spectrum-decay
At its defaults it draws what a rank-10 approximation can achieve, by spectrum. A semi-logarithmic plot of singular value against index for three spectra — geometric decay, algebraic decay, and flat — with the rank-ten approximation error marked on each.
spectrum-decay is one function in lib/figures/randomised.js —
randomised — a bound that holds with a probability, and the seed that moves. 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.
A semi-logarithmic plot of singular value against index for three spectra — geometric decay, algebraic decay, and flat — with the rank-ten approximation error marked on each.
decay: 1
The arguments are the ones A bound that holds with probability 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.
A semi-logarithmic plot of singular value against index for three spectra — geometric decay, algebraic decay, and flat — with the rank-ten approximation error marked on each.
decay: 2.5
The arguments are the ones Randomisation does not create structure 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.
A semi-logarithmic plot of singular value against index for three spectra — geometric decay, algebraic decay, and flat — with the rank-ten approximation error marked on each.
decay: 0.5
The arguments are the ones The dimension does not appear 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.
A semi-logarithmic plot of singular value against index for three spectra — geometric decay, algebraic decay, and flat — with the rank-ten approximation error marked on each.
decay: 2
The arguments are the ones The matrix that is one row 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.
A semi-logarithmic plot of singular value against index for three spectra — geometric decay, algebraic decay, and flat — with the rank-ten approximation error marked on each.
decay: 0
The arguments are the ones When the answer is a choice 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.
A semi-logarithmic plot of singular value against index for three spectra — geometric decay, algebraic decay, and flat — with the rank-ten approximation error marked on each.
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.
11 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.
the rank-10 error equals σ_{k+1} on the algebraic, j^−1.00 spectrum — asserted 5 times
and on the flat spectrum that optimum is the whole of σ₁
matmul shapes agree
so the spectrum, not the algorithm, decides what is achievable
the rank-10 error equals σ_{k+1} on the flat spectrum
the rank-10 error equals σ_{k+1} on the geometric, 0.85ʲ spectrum
while a geometric spectrum is genuinely approximable
Against the rule
It draws a decomposition and prints its residual. It calls
svd, lowRank,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
Across the library: the rule bites on 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 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 bound that holds with probability
Every other guarantee on this site is deterministic. The randomised low-rank approximation offers one that holds with a probability, the seed changes the answer, and the honest figure is a band rather than a line.
Randomised, and the guarantee that changes kindRandomisation does not create structure
On a matrix whose singular values are all equal, a rank-ten randomised approximation has error 1.0 — and so does the optimal deterministic one. Neither achieved anything, and only one of them is usually sold with the implication that it might.
Randomised, and the guarantee that changes kindThe dimension does not appear
A random projection preserves the lengths of a set of vectors to within a distortion that depends on how many vectors there are and not on how many coordinates each one has. That is the fact the whole field rests on, and it is genuinely surprising.
Structure, and the solver that cannot see itThe matrix that is one row
A circulant of size 16 is sixteen numbers, has no zero entry anywhere, and hands over its entire spectrum in closed form — the discrete Fourier transform of its first column, exactly. An eigensolver spends a sweep of Jacobi rotations over 256 entries arriving at the same answer, and agrees to 1.2·10⁻¹⁵.
Regularisation, and the answer that is chosenWhen the answer is a choice
A backward-stable least-squares solve of this problem returns an answer whose relative error is 5.5·10⁸. Nothing went wrong. The singular values decay exponentially with no gap anywhere in them, the data does not determine the answer, and something outside the data has to choose — which is the computation rather than a preliminary to it.