Parameter choice — where it appears
Named by 21 essays across 4 fields — each of them below, with the objects they name alongside it.
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.
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.
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%.
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.
A ranking that is an eigenvector
PageRank is the stationary vector of a walk that follows links with probability α and jumps at random otherwise. The iteration and the elimination agree to 4·10⁻¹⁷. What α is set to changes which pages come third, fourth and fifth.
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 accuracy worth paying for
Used as a preconditioner, a hierarchical representation gets better at every accuracy — the iteration count falls monotonically all the way to the tightest tolerance. The total work does not. Its minimum sits at a rank-one preconditioner on an easy problem and six decades further along on a hard one.
Thirty-two coefficients instead of a noise level
The discrepancy principle has to be told the noise, and told too little it does not degrade — it falls off a cliff, at 0.80 of the truth when the noise is 10% and at 0.58 when it is 0.001%, exactly where the understatement forces the filter past its best truncation. The missing number is in the data. The root mean square of the last thirty-two coefficients never sends the rule over the cliff at or below 1% noise in four hundred draws, where eight coefficients with the same median do so thirty-five times.
A ranking whose order is not determined
Of the fourteen adjacent comparisons in a top-fifteen, fourteen survive perturbing the arithmetic at the rounding level, eleven survive moving the teleportation parameter across its usual range, and three survive removing one link. The computation is the strongest part of the answer.
A stopping rule that follows the run it is given
A preconditioner that reaches the answer four times sooner leaves four steps within 10% of its best instead of sixteen, and a rule that stops by the residual ought to miss so narrow a window more often. Over forty draws of the noise it misses it less: the discrepancy principle stops at 1.030 times the preconditioned run's best against 1.073 times the plain run's. And past the edge it stops within a factor of 1.7 of a run whose own best is 5.5 times Tikhonov's — faithful to a run that has already failed.
The knob that moved two things
Decide how many digits the answer needs, divide by the condition number, and compress to that. It is the one rule licensed in advance here, and its two factors are not the independent inputs it reads as: the partition's leaf moves neither of them and moves the answer by nearly a factor of three, and the only knob here that raises κ halves the ranks while it does so.
Where the grid hands over to λ
An unregularised solve on a coarse grid comes within a tenth of the best Tikhonov answer on a fine one, and the pair of a grid and a λ was left unmeasured. Measured, the two do not trade. On every grid up to the best unregularised one no λ helps at all. On every grid of 40 points and more the best λ is the same to within a quarter of a decade — 3.2·10⁻² at 1% noise per sample, 10⁻³ at 0.01% — and the 96-point grid with it beats the best coarse grid by 7, 9 and 13 per cent. The grids between the two, given their own λ, land between them.
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.
A count that marks the edge and not the pace
The number of directions a truncated preconditioner divides is counted for free when it is built, and it was proposed as a stand-in for the stride it buys. On three blurs it is not one — the runs leave the answer at strides of 5.71, 3.44 and 2.41. What the count does predict is the edge: on all three blurs, at two noise levels, a run stops landing on the answer's path within five per cent of the point where the count reaches the answer's own effective dimension. And the halved cutoff rule, measured on one blur, crosses that line on the narrowest.
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.
The minimum on the right
Generalised cross-validation's worst draws on a fine grid were all one mistake: a second dip in its function at λ near zero, deeper than the real minimum. The proposed repair was a residual threshold, one number, refusing any λ whose residual falls too far below the real minimum's. Measured over 528 draws, a threshold of one half still lets two hundredfold misses through; only the extreme value, which is no threshold at all but the rule 'take the rightmost local minimum', removes all eleven. It costs nothing on the coarse grids where the dip is the right answer, and on the collection's own problem over five thousand draws it turns 243 tenfold misses into 17.
More samples take the floor and leave the dip
GCV's catastrophic misses on fine grids were blamed on squareness: on an n × n system the residual and n − t both reach zero as λ does, and their ratio can dip there. With more samples than unknowns neither reaches zero, and the prediction was that the dip would be gone by construction. Half of it is. The minimum at the floor of the scale, 21 draws in 240 on square systems, is gone at every ratio. The interior dip is not — 28, 20, 13, 8 and 4 draws at one to four samples per unknown — and at four per unknown one draw still misses the oracle by 877 times.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Tikhonov regularisationDiscrepancy principleRegularisationGeneralised cross-validationL-curveFilter factorsIll-posed problemOracleNoise floorPreconditioningCondition numberConjugate gradients