projected-floor
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 drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run 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: 32
The arguments are the ones Changing the condition number on purpose 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.
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 6.82·10⁻⁴; the noise divided by it, and the unregularised error, rise together and stay a factor of about 0.23 apart over four decades.
steps: 28
The arguments are the ones Randomisation does not create structure 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.
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 0.00282; the noise divided by it, and the unregularised error, rise together and stay a factor of about 0.21 apart over four decades.
steps: 48
The arguments are the ones The condition number is an amplifier 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.
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: 36
The arguments are the ones The projection and the right angle 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.
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 8.05·10⁻⁵; the noise divided by it, and the unregularised error, rise together and stay a factor of about 0.22 apart over four decades.
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.
29 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.
the error stays under ‖e‖/σ_min at k = 24 — asserted 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 70
of 151 generators —
55 print a residual and
15 are exempt with a published reason;
81 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%.
Iterating, instead of factorisingChanging the condition number on purpose
Preconditioning is usually introduced as a trick that makes an iteration converge faster. It is not a trick. It is solving a different system with the same solution and a condition number chosen rather than inherited, and the new condition number is computable.
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.
Two errors, and whose fault they areThe condition number is an amplifier
κ is usually introduced as a definition and then quoted. It is a measurement: perturb the input by a known amount, look at how much the output moves, and the largest ratio you can find is the number.
Least squares, and the road not to takeThe projection and the right angle
The least-squares solution is the one whose residual is perpendicular to everything the columns can reach. That is not a mnemonic — it is an equation, Aᵀr = 0, and the computed answer satisfies it to 10⁻¹⁶.
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.
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.