filter-factors
At its defaults it draws two filters on one sum, λ = 0.01. The weight each term of the solution is given, against its index. Truncation is a step: one for the first 26 terms and zero after. Tikhonov is σ²/(σ² + λ²), which falls smoothly through the same place. The unregularised solution is the constant one, which is why it divides noise by a σ of 1.7·10⁻¹³.
filter-factors 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.
The weight each term of the solution is given, against its index. Truncation is a step: one for the first 26 terms and zero after. Tikhonov is σ²/(σ² + λ²), which falls smoothly through the same place. The unregularised solution is the constant one, which is why it divides noise by a σ of 1.7·10⁻¹³.
lambda: 0.001
The arguments are the ones Choosing without knowing 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 weight each term of the solution is given, against its index. Truncation is a step: one for the first 32 terms and zero after. Tikhonov is σ²/(σ² + λ²), which falls smoothly through the same place. The unregularised solution is the constant one, which is why it divides noise by a σ of 1.7·10⁻¹³.
lambda: 0.01
The arguments are the ones The valley with no bottom 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 weight each term of the solution is given, against its index. Truncation is a step: one for the first 26 terms and zero after. Tikhonov is σ²/(σ² + λ²), which falls smoothly through the same place. The unregularised solution is the constant one, which is why it divides noise by a σ of 1.7·10⁻¹³.
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.
69 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.
the filter is monotone at k = 1 — asserted 63 times
a regularisation parameter that regularises
and both beat doing nothing by orders of magnitude
and nearly none of the last is kept
the first filter factor matches its closed form
the matched truncation keeps some terms and drops some
Tikhonov keeps more than half of the last term the truncation keeps
Against the rule
It draws a decomposition and prints its residual. It calls
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 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.
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.
Least squares, and the road not to takeThe valley with no bottom
A degree-nine fit's coefficients can be moved by a third of their own size before the residual changes in the sixth significant figure. The arithmetic did not lose those digits. The data never contained them.
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.
Regularisation, and the answer that is chosenWhere the answer stops being in the data
The Picard condition finds the index where a noisy right-hand side stops carrying signal, from the data alone, with no knowledge of the answer. It lands at 32 where the truncation that actually minimises the error is 28 — and at 45 where the best is 38. It overshoots at every noise level from 1% to 0.001%, and it overshoots for a reason.