Generalised cross-validation — where it appears
Named by 13 essays across 4 fields — each of them below, with the objects they name alongside it.
Counting what cannot be looked at
The trace is n additions and one of the most expensive quantities in the subject to estimate, because the matrices whose trace is wanted are never stored. Hutchinson's estimator is unbiased with one line of algebra — and its variance depends on which random vector is used, by a factor that is a property of the matrix, and on a diagonal matrix one choice is exact from the first probe and the other is not.
A second blur, narrower than the first
A regularised answer is not the truth with the noise taken out. It is the truth seen through a second blur, V F Vᵀ, which depends on the operator and λ and on nothing that was measured. At the best λ for 0.1% noise its rows are 2.82 points wide against the instrument's 5.89, they dip to −0.075 on either side, and their width times the number of components kept stays between 1.10n and 1.27n across seven decades of λ. Two spikes four points apart come back as two; three apart, as one.
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.
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.
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.
The degree that is safe to overshoot
The rules that choose a Tikhonov parameter miss by factors of millions on one draw in twenty. Transplanted to the degree of a polynomial fit, in a basis orthonormal on the data, the same rules never cost more than 2.7 times the best degree's error in three hundred draws. The reason is the shape of the valley they search: six degrees too few costs from 44 to 16,000 times the best error, forty degrees too many costs about twice it. The one rule with a tail, the discrepancy principle, has its threshold half a standard deviation above the residual it is waiting for.
An estimate that shares the dip's luck
GCV's interior dips form where the residual per remaining degree of freedom is low, and the proposal was a guard that estimates the noise from the least-squares residual and refuses a minimum whose residual is implausibly small. Over 960 draws holding 45 dips it refuses none, and picks plain GCV's minimum on every draw. At the dip ρ is 0.84 with the true noise and 0.98 with the estimate, because the estimate is made from the residual at the dip's own end of the scale and inherits its luck. Told the true noise instead, the guard can refuse 14 dips only by refusing 78 real minima. Handed the same estimate, the discrepancy principle misses by 570,000 times.
Named alongside it
The objects these essays reach for when they reach for this one.
Discrepancy principleParameter choiceTikhonov regularisationRegularisationL-curveInfluence matrixDeconvolutionDiscretisationOracleFilter factorsIll-posed problemLeast-squares