Stochastic determination of matrix determinants

Matrix determinants play an important role in data analysis, in particular when Gaussian processes are involved. Due to currently exploding data volumes, linear operations—matrices—acting on the data are often not accessible directly but are only represented indirectly in form of a computer routine. Such a routine implements the transformation a data vector undergoes under matrix multiplication. While efficient probing routines to estimate a matrix's diagonal or trace, based solely on such computationally affordable matrix-vector multiplications, are well known and frequently used in signal inference, there is no stochastic estimate for its determinant. We introduce a probing method for the logarithm of a determinant of a linear operator. Our method rests upon a reformulation of the log-determinant by an integral representation and the transformation of the involved terms into stochastic expressions. This stochastic determinant determination enables large-size applications in Bayesian inference, in particular evidence calculations, model comparison, and posterior determination.

Figure 1

Illustration of the matrices $A_2$ (top) and $A_4$ (bottom) in position space with linear color bars.

Figure 2

The integrand of Eq. (8) (lower panel) and $\mathrm{\Delta }\left(x\right)$ (upper panel) for explicit and implicit representations of $A_4$.

Figure 3

The integrand of Eq. (8) (lower panel) and $\mathrm{\Delta }\left(x\right)$ (upper panel) for explicit and implicit representations of $A_2$ with only $m=10$ steps in pseudotime.

Figure 4

Dependency of the determinant's result on the discretization of the integration interval into $m$ parts, using $A_4$.

Figure 5

Logarithmic posterior of the calibration amplitude parameter $\gamma$ using implicit and explicit representations of the involved operators, see Eq. (21) and Eq. (22) for details. The abbreviation $\mathrm{\Delta }$ denotes the logarithm of the term given by Eq. (22).