Singular values of a rank-4 matrix with noise of relative size 10⁻⁸
At its defaults it draws singular values of a rank-4 matrix with noise of relative size 10⁻⁸. Ten singular values on a logarithmic axis. The first four sit near one; the rest sit at the noise level, and the vertical distance between the two groups is the evidence for the rank.
rank-decision is one function in lib/figures/spectra.js —
spectra — sensitivity, the symmetric easy case, and rank as a decision. 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.
Ten singular values on a logarithmic axis. The first four sit near one; the rest sit at the noise level, and the vertical distance between the two groups is the evidence for the rank.
logNoise: -8
The arguments are the ones Rank is a decision passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Ten singular values on a logarithmic axis. The first four sit near one; the rest sit at the noise level, and the vertical distance between the two groups is the evidence for the rank.
logNoise: -14
The arguments are the ones Rank is a decision passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Ten singular values on a logarithmic axis. The first four sit near one; the rest sit at the noise level, and the vertical distance between the two groups is the evidence for the rank.
logNoise: -1
The arguments are the ones Rank is a decision passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Ten singular values on a logarithmic axis. The first four sit near one; the rest sit at the noise level, and the vertical distance between the two groups is the evidence for the rank.
logNoise: -5
The arguments are the ones Rank is a decision passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Ten singular values on a logarithmic axis. The first four sit near one; the rest sit at the noise level, and the vertical distance between the two groups is the evidence for the rank.
logNoise: -4
The arguments are the ones Rank is a decision passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Ten singular values on a logarithmic axis. The first four sit near one; the rest sit at the noise level, and the vertical distance between the two groups is the evidence for the rank.
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.
5 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.
and the gap is inversely proportional to the noise
matmul shapes agree
the added noise is the size the caption states
the fourth singular value is above the fifth
the noiseless matrix has the rank it was built with
Against the rule
It calls a factoriser without drawing a factorisation
(svd),
so the rule is written down as not applying, with the reason:
the figure is the singular value spectrum itself
The exemption list is the interesting half of the rule rather than an escape hatch — it is
where a decision about a figure had to be argued in one line. residualcheck
refuses an exemption that is not doing work, and rejected ten of the fifteen written for the
expansion's figures on exactly that ground: a figure whose vertical axis is a residual
satisfies the rule by construction, and touching a factoriser does not by itself require an
entry.
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.
Rank is a decision
A floating-point matrix does not have a rank. It has a spectrum of singular values, and somewhere in that spectrum is a place where the values stop being signal and start being noise. Deciding where is a judgement, and the evidence for it is a gap.
Eigenvalues, singular values, rankThe largest gap is inside the null space
The rule recommended for counting a pencil's infinite eigenvalues is to cut at the largest gap in the singular values of B. On integer pencils, with no perturbation anywhere and an exact answer available from the characteristic polynomial, it returns the wrong count on nine of twenty-five — because the singular values that are mathematically zero come back spread over a hundred and forty orders of magnitude, and the largest ratio in the list is between two of them.