L-curve — where it appears
Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
Noise that spares the answer and fools the rules
Make each noise sample remember the last one, keep its size fixed, and the best answer available gets slightly better — 0.1056 to 0.1010 — because slow noise hides in the directions where dividing by σ costs nothing. The Picard crossing still lands two dozen indices past the best truncation. What breaks is the rules. Generalised cross-validation more than doubles the best error on 14 draws of 48 instead of 3, the discrepancy principle's typical cost triples, and the two miss in opposite directions. Whitening by the covariance takes GCV back to 3.
One draw in twenty
Sixteen draws gave generalised cross-validation a worst case of 12%. A thousand draws at each of five noise levels give it a second answer on four to six in every hundred, ten to seven million times worse than the oracle, while its median stays among the best of five rules. The quasi-optimality criterion, told nothing either, never costs more than 1.41 in five thousand draws. The share settles by a thousand draws, and letting the search look further down more than triples it.
The corner reads the norm it is drawn in
The L-curve was the costliest rule this field scored, and the cost was not the rule's. On the same sixty draws, with the same best achievable error, the corner of ‖x‖ against the residual costs 1.53 times the oracle and the corner of ‖L₁x‖ costs 1.003. Across five signals and three penalties the corner lands wherever amplified noise is between a tenth and a fifth of the norm being plotted, and it finds the oracle only when the oracle happens to sit there — twenty-nine times too costly on a smooth signal under ‖x‖, within half a per cent on four spikes.
The rule that is wrong in the right direction
The preconditioner's cutoff is not a new parameter. It is the regularisation parameter this field already knows how to choose, halved — and the rule criticised for choosing λ a factor of two or three too large is the one whose cutoff keeps the floor on every draw, where the rule that chooses λ to within 3% has a worst draw two hundred times off it.
A rule that has to be told how good its answer will be
The L-curve's corner reads a noise share of 0.10 to 0.21 across fifteen pairings of signal and penalty, and the share the best λ sits at runs from 0.0037 to 0.43 — a factor of a hundred and fifteen. A rule aimed at the right share is within a few per cent of the oracle on every one of them. The right share is about a third to four-fifths of the relative error that λ will achieve, which is the number the answer was wanted for.
A second penalty is not a second parameter
Penalise ‖x‖ and ‖L₁x‖ at once and there are two λ to choose. Over ten draws on five signals the best pair beats the better single penalty by between 0.00% and 3.1%, and one of its two parameters is exactly zero on 30 to 70 per cent of draws. Choosing the wrong one of the two costs up to 54%. The surface is a choice between two curves with a knob nobody needs.
The data count their dimensions, not the step's
Every grid in the deconvolution essays was chosen with the answer in hand, and so was every λ. From the data alone, the discrepancy principle's worst draw is within 16 per cent of the oracle on every grid from 16 points to 96; generalised cross-validation is better on the median draw and, on grids of thirty points and more, has draws thousands of times worse. And the data can say how many dimensions they carry — about 20, 25 and 29 at three noise levels, one number once the grid exceeds it — but not how many more the step needs: the grid that count chooses is 14 to 19 per cent worse than forty points at the lower two.
A third penalty on a flat floor
Two penalties at once were worth three per cent at most, and the explanation offered was that one penalty already does the work — which predicts that a third buys less still and that the best choice sits on a face of the parameter cube. The third buys a median of exactly nothing on all five signals and 0.66% on its best draw. But on twelve draws of forty the best triple does use all three, and on every one of them the nearest point with a penalty switched off is within that same 0.66%. The minimum is not on a face; it is on a floor so flat that where it lands is noise. Choosing the right single penalty is worth a factor of two.
A corner chosen without the iteration
The ramp — conjugate gradients' first-step filter kept as a fixed method — matches the iteration's error at the same position, but every comparison placed it at the iterate's position, which needs the iteration. Placed instead by the iteration's own stopping rules, it was predicted to land within the same tenth of its best and earlier. The discrepancy principle does exactly that, earlier on 61 draws of 72 and dearer by a few hundredths on 71. The L-curve finds the ramp's corner as well as Tikhonov's and far better than the iteration's: twice the best on 26 draws against 44, worst 16 against 896. And GCV fails the ramp in a way it never fails the iteration — choosing a solution that discards three quarters of one direction, 696 times its best — until its first minimum from the regularised end is taken, after which the ramp is within a tenth of its best on 64 draws of 72 and never worse than 1.25.
Named alongside it
The objects these essays reach for when they reach for this one.
Parameter choiceTikhonov regularisationDiscrepancy principleGeneralised cross-validationRegularisationFilter factorsNoise floorOracleGeneral form regularisationConjugate gradientsDeconvolutionIll-posed problem