Generator

train-dimension

One function in the ttrain library, called 11 times across 7 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 8 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 entries against numbers stored, for sin of a sum on 6 points a side, as indices are added. The upper line is the tensor: 6^d entries, which is a straight line on a logarithmic axis and reaches 46,656 at d = 6. The lower one is the train, which for this family is 4n(d − 1) exactly — 24, 48, 72, 96, 120 — a straight line on a *linear* axis and therefore a logarithm on this one. Its fitted slope against d is 24.0, which is 4n. The two are the same object to within 1.39·10⁻¹⁵, so nothing has been given up: the ratio at d = 6 is 389, and it grows by a factor of n with every index added.

train-dimension is one function in lib/figures/ttrain.js — the train — one rank per cut of the index list, and the iterate that must be cut back. 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 upper line is the tensor: 6^d entries, which is a straight line on a logarithmic axis and reaches 46,656 at d = 6. The lower one is the train, which for this family is 4n(d − 1) exactly — 24, 48, 72, 96, 120 — a straight line on a *linear* axis and therefore a logarithm on this one. Its fitted slope against d is 24.0, which is 4n. The two are the same object to within 1.39·10⁻¹⁵, so nothing has been given up: the ratio at d = 6 is 389, and it grows by a factor of n with every index added.

`n: 6`

The arguments are the ones A block nobody can call sparse 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.

`n: 3`

The arguments are the ones The format that does not notice the dimension 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 upper line is the tensor: 3^d entries, which is a straight line on a logarithmic axis and reaches 729 at d = 6. The lower one is the train, which for this family is 4n(d − 1) exactly — 12, 24, 36, 48, 60 — a straight line on a *linear* axis and therefore a logarithm on this one. Its fitted slope against d is 12.0, which is 4n. The two are the same object to within 3.58·10⁻¹⁵, so nothing has been given up: the ratio at d = 6 is 12, and it grows by a factor of n with every index added.

`n: 4`

The upper line is the tensor: 4^d entries, which is a straight line on a logarithmic axis and reaches 4,096 at d = 6. The lower one is the train, which for this family is 4n(d − 1) exactly — 16, 32, 48, 64, 80 — a straight line on a *linear* axis and therefore a logarithm on this one. Its fitted slope against d is 16.0, which is 4n. The two are the same object to within 8.25·10⁻¹⁶, so nothing has been given up: the ratio at d = 6 is 51, and it grows by a factor of n with every index added.

`n: 5`

The upper line is the tensor: 5^d entries, which is a straight line on a logarithmic axis and reaches 15,625 at d = 6. The lower one is the train, which for this family is 4n(d − 1) exactly — 20, 40, 60, 80, 100 — a straight line on a *linear* axis and therefore a logarithm on this one. Its fitted slope against d is 20.0, which is 4n. The two are the same object to within 2.45·10⁻¹⁵, so nothing has been given up: the ratio at d = 6 is 156, and it grows by a factor of n with every index added.

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.

8 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.

every cut at d = 2 has rank two — asserted 5 times

a grid the tensors can be formed at

and the storage is 4n(d − 1)

so the storage grows by 4n per index

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.

Neither sparse nor dense

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.

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 solve that is d decompositions

A Kronecker sum is closed under nothing useful — its inverse is not a Kronecker sum and no factorisation of it is one. What it has instead is eigenvectors that are Kronecker products, so a solve with 1,728 unknowns takes one decomposition of a 12 × 12 matrix and nothing else.

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.

Iterating, instead of factorising

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.

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.

Where the flop count stopped predicting the time

The order the products are taken in

The sparsity field's first essay says the elimination order decides the memory. This is the same sentence about arithmetic: a contraction of several tensors over shared indices has one value and many evaluation orders, and on the inner product of two trains they differ by a factor of two million.

The whole library · All essays · What must fail