rank-band
At its defaults it draws a randomised rank-k solve, 4 seeds a rank. Relative error against the rank kept, on a logarithmic vertical axis, with a vertical bar at each rank spanning the seeds. The band is 1.84 wide at rank 8, where the method is at its worst, and 1.022 wide at rank 24, where it is at its best. The same computation on the same data returns a different answer each time.
rank-band 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 rank kept, on a logarithmic vertical axis, with a vertical bar at each rank spanning the seeds. The band is 1.84 wide at rank 8, where the method is at its worst, and 1.022 wide at rank 24, where it is at its best. The same computation on the same data returns a different answer each time.
seeds: 4
The arguments are the ones A bound that holds with probability 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 rank kept, on a logarithmic vertical axis, with a vertical bar at each rank spanning the seeds. The band is 1.84 wide at rank 8, where the method is at its worst, and 1.022 wide at rank 24, where it is at its best. The same computation on the same data returns a different answer each time.
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.
6 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 size the randomised solve is affordable at
and barely at all at the optimum
enough seeds for a band and few enough to draw
matmul shapes agree
the seed changes the answer somewhere in the range
with the spread worst where the method is worst
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 bound that holds with probability
Every other guarantee on this site is deterministic. The randomised low-rank approximation offers one that holds with a probability, the seed changes the answer, and the honest figure is a band rather than a line.
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.
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.
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.