sketch-distortion
At its defaults it draws sketch distortion against sketch width, for 60 vectors. A log-log plot of the worst relative change in vector length against the number of rows in the sketch, for vectors of two dimensions a factor of four apart. The two curves lie almost on top of one another and both fall steadily.
sketch-distortion is one function in lib/figures/randomised.js —
randomised — a bound that holds with a probability, and the seed that moves. 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.
A log-log plot of the worst relative change in vector length against the number of rows in the sketch, for vectors of two dimensions a factor of four apart. The two curves lie almost on top of one another and both fall steadily.
vectors: 60
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.
A log-log plot of the worst relative change in vector length against the number of rows in the sketch, for vectors of two dimensions a factor of four apart. The two curves lie almost on top of one another and both fall steadily.
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.
7 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.
at 25 rows the distortion barely notices a fourfold change of dimension — asserted 5 times
and a wider sketch distorts less
at a rate near the inverse square root of the sketch width
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 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.
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.
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.
Randomised, and the guarantee that changes kindThe dimension does not appear
A random projection preserves the lengths of a set of vectors to within a distortion that depends on how many vectors there are and not on how many coordinates each one has. That is the fact the whole field rests on, and it is genuinely surprising.
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.