Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

General form regularisation — where it appears

Tikhonov regularisation with a penalty on ‖Lx‖ rather than on ‖x‖, for an operator L that says what a smooth answer looks like. L is usually a difference operator, so the penalty charges for roughness instead of for size, and the filter it produces lives in the generalised singular basis of the pair rather than in the operator's own.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

where each λ sitsoracle's share0.0037corner's share0.1corner's cost25and what a target costsbest fixed target0.0035its cost1estimate ÷ truth at the oracle0.9210⁻⁸10⁻⁶10⁻⁴10⁻²110⁻³10⁻²10⁻¹110¹10²10³10⁴λnoise part ÷ signal parttarget 0.0035oracle λ: share 0.0037corner: share 0.102solid: the share on this draw · dashed: the share a rule can estimatethe rule is a level set of the dashed curve

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.

regularisation · Parameter choice what each reachesbest pair0.0034‖x‖ alone0.0048‖L₁x‖ alone0.0035λ₁, on ‖x‖λ₂, on ‖L₁x‖ — log₁₀0-6-5-4-3-2-100-6-5-4-3-2-10outlined row and column: one penalty switched offred cell: the best pair, 0.00343darker is worse, on a logarithmic scalethe optimum sits on or beside an edge

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.

regularisation · Parameter choice

Named alongside it

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

Filter factorsL-curveOracleParameter choiceRegularisationTikhonov regularisationCondition numberDiscrepancy principleGeneralised cross-validationNoise levelNormal equationsQR factorisation

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