knob-floor
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 A parameter that counts steps 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.
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.
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.
8 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 every one of the four methods is affordable at
and the four best errors agree
CGLS step has an interior optimum
matmul shapes agree
randomised rank has an interior optimum
Tikhonov λ has an interior optimum
truncation K has an interior optimum
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 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.
A parameter that counts steps
The regularisation field's knob is a positive real number chosen by one of three rules. The iterative field's is an integer nobody called a knob — where to stop. On the same problem the best step is 20 and the best λ is 0.025, and they reach 0.1426 and 0.1406.
Methods that were designed apartAn 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.
Regularisation, and the answer that is chosenChoosing 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.
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.
Randomised, and the guarantee that changes kindRandomisation does not create structure
On a matrix whose singular values are all equal, a rank-ten randomised approximation has error 1.0 — and so does the optimal deterministic one. Neither achieved anything, and only one of them is usually sold with the implication that it might.
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.
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.