Improving stochastic estimates with inference methods: Calculating matrix diagonals

Estimating the diagonal entries of a matrix, that is not directly accessible but only available as a linear operator in the form of a computer routine, is a common necessity in many computational applications, especially in image reconstruction and statistical inference. Here, methods of statistical inference are used to improve the accuracy or the computational costs of matrix probing methods to estimate matrix diagonals. In particular, the generalized Wiener filter methodology, as developed within information field theory, is shown to significantly improve estimates based on only a few sampling probes, in cases in which some form of continuity of the solution can be assumed. The strength, length scale, and precise functional form of the exploited autocorrelation function of the matrix diagonal is determined from the probes themselves. The developed algorithm is successfully applied to mock and real world problems. These performance tests show that, in situations where a matrix diagonal has to be calculated from only a small number of computationally expensive probes, a speedup by a factor of 2 to 10 is possible with the proposed method.

Figure 1

Result from the trivial case: (a) the exact matrix diagonal, (b) the probing estimate, and (c) the Bayesian estimate started with four probes, both after around $0.3$ s.

Figure 2

$L^2$ norm of the error (divided by the number of pixels $r$) as a function of CPU time for the trivial case: the evolution of the probing estimator (solid), its theoretical prediction $\propto 1/\sqrt{A}$ (dotted) and the Bayesian estimators starting with four (dashed) and nine samples (dashed dotted) are shown.

Figure 3

Realistic case: (a) the matrix diagonal of the mock noise covariance $N$ and (b) the “true” diagonal of the propagator $D={(S^{-1}+N^{-1})}^{-1}$.

Figure 4

Same as Fig. 2, only for the propagator for the realistic case.

Figure 5

Diagonal of the propagator for the reconstruction of the galactic Faraday depth. Panel (a) shows the result of the exact calculation according to Eq. (11), panel (b) the probing result after ten iterations, and panel (c) the result of the Bayesian estimator, using ten random vectors as well.

Figure 6

Same as Fig. 2, only for the propagator for the reconstruction of the galactic Faraday depth.