Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

tucker-error

One function in the hosvd library, called 17 times across 10 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 15 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

At its defaults it draws the truncation error of a smooth tensor against the rank kept, between the two bounds the theorem gives. The middle curve is the measured error of the projection; the upper dashed one is √(Σ_k tail_k²), which the theorem says it cannot exceed, and the lower one is max_k tail_k, which the best possible error cannot fall below. They are a factor of √3 apart. The measurement is that the projection sits on the upper one, and not between them: the ratio of error to bound runs 0.688, 0.717, 0.879, 0.933 … 0.999982, so by rank 10 the bound is attained to five decimals and the ratio to the lower bound is 1.7320 against √3 = 1.7321. That reads as a bad result and is not one — what it says is that the lower bound is weak, which only a second measurement can establish.

tucker-error is one function in lib/figures/hosvd.js — the higher-order svd — d matrix decompositions, and a core that cannot be diagonal. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.

The truncation error of a smooth tensor against the rank kept, between the two bounds the theorem givesThe middle curve is the measured error of the projection; the upper dashed one is √(Σ_k tail_k²), which the theorem says it cannot exceed, and the lower one is max_k tail_k, which the best possible error cannot fall below. They are a factor of √3 apart. The measurement is that the projection sits on the upper one, and not between them: the ratio of error to bound runs 0.688, 0.717, 0.879, 0.933 … 0.999982, so by rank 10 the bound is attained to five decimals and the ratio to the lower bound is 1.7320 against √3 = 1.7321. That reads as a bad result and is not one — what it says is that the lower bound is weak, which only a second measurement can establish.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

The middle curve is the measured error of the projection; the upper dashed one is √(Σ_k tail_k²), which the theorem says it cannot exceed, and the lower one is max_k tail_k, which the best possible error cannot fall below. They are a factor of √3 apart. The measurement is that the projection sits on the upper one, and not between them: the ratio of error to bound runs 0.688, 0.717, 0.879, 0.933 … 0.999982, so by rank 10 the bound is attained to five decimals and the ratio to the lower bound is 1.7320 against √3 = 1.7321. That reads as a bad result and is not one — what it says is that the lower bound is weak, which only a second measurement can establish.

family: "smooth"

The arguments are the ones A decomposition made only of SVDs passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

The truncation error of a smooth tensor against the rank kept, between the two bounds the theorem givesThe middle curve is the measured error of the projection; the upper dashed one is √(Σ_k tail_k²), which the theorem says it cannot exceed, and the lower one is max_k tail_k, which the best possible error cannot fall below. They are a factor of √3 apart. The measurement is that the projection sits on the upper one, and not between them: the ratio of error to bound runs 0.688, 0.717, 0.879, 0.933 … 0.999982, so by rank 10 the bound is attained to five decimals and the ratio to the lower bound is 1.7320 against √3 = 1.7321. That reads as a bad result and is not one — what it says is that the lower bound is weak, which only a second measurement can establish.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

The middle curve is the measured error of the projection; the upper dashed one is √(Σ_k tail_k²), which the theorem says it cannot exceed, and the lower one is max_k tail_k, which the best possible error cannot fall below. They are a factor of √3 apart. The measurement is that the projection sits on the upper one, and not between them: the ratio of error to bound runs 0.688, 0.717, 0.879, 0.933 … 0.999982, so by rank 10 the bound is attained to five decimals and the ratio to the lower bound is 1.7320 against √3 = 1.7321. That reads as a bad result and is not one — what it says is that the lower bound is weak, which only a second measurement can establish.

family: "hilbert"

The arguments are the ones A decomposition made only of SVDs passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

The truncation error of a hilbert tensor against the rank kept, between the two bounds the theorem givesThe middle curve is the measured error of the projection; the upper dashed one is √(Σ_k tail_k²), which the theorem says it cannot exceed, and the lower one is max_k tail_k, which the best possible error cannot fall below. They are a factor of √3 apart. The measurement is that the projection sits on the upper one, and not between them: the ratio of error to bound runs 0.689, 0.829, 0.906, 0.950 … 0.999812, so by rank 10 the bound is attained to five decimals and the ratio to the lower bound is 1.7317 against √3 = 1.7321. That reads as a bad result and is not one — what it says is that the lower bound is weak, which only a second measurement can establish.024681010⁻¹²10⁻⁹10⁻⁶10⁻³1rank kept in every moderelative errordashes above: √(Σ tail²), the upper bounddashes below: max tail, a floor under the bestsolid: what the projection returnshilbert: pinned to the upper boundrank 10 error1.4·10⁻¹²its upper bound1.4·10⁻¹²the lower bound8.3·10⁻¹³error ⁄ bound1error ⁄ lower1.7inside the boundand sitting on it

