Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

inner-rule-map

One function in the hybrid library, called 8 times across 7 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 8 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

At its defaults it draws two inner rules, and the quantity one of them divides by. Three quantities against the subspace size on a logarithmic vertical axis. The trace in GCV's denominator climbs from 1.01 to 26.98, a factor of 27; the λ it selects moves by 2.15. The discrepancy principle has no answer below 8 steps, where no parameter brings the residual down to the noise, and returns 0.0681 at every size above it.

inner-rule-map 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.

Two inner rules, and the quantity one of them divides byThree quantities against the subspace size on a logarithmic vertical axis. The trace in GCV's denominator climbs from 1.01 to 26.98, a factor of 27; the λ it selects moves by 2.15. The discrepancy principle has no answer below 8 steps, where no parameter brings the residual down to the noise, and returns 0.0681 at every size above it.27121722273237424710⁻²10⁻¹110¹subspace size kvalueno answer below 8GCV's traceλ from GCVλ from the residuala denominator that is not the problem'strace at k = 41trace at k = 4827λ range across the run2.2subspace before an answer8the divisor moves by twenty-sevenand the answer does not move

Three quantities against the subspace size on a logarithmic vertical axis. The trace in GCV's denominator climbs from 1.01 to 26.98, a factor of 27; the λ it selects moves by 2.15. The discrepancy principle has no answer below 8 steps, where no parameter brings the residual down to the noise, and returns 0.0681 at every size above it.

noise: 0.02

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.

Two inner rules, and the quantity one of them divides byThree quantities against the subspace size on a logarithmic vertical axis. The trace in GCV's denominator climbs from 1.01 to 28.31, a factor of 28; the λ it selects moves by 2.15. The discrepancy principle has no answer below 6 steps, where no parameter brings the residual down to the noise, and returns 0.0681 at every size above it.27121722273237424710⁻¹110¹subspace size kvalueno answer below 6GCV's traceλ from GCVλ from the residuala denominator that is not the problem'strace at k = 41trace at k = 4828λ range across the run2.2subspace before an answer6the divisor moves by twenty-sevenand the answer does not move

Three quantities against the subspace size on a logarithmic vertical axis. The trace in GCV's denominator climbs from 1.01 to 28.31, a factor of 28; the λ it selects moves by 2.15. The discrepancy principle has no answer below 6 steps, where no parameter brings the residual down to the noise, and returns 0.0681 at every size above it.

noise: 0.01

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.

Two inner rules, and the quantity one of them divides byThree quantities against the subspace size on a logarithmic vertical axis. The trace in GCV's denominator climbs from 1.01 to 26.98, a factor of 27; the λ it selects moves by 2.15. The discrepancy principle has no answer below 8 steps, where no parameter brings the residual down to the noise, and returns 0.0681 at every size above it.27121722273237424710⁻²10⁻¹110¹subspace size kvalueno answer below 8GCV's traceλ from GCVλ from the residuala denominator that is not the problem'strace at k = 41trace at k = 4827λ range across the run2.2subspace before an answer8the divisor moves by twenty-sevenand the answer does not move

Three quantities against the subspace size on a logarithmic vertical axis. The trace in GCV's denominator climbs from 1.01 to 26.98, a factor of 27; the λ it selects moves by 2.15. The discrepancy principle has no answer below 8 steps, where no parameter brings the residual down to the noise, and returns 0.0681 at every size above it.

noise: 0.1

The arguments are the ones An answer that changes with the seed 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.

Two inner rules, and the quantity one of them divides byThree quantities against the subspace size on a logarithmic vertical axis. The trace in GCV's denominator climbs from 1.06 to 33.01, a factor of 31; the λ it selects moves by 2.15. The discrepancy principle has no answer below 4 steps, where no parameter brings the residual down to the noise, and returns 0.215 at every size above it.27121722273237424710⁻¹110¹10²subspace size kvalueno answer below 4GCV's traceλ from GCVλ from the residuala denominator that is not the problem'strace at k = 41.3trace at k = 4833λ range across the run2.2subspace before an answer4the divisor moves by twenty-sevenand the answer does not move

Three quantities against the subspace size on a logarithmic vertical axis. The trace in GCV's denominator climbs from 1.06 to 33.01, a factor of 31; the λ it selects moves by 2.15. The discrepancy principle has no answer below 4 steps, where no parameter brings the residual down to the noise, and returns 0.215 at every size above it.

noise: 0.005

The arguments are the ones The valley with no bottom 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.

Two inner rules, and the quantity one of them divides byThree quantities against the subspace size on a logarithmic vertical axis. The trace in GCV's denominator climbs from 1.01 to 25.73, a factor of 26; the λ it selects moves by 3.16. The discrepancy principle has no answer below 12 steps, where no parameter brings the residual down to the noise, and returns 0.0464 at every size above it.27121722273237424710⁻²10⁻¹110¹subspace size kvalueno answer below 12GCV's traceλ from GCVλ from the residuala denominator that is not the problem'strace at k = 41trace at k = 4826λ range across the run3.2subspace before an answer12the divisor moves by twenty-sevenand the answer does not move

Three quantities against the subspace size on a logarithmic vertical axis. The trace in GCV's denominator climbs from 1.01 to 25.73, a factor of 26; the λ it selects moves by 3.16. The discrepancy principle has no answer below 12 steps, where no parameter brings the residual down to the noise, and returns 0.0464 at every size above it.

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 5 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 the sweep is affordable at

a subspace smaller than the problem

and at this noise level it hardly moves at all

and available over most of it

and the λ it divides into moves less than it does

the discrepancy principle is undefined somewhere on this axis

the trace moves by an order of magnitude

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.

Randomised, and the guarantee that changes kind

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 apart

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 apart

An 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 chosen

Choosing 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 apart

Four 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.

 Methods that were designed apart

The 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 take

The 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.

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