MMP.12:4.4 - Select strength from the required sensitivity and tolerated loss

Work out how the added structure changes recovery before choosing its numerical strength. For L=I, x0=0 and one nonzero singular direction, quadratic regularization replaces division by sigma with multiplication by

sigma/(sigma^2 + lambda).

This reduces noise amplification. With exact data, it also multiplies the true component by sigma^2/(sigma^2+lambda), shrinking it toward zero. A component invisible to A is selected through the regularizer, not recovered from the records.

Choose lambda using the error account and the receiving use. A bound on acceptable noise amplification, together with a bound on tolerated shrinkage, can determine an interval of useful values. Section :5.1 computes such a choice. A supported discrepancy level can instead guide a parameter search: compare each candidate’s residual with the level warranted by the observation and model errors. An empirical choice rule needs evidence appropriate to its own claim; fitting the available records best is not a general parameter-selection argument.

Evaluate both sides of the trade-off. Compare the changed target when the data are perturbed within their error account and when the reference, penalty or strength changes within its justified range. Return consequential dependence rather than hiding it behind one selected value. A few numerical trials can reveal a failure; they establish a bound only when an argument covers the claimed variation.

If no supported strength gives the needed sensitivity and tolerable loss, narrow the target or return the missing contribution. A stronger penalty can make outputs nearly constant while leaving them useless for the question.