Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Unfolding — where it appears

A tensor reshaped into a matrix by putting one index on the rows and the rest on the columns. Every rank in a tensor format is a rank of one of these, which is what makes the formats computable at all.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

03672108144180216024681012eigenvalues in orderλthe closed formmarks: the assembled matrix, decomposeda spectrum nobody computedrows of the matrix216numbers that describe it108λ smallest0.59λ largest11worst |computed − exact|7.1·10⁻¹³the matrix is never neededand neither is its decomposition

An index that is a pair

A discretisation on a two-dimensional grid of n points a side has n² unknowns and a matrix with n⁴ entries — 10⁸ at n = 100. What that matrix is instead is two Kronecker products of an n × n matrix, which is 2n² numbers, and nothing has been approximated: assembling it was the mistake.

tensor · Kronecker 110¹10²10³10⁻³10⁻²10⁻¹110¹10²10³n‖Aₙ − A‖ and the largest term's norm‖Aₙ − A‖the larger of its two termsan infimum that is not attainedn1024‖Aₙ − A‖0.0017largest term1024their product1.7√31.7the distance goes to zeroand nothing reaches it

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.

tensor · Tensor rank 10¹10²10³10⁴00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951drawsshare with real rank twoπ/4 = 0.78539815,705 of 19,953 have rank twoa probability with a closed formdraws2·10⁴rank two1.6·10⁴share0.79π/40.79standard errors out0.59two typical ranksand the split is π/4

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.

tensor · Tensor rank 024681010⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹rank kept in every moderelative errordashes above: √(Σ tail²), the upper bounddashes below: max tail, a floor under the bestsolid: what the projection returnssmooth: pinned to the upper boundrank 10 error1.1·10⁻¹¹its upper bound1.1·10⁻¹¹the lower bound6.3·10⁻¹²error ⁄ bound1error ⁄ lower1.7inside the boundand sitting on it

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.

tensor · Multilinear rank smooth97.7%hilbert94.5%wave26.6%noise0.4%share of the core's energy on its 6 superdiagonal entrieskeeping only them: 0.152 against 7.94·10⁻⁶keeping only them: 0.235 against 1.77·10⁻⁶keeping only them: 0.857 against 1.34·10⁻¹⁵keeping only them: 0.998 against 0.842orthogonal, and not diagonalsmooth on-diagonal0.98hilbert on-diagonal0.94wave on-diagonal0.27noise on-diagonal0.0044worst slice pair3.5·10⁻¹⁶the slices are orthogonalthe core is not diagonal

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.

tensor · Multilinear rank 123456710¹10²10³10⁴10⁵number of indicesnumbersentries: 6^dstored: 4n(d − 1)exponential against linearentries at d = 64.7·10⁴numbers stored120ratio389slope against d24‖T − Tₜₜ‖ ⁄ ‖T‖1.4·10⁻¹⁵one line is n^dthe other is a constant per index

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.

tensor · Tensor train 6 × 6 × 6, rank 3 · 9 ≥ 81.00002 × 2 × 2, rank 3 · 6 < 80.02076 × 6 matrix, rank 3 · no condition0.1089worst agreement between two runs about the factors, 0 to 1every run fits to 3·10⁻¹³every run fits to 1.2·10⁻¹¹every run factorises to 2.4·10⁻¹⁵ and none agrees with anotherthe only thing that improvestensor, Kruskal holds1tensor, Kruskal fails0.021matrix0.11worst residual1.2·10⁻¹¹three successful fitsone recoverable answer

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.

tensor · Uniqueness 15913172110¹10⁶10¹¹10¹⁶10²¹10²⁶number of indicesnumbers storedthe tensor: n^dthe core: r^d + dnrthe train: (d − 2)nr² + 2nrthe curse, movedentries at d = 2010²⁶core1.1·10¹²train5920core ⁄ train1.9·10⁸entries ⁄ core9.5·10¹³the definition is repairedthe size is not

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.

tensor · Multilinear rank -13-11-9-7-5-3-102468log₁₀ of the accuracy asked forlargest train rank0.30 of rank per decadea cost that is typed inentries216stored at 10⁻²90stored at 10⁻¹²288rank per decade0.3worst error ⁄ tolerance0.83the storage is chosena constant of rank a decade

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.

tensor · Tensor train

Named alongside it

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

Low-rank approximationTruncated SVDCurse of dimensionalityEckart–YoungExact ground truthHigher-order SVDMultilinear rankSeparabilityTensor rankTucker decompositionNumerical rankOrthogonality

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