The recovered signal at three λ, 0.10% noise
At its defaults it draws the recovered signal at three λ, 0.10% noise. Three recovered signals against the truth. At the best λ the two bumps are recovered and the step is rounded off. Fifty times smaller and the answer is dominated by amplified noise; fifty times larger and the step has gone entirely. The step is the feature that costs the most to represent and it is the first thing any filter loses.
recovered-signal is one function in lib/figures/regular.js —
regularisation — the filter, the corner, and the answer that is a choice. 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.
Three recovered signals against the truth. At the best λ the two bumps are recovered and the step is rounded off. Fifty times smaller and the answer is dominated by amplified noise; fifty times larger and the step has gone entirely. The step is the feature that costs the most to represent and it is the first thing any filter loses.
noise: 0.001
The arguments are the ones Choosing without knowing passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Three recovered signals against the truth. At the best λ the two bumps are recovered and the step is rounded off. Fifty times smaller and the answer is dominated by amplified noise; fifty times larger and the step has gone entirely. The step is the feature that costs the most to represent and it is the first thing any filter loses.
noise: 0.01
The arguments are the ones Choosing without knowing passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Three recovered signals against the truth. At the best λ the two bumps are recovered and the step is rounded off. Fifty times smaller and the answer is dominated by amplified noise; fifty times larger and the step has gone entirely. The step is the feature that costs the most to represent and it is the first thing any filter loses.
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 3 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 noise level small enough to be noise
and so is too much
and the over-regularised one too small
the under-regularised answer is too big
too little regularisation is worse
Against the rule
It draws a decomposition and prints its residual. It calls
svd, lstsqQR,
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.
Choosing without knowing
Three published rules for choosing a regularisation parameter, scored against an oracle that requires the exact answer and is therefore not a method. Generalised cross-validation lands on the oracle's λ exactly; the discrepancy principle costs 6%; the L-curve costs 129%. And told a noise level ten times too small, the discrepancy principle's error goes from 0.112 to 10,449.
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.