The middle curve is the measured error of the projection; the upper dashed one is √(Σ_k tail_k²), which the theorem says it cannot exceed, and the lower one is max_k tail_k, which the best possible error cannot fall below. They are a factor of √3 apart. The measurement is that the projection sits on the upper one, and not between them: the ratio of error to bound runs 0.689, 0.829, 0.906, 0.950 … 0.999812, so by rank 10 the bound is attained to five decimals and the ratio to the lower bound is 1.7317 against √3 = 1.7321. That reads as a bad result and is not one — what it says is that the lower bound is weak, which only a second measurement can establish.

family: "noise"

The arguments are the ones A decomposition made only of SVDs passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

The truncation error of a noise tensor against the rank kept, between the two bounds the theorem givesThe middle curve is the measured error of the projection; the upper dashed one is √(Σ_k tail_k²), which the theorem says it cannot exceed, and the lower one is max_k tail_k, which the best possible error cannot fall below. They are a factor of √3 apart. The measurement is that the projection sits on the upper one, and not between them: the ratio of error to bound runs 0.618, 0.648, 0.687, 0.725 … 0.927736, so by rank 10 the bound is attained to five decimals and the ratio to the lower bound is 1.5839 against √3 = 1.7321. That reads as a bad result and is not one — what it says is that the lower bound is weak, which only a second measurement can establish.024681010⁻¹rank kept in every moderelative errordashes above: √(Σ tail²), the upper bounddashes below: max tail, a floor under the bestsolid: what the projection returnsnoise: pinned to the upper boundrank 10 error0.5its upper bound0.54the lower bound0.31error ⁄ bound0.93error ⁄ lower1.6inside the boundand sitting on it

The middle curve is the measured error of the projection; the upper dashed one is √(Σ_k tail_k²), which the theorem says it cannot exceed, and the lower one is max_k tail_k, which the best possible error cannot fall below. They are a factor of √3 apart. The measurement is that the projection sits on the upper one, and not between them: the ratio of error to bound runs 0.618, 0.648, 0.687, 0.725 … 0.927736, so by rank 10 the bound is attained to five decimals and the ratio to the lower bound is 1.5839 against √3 = 1.7321. That reads as a bad result and is not one — what it says is that the lower bound is weak, which only a second measurement can establish.

family: "wave"

The arguments are the ones A decomposition made only of SVDs passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

The truncation error of a wave tensor against the rank kept, between the two bounds the theorem givesThe middle curve is the measured error of the projection; the upper dashed one is √(Σ_k tail_k²), which the theorem says it cannot exceed, and the lower one is max_k tail_k, which the best possible error cannot fall below. They are a factor of √3 apart. The measurement is that the projection sits on the upper one, and not between them: the ratio of error to bound runs 0.780, 1.103, 1.292, 1.530 … 0.779766, so by rank 10 the bound is attained to five decimals and the ratio to the lower bound is 7.1766 against √3 = 1.7321. That reads as a bad result and is not one — what it says is that the lower bound is weak, which only a second measurement can establish.024681010⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹rank kept in every moderelative errordashes above: √(Σ tail²), the upper bounddashes below: max tail, a floor under the bestsolid: what the projection returnswave: pinned to the upper boundrank 1 error0.89its upper bound1.1the lower bound0.69error ⁄ bound0.78error ⁄ lower1.3inside the boundand sitting on it

The middle curve is the measured error of the projection; the upper dashed one is √(Σ_k tail_k²), which the theorem says it cannot exceed, and the lower one is max_k tail_k, which the best possible error cannot fall below. They are a factor of √3 apart. The measurement is that the projection sits on the upper one, and not between them: the ratio of error to bound runs 0.780, 1.103, 1.292, 1.530 … 0.779766, so by rank 10 the bound is attained to five decimals and the ratio to the lower bound is 7.1766 against √3 = 1.7321. That reads as a bad result and is not one — what it says is that the lower bound is weak, which only a second measurement can establish.

What it checked while drawing

Every figure above asserted its own claims on the way to being drawn, and a claim that failed would have failed the build rather than drawn a wrong picture. Those assertions used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

15 distinct claims across 5 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

the error at rank 1 is inside its bound — asserted 8 times

a family this site defines

and inside √d of the lower bound

hilbert is not reproduced exactly at every rank drawn

matmul shapes agree

noise is not reproduced exactly at every rank drawn

smooth is not reproduced exactly at every rank drawn

wave is not reproduced exactly at every rank drawn

Against the rule

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

Across the library: the rule bites on 146 of 287 generators — 131 print a residual and 15 are exempt with a published reason; 141 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

When the index is a tuple

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.

 When the index is a tuple

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.

 When the index is a tuple

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.

 When the index is a tuple

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.

 When the index is a tuple

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.

 When the index is a tuple

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.

 Randomised, and the guarantee that changes kind

Sketching what is never unfolded

A range finder multiplies its matrix by a few random vectors. For a mode-k unfolding those vectors have n^{d−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.

 Eigenvalues, singular values, rank

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.

 When the index is a tuple

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.

 When the index is a tuple

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.

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