What a rank-10 approximation can achieve, by spectrum
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: 0.25
The arguments are the ones Randomisation does not create structure passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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: 2.25
The arguments are the ones Randomisation does not create structure passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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: 1
The arguments are the ones Randomisation does not create structure passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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: 0.75
The arguments are the ones Randomisation does not create structure passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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: 1.75
The arguments are the ones Randomisation does not create structure passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked 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 checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks 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.
12 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 σₖ₊₁ on the algebraic, j^−1.00 spectrum — checked 6 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 σₖ₊₁ on the flat spectrum
the rank-10 error equals σₖ₊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 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 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.
Randomisation 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.
Eigenvalues, singular values, rankThe vector was what was wanted
Nobody who computes a matrix exponential wants the matrix. They want eᴬᵗb — one vector, the state of a system at a later time. Twenty matrix–vector products get it to sixteen digits on a hundred-by-hundred problem, without ever forming a hundred-by-hundred exponential, and the exponential that does get computed is twenty by twenty.
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.