On w-Optimