filter-comparison
At its defaults it draws three filters at their own optima, on one problem. Filter factors against the singular-value index. Truncation at K = 21 is a step from one to zero; Tikhonov at λ = 0.0215 is a smooth descent; the 20-step conjugate gradient filter rises above one and changes direction 5 times. The three reach errors within a few per cent of each other while differing by up to 0.99 in individual factors.
filter-comparison is one function in lib/figures/compose.js —
four knobs — one floor under them, and a fifth that is not a knob at all. 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.
Filter factors against the singular-value index. Truncation at K = 21 is a step from one to zero; Tikhonov at λ = 0.0215 is a smooth descent; the 20-step conjugate gradient filter rises above one and changes direction 5 times. The three reach errors within a few per cent of each other while differing by up to 0.99 in individual factors.
noise: 0.01
The arguments are the ones An answer that changes with the seed 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.
Filter factors against the singular-value index. Truncation at K = 21 is a step from one to zero; Tikhonov at λ = 0.0215 is a smooth descent; the 20-step conjugate gradient filter rises above one and changes direction 5 times. The three reach errors within a few per cent of each other while differing by up to 0.99 in individual factors.
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.
4 distinct claims across 2 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
a size the SVD is affordable at
and the CGLS filter is not monotone
the filters differ substantially at their own optima
Against the rule
It draws a decomposition and prints its residual. It calls
cgls,
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 66
of 131 generators —
51 print a residual and
15 are exempt with a published reason;
65 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.
An answer that changes with the seed
A randomised rank-k solve is a truncation computed in a random subspace, and it reaches the same floor as the deterministic ones. What it does not do is return the same answer twice — a factor of 1.84 across four seeds at rank 8, and 1.02 at the rank where the method is best.
Methods that were designed apartFour knobs and one floor
A truncation, a Tikhonov parameter, a step count and a randomised rank, on one problem with an answer that is known. Their best errors are 0.1445, 0.1406, 0.1426 and 0.1449 — a spread of 3% across four methods that share no arithmetic.
Eigenvalues, singular values, rankRank 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, rankRestarting is a filter
A restart throws away the Ritz values it does not want and begins again from a new starting vector. Written in the eigenbasis, that vector's components have been multiplied by a polynomial with its roots at the discarded values — measured component by component, and agreeing with the polynomial to rounding.
Eigenvalues, singular values, rankThe best approximation there is
The error of the best rank-k approximation is not bounded by the next singular value. It is equal to it. That is an unusually sharp theorem, and it makes the theorem itself usable as an independent check on the computation.
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.