Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Picard condition — where it appears

The requirement that the coefficients uₖᵀb of a right-hand side fall faster than the singular values σₖ, so every term of the solution's expansion stays bounded. With noise the coefficients flatten at the noise level, and the index where they flatten marks where the data stops carrying signal.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

081624324048566410⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹index kmagnitudethe floor: k = 32best truncation: k = 28σₖ|uₖᵀb| exact|uₖᵀb| with noisetwo different indicesthe crossing, from the data alone32the truncation that is actually best28relative error there0.11the exact coefficients never flattenthe noisy ones stop at ‖e‖/√n

Where the answer stops being in the data

The Picard condition finds the index where a noisy right-hand side stops carrying signal, from the data alone, with no knowledge of the answer. It lands at 32 where the truncation that actually minimises the error is 28 — and at 45 where the best is 38. It overshoots at every stop from 10% noise to 0.0001%, and it overshoots for a reason. The best truncation walks up the spectrum in a straight line, six or seven indices a decade; the crossing climbs in jumps of 11, 0, 8, 5 and 1.

regularisation · Regularisation 081624324048566410⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹index kmagnitudethe floor: k = 43best truncation: k = 31σₖ|uₖᵀb| exact|uₖᵀb| with noise|uₖᵀe|, smoothedcorrelation ρ = 0.9noise energy, first 28 directions0.98tilt of the noise floor, decades1.1the crossing43the best truncation31the floor tilts, and is still a floorρ = 0.9

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.

regularisation · Regularisation 1110¹10²10³10⁴10⁵the noise level it is told ÷ the true oneerror ÷ the oracle's0.40.50.71.523twice the oraclecliff, ρ = 0.69told the truthworst of 400middle 80%median draweach draw against its own oraclecliff, median ρ0.69told the truth, median1.1told a half, median3201shaded: the middle 80% of drawsbelow the cliff the error has doubled

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.

regularisation · Parameter choice

Named alongside it

The objects these essays reach for when they reach for this one.

Noise floorRegularisationDiscrepancy principleFilter factorsIll-posed problemParameter choiceSingular value decompositionTikhonov regularisationCorrelated noiseGeneralised cross-validationL-curveOracle

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