Truncating the singular value decomposition for ILL-posed problems
Bert W. Rust · 1998
Discretizing the rst-kind integral equations which model many physical measurement processes yields an ill-conditioned linear regression model b = Ax + , where x is a vector representation of the function being measured, A is an instrument response matrix, b is a vector of measurements, and is a vector of unknown, random measuring errors.Least squares estimation usually gives a sum of squared residuals much smaller than the expected value and a wildly oscillating, physically implausible estimate of x .These symptoms suggest that the least squares estimate captures part of the variance that properly belongs in the residuals.One strategy for shifting some of this variance to the residuals and simultaneously stabilizing the estimate is to truncate the singular value decomposition A = UV T where U and V are orthogonal matrices and is a diagonal matrix of singular values.All of the singular values below some threshold value are reset to zero to give a new matrix tr , and the estimated solution is calculated from the generalized inverse of the matrix U tr V T .The most delicate part of this procedure is the determination of the truncation threshold.Conventionally this has been regarded as a problem of determining the umerical rank" of A, but in most cases A is clearly not rank-de cient.This paper suggests an alternate strategy which uses the variances of the measuring errors to specify a truncation for the elements of the rotated measurement vector U T b.The idea is to zero all of the components that are dominated by the measurement errors and compute the estimate using the full rank matrix.The problem of setting the truncation threshold becomes one of deciding whether or not a measured value is signi cantly dierent from zero, a procedure familiar to most experimentalists.The paper also develops some new diagnostics for the residuals which are useful not only for choosing the truncation level for the (U T b) i , but also for assessing the quality of an estimate obtained by any procedure.