Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

core-energy

One function in the hosvd library, called 14 times across 6 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 7 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 share of a hosvd core's energy on its superdiagonal, by family, at rank 6. A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 6 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.4%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.998 in relative error against the full core's 7.94·10⁻⁶, 1.77·10⁻⁶, 1.34·10⁻¹⁵, 0.842.

core-energy 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 share of a HOSVD core's energy on its superdiagonal, by family, at rank 6A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 6 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.4%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.998 in relative error against the full core's 7.94·10⁻⁶, 1.77·10⁻⁶, 1.34·10⁻¹⁵, 0.842.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

A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 6 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.4%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.998 in relative error against the full core's 7.94·10⁻⁶, 1.77·10⁻⁶, 1.34·10⁻¹⁵, 0.842.

r: 4

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 share of a HOSVD core's energy on its superdiagonal, by family, at rank 4A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 4 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.4%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.998 in relative error against the full core's 8.81·10⁻⁴, 4.67·10⁻⁴, 1.33·10⁻¹⁵, 0.925.smooth97.7%hilbert94.5%wave26.6%noise0.4%share of the core's energy on its 4 superdiagonal entrieskeeping only them: 0.152 against 8.81·10⁻⁴keeping only them: 0.235 against 4.67·10⁻⁴keeping only them: 0.857 against 1.33·10⁻¹⁵keeping only them: 0.998 against 0.925orthogonal, and not diagonalsmooth on-diagonal0.98hilbert on-diagonal0.94wave on-diagonal0.27noise on-diagonal0.0039worst slice pair3.5·10⁻¹⁶the slices are orthogonalthe core is not diagonal

A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 4 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.4%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.998 in relative error against the full core's 8.81·10⁻⁴, 4.67·10⁻⁴, 1.33·10⁻¹⁵, 0.925.

r: 6

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 share of a HOSVD core's energy on its superdiagonal, by family, at rank 6A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 6 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.4%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.998 in relative error against the full core's 7.94·10⁻⁶, 1.77·10⁻⁶, 1.34·10⁻¹⁵, 0.842.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

A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 6 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.4%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.998 in relative error against the full core's 7.94·10⁻⁶, 1.77·10⁻⁶, 1.34·10⁻¹⁵, 0.842.

r: 2

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 share of a HOSVD core's energy on its superdiagonal, by family, at rank 2A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 2 entries on the superdiagonal: smooth 97.6%, hilbert 94.5%, wave 26.6%, noise 0.2%. Keeping only those entries costs 0.154, 0.235, 0.857, 0.999 in relative error against the full core's 0.0671, 0.0447, 1.32·10⁻¹⁵, 0.971.smooth97.6%hilbert94.5%wave26.6%noise0.2%share of the core's energy on its 2 superdiagonal entrieskeeping only them: 0.154 against 0.0671keeping only them: 0.235 against 0.0447keeping only them: 0.857 against 1.32·10⁻¹⁵keeping only them: 0.999 against 0.971orthogonal, and not diagonalsmooth on-diagonal0.98hilbert on-diagonal0.94wave on-diagonal0.27noise on-diagonal0.0017worst slice pair3.5·10⁻¹⁶the slices are orthogonalthe core is not diagonal

A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 2 entries on the superdiagonal: smooth 97.6%, hilbert 94.5%, wave 26.6%, noise 0.2%. Keeping only those entries costs 0.154, 0.235, 0.857, 0.999 in relative error against the full core's 0.0671, 0.0447, 1.32·10⁻¹⁵, 0.971.

r: 10

The arguments are the ones The orthogonality that cannot be diagonal 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 share of a HOSVD core's energy on its superdiagonal, by family, at rank 10A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 10 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.8%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.996 in relative error against the full core's 1.09·10⁻¹¹, 1.44·10⁻¹², 1.33·10⁻¹⁵, 0.498.smooth97.7%hilbert94.5%wave26.6%noise0.8%share of the core's energy on its 10 superdiagonal entrieskeeping only them: 0.152 against 1.09·10⁻¹¹keeping only them: 0.235 against 1.44·10⁻¹²keeping only them: 0.857 against 1.33·10⁻¹⁵keeping only them: 0.996 against 0.498orthogonal, and not diagonalsmooth on-diagonal0.98hilbert on-diagonal0.94wave on-diagonal0.27noise on-diagonal0.0081worst slice pair3.5·10⁻¹⁶the slices are orthogonalthe core is not diagonal

A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 10 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.8%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.996 in relative error against the full core's 1.09·10⁻¹¹, 1.44·10⁻¹², 1.33·10⁻¹⁵, 0.498.

r: 3

The arguments are the ones The orthogonality that cannot be diagonal 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 share of a HOSVD core's energy on its superdiagonal, by family, at rank 3A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 3 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.4%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.998 in relative error against the full core's 0.0122, 0.00526, 1.32·10⁻¹⁵, 0.951.smooth97.7%hilbert94.5%wave26.6%noise0.4%share of the core's energy on its 3 superdiagonal entrieskeeping only them: 0.152 against 0.0122keeping only them: 0.235 against 0.00526keeping only them: 0.857 against 1.32·10⁻¹⁵keeping only them: 0.998 against 0.951orthogonal, and not diagonalsmooth on-diagonal0.98hilbert on-diagonal0.94wave on-diagonal0.27noise on-diagonal0.0038worst slice pair3.5·10⁻¹⁶the slices are orthogonalthe core is not diagonal

A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 3 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.4%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.998 in relative error against the full core's 0.0122, 0.00526, 1.32·10⁻¹⁵, 0.951.

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.

7 distinct claims across 6 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.

a rank the core is read at

and a diagonal core is not better

matmul shapes agree

the hilbert core's slices are mutually orthogonal

the noise core's slices are mutually orthogonal

the smooth core's slices are mutually orthogonal

the wave core's slices are mutually orthogonal

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 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 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.

 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.

The whole library · All essays · What must fail