The smallest singular value the subspace has let in
At its defaults it draws the smallest singular value the subspace has let in. Three quantities against the number of steps, on a logarithmic vertical axis. The smallest singular value of the projected problem falls from 0.724 to 1.28·10⁻⁷; the noise divided by it, and the unregularised error, rise together and stay a factor of about 0.21 apart over four decades.
projected-floor is one function in lib/figures/hybrid.js —
hybrid regularisation — a penalty inside the subspace, and the rule that transfers to it. 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.
Three quantities against the number of steps, on a logarithmic vertical axis. The smallest singular value of the projected problem falls from 0.724 to 1.28·10⁻⁷; the noise divided by it, and the unregularised error, rise together and stay a factor of about 0.21 apart over four decades.
steps: 40
The arguments are the ones A parameter chosen on a smaller problem passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Three quantities against the number of steps, on a logarithmic vertical axis. The smallest singular value of the projected problem falls from 0.724 to 1.31·10⁻⁵; the noise divided by it, and the unregularised error, rise together and stay a factor of about 0.23 apart over four decades.
steps: 48
The arguments are the ones The step that stops mattering passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Three quantities against the number of steps, on a logarithmic vertical axis. The smallest singular value of the projected problem falls from 0.724 to 1.28·10⁻⁷; the noise divided by it, and the unregularised error, rise together and stay a factor of about 0.21 apart over four decades.
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.
29 distinct claims across 3 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.
the error stays under ‖e‖/σₘᵢₙ at k = 24 — checked 25 times
a size the reference SVD is affordable at
a subspace smaller than the problem and large enough to go bad
and is a fixed fraction of it from twenty-four steps on
and the projected condition number never approaches the problem's
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.
A parameter chosen on a smaller problem
Inside a hybrid method the regularisation parameter is chosen on a 25×24 problem rather than a 64×64 one. The rule that reads a residual transfers exactly; the rule that reads a trace is biased by exactly two grid steps at twenty-four steps and one at forty, at every noise level from 10% to 0.1%.
Methods that were designed apartThe step that stops mattering
Regularise the problem the iteration has built rather than the problem it was given, and the error curve stops turning. The unregularised run ends 1,127 times above its own best; the same run with a penalty inside it ends 1.000000000003 times above.