Four knobs on one problem at 1.0% noise
At its defaults it draws four knobs on one problem at 1.0% noise. Relative error against the fraction of each method's own range, on a logarithmic vertical axis. A truncation, a Tikhonov parameter, a conjugate gradient step count and a randomised rank each have an interior minimum, and the four minima are 0.1445, 0.1406, 0.1426, 0.1449 — a spread of 3%. The horizontal line is the best of them.
knob-floor 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.
Relative error against the fraction of each method's own range, on a logarithmic vertical axis. A truncation, a Tikhonov parameter, a conjugate gradient step count and a randomised rank each have an interior minimum, and the four minima are 0.1445, 0.1406, 0.1426, 0.1449 — a spread of 3%. The horizontal line is the best of them.
noise: 0.01
The arguments are the ones Four knobs and one floor passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Relative error against the fraction of each method's own range, on a logarithmic vertical axis. A truncation, a Tikhonov parameter, a conjugate gradient step count and a randomised rank each have an interior minimum, and the four minima are 0.1445, 0.1406, 0.1426, 0.1449 — a spread of 3%. The horizontal line is the best of them.
noise: 0.1
The arguments are the ones Four knobs and one floor passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Relative error against the fraction of each method's own range, on a logarithmic vertical axis. A truncation, a Tikhonov parameter, a conjugate gradient step count and a randomised rank each have an interior minimum, and the four minima are 0.1630, 0.1881, 0.1675, 0.1672 — a spread of 15%. The horizontal line is the best of them.
noise: 0.001
The arguments are the ones Four knobs and one floor passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Relative error against the fraction of each method's own range, on a logarithmic vertical axis. A truncation, a Tikhonov parameter, a conjugate gradient step count and a randomised rank each have an interior minimum, and the four minima are 0.1058, 0.1052, 0.1050, 0.1058 — a spread of 1%. The horizontal line is the best of them.
noise: 0.02
The arguments are the ones Four knobs and one floor passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Relative error against the fraction of each method's own range, on a logarithmic vertical axis. A truncation, a Tikhonov parameter, a conjugate gradient step count and a randomised rank each have an interior minimum, and the four minima are 0.1494, 0.1513, 0.1498, 0.1506 — a spread of 1%. The horizontal line is the best of them.
noise: 0.005
The arguments are the ones Four knobs and one floor passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Relative error against the fraction of each method's own range, on a logarithmic vertical axis. A truncation, a Tikhonov parameter, a conjugate gradient step count and a randomised rank each have an interior minimum, and the four minima are 0.1297, 0.1244, 0.1217, 0.1304 — a spread of 7%. The horizontal line is the best of them.
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.
27 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 past the qualification the gap widens with ν, at ν = 2 — checked 2 times
iterated2 sits at its ceiling at ν = 4 — checked 2 times
a noise level small enough to be noise
a size every one of the four methods is affordable at
a size the sweeps are affordable at
a size the three sweeps are affordable at
a source condition of a order the arithmetic can hold
a truth this figure knows how to build
and the ceilings are in order
and the four best errors agree
at ν = 0.25 no method has reached a ceiling and the five agree
CGLS step has an interior optimum
matmul shapes agree
on the constructed signal the two stay together
past it Tikhonov's rate stops and the others' do not
past the qualification Tikhonov falls behind the truncation
randomised rank has an interior optimum
smoothness either side of every ceiling drawn
the constructed signal gives cgls almost no rate
the constructed signal gives tikhonov almost no rate
the constructed signal gives truncation almost no rate
tikhonov sits at its ceiling at ν = 4
Tikhonov λ has an interior optimum
truncation K has an interior optimum
truths either side of the qualification
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
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.
Four 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.
Methods that were designed apartThe method that cannot use a smooth answer
On the collection's own signal four regularisers reach the same floor to a few per cent. Score them instead against answers of increasing smoothness and one stops improving. Across six decades of noise Tikhonov's error falls with a fitted slope of 0.70 whether the answer is twice or four times as smooth, while truncation's rises to 0.94 — and at 10⁻⁶ noise Tikhonov's best is 38 times truncation's.