stretch — on six problems, Tikhonov and truncation reach the iteration's error at the iteration's dimension to within 7.2 per cent from 0.7 of the answer to its top — and the two separate on either side. But the curve is shared by a trade, not by an identical answer: at the same dimension Tikhonov carries 22 to 42 per cent more noise than the iteration and up to five per cent less bias.
The overshoot was the lead
Past its best, conjugate gradients' error rose more slowly than Tikhonov's at the same effective dimension — on the narrow blur at 0.1% noise Tikhonov was 2.35 times worse at 1.3 of the answer — and the reading was that the iteration spends its dimension where the data has content. Count admitted directions instead of summing factors, so that a factor of 1.81 counts once, and the lead is gone: 0.96 on that problem, and within 0.17 of one on all six from 1.1 to 1.3. Tikhonov now carries less noise than the iteration there. Below half the answer, where the three filters also disagree, the count changes nothing.
A tail from Tikhonov and a corner from truncation
Below half the answer, conjugate gradients beat Tikhonov at a matched count of directions by up to 69 per cent, and the proposed measurement was the sharpness p of a roll-off between Tikhonov and truncation that matches the iteration there. Any p from 2 to 4 closes the gap to a tenth; truncation, the family's limit, reopens it to a third. But no p describes the iteration. Its filter has Tikhonov's slope exactly in its tail and a local sharpness of 2 to 5 on its shoulder, and a single fitted p is a compromise that drifts from 2.3 at the first step to 1.7 at the answer.
Named alongside it
The objects these essays reach for when they reach for this one.
Filter factorsLow-rank approximationIll-posed problemDeconvolutionTikhonov regularisationEckart–YoungRegularisationUnfoldingIterative regularisationCondition numberConjugate gradientsHigher-order SVD