tucker-storage
At its defaults it draws numbers stored against the number of indices, at n = 20 and rank 4: the tensor, its core, and a train. The tensor is n^d, which at d = 20 is 1.05·10²⁶. A Tucker representation of it is r^d + d·n·r — the core is still exponential in d, so at rank 4 it is 1.1·10¹², smaller than the tensor by 9.54·10¹³ and still unstorable. A train is (d − 2)nr² + 2nr, which is 5,920 — linear in d. The three lines are the field's whole argument: fixing the definition of the decomposition does not fix the size of what it returns, and the second fix is the same projection cut in a different place.
tucker-storage 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 tensor is n^d, which at d = 20 is 1.05·10²⁶. A Tucker representation of it is r^d + d·n·r — the core is still exponential in d, so at rank 4 it is 1.1·10¹², smaller than the tensor by 9.54·10¹³ and still unstorable. A train is (d − 2)nr² + 2nr, which is 5,920 — linear in d. The three lines are the field's whole argument: fixing the definition of the decomposition does not fix the size of what it returns, and the second fix is the same projection cut in a different place.
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 tensor is n^d, which at d = 20 is 1.05·10²⁶. A Tucker representation of it is r^d + d·n·r — the core is still exponential in d, so at rank 4 it is 1.1·10¹², smaller than the tensor by 9.54·10¹³ and still unstorable. A train is (d − 2)nr² + 2nr, which is 5,920 — linear in d. The three lines are the field's whole argument: fixing the definition of the decomposition does not fix the size of what it returns, and the second fix is the same projection cut in a different place.
r: 8
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 tensor is n^d, which at d = 20 is 1.05·10²⁶. A Tucker representation of it is r^d + d·n·r — the core is still exponential in d, so at rank 8 it is 1.15·10¹⁸, smaller than the tensor by 9.09·10⁷ and still unstorable. A train is (d − 2)nr² + 2nr, which is 23,360 — linear in d. The three lines are the field's whole argument: fixing the definition of the decomposition does not fix the size of what it returns, and the second fix is the same projection cut in a different place.
r: 3
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 tensor is n^d, which at d = 20 is 1.05·10²⁶. A Tucker representation of it is r^d + d·n·r — the core is still exponential in d, so at rank 3 it is 3.49·10⁹, smaller than the tensor by 3.01·10¹⁶ and still unstorable. A train is (d − 2)nr² + 2nr, which is 3,360 — linear in d. The three lines are the field's whole argument: fixing the definition of the decomposition does not fix the size of what it returns, and the second fix is the same projection cut in a different place.
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 tensor is n^d, which at d = 20 is 1.05·10²⁶. A Tucker representation of it is r^d + d·n·r — the core is still exponential in d, so at rank 2 it is 1.05·10⁶, smaller than the tensor by 9.99·10¹⁹ and still unstorable. A train is (d − 2)nr² + 2nr, which is 1,520 — linear in d. The three lines are the field's whole argument: fixing the definition of the decomposition does not fix the size of what it returns, and the second fix is the same projection cut in a different place.
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.
3 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.
a rank the storage is counted at
and at two indices they are the same object
the core outgrows the train
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.
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 tupleA 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 tupleAn 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 tupleThe 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 tupleThe 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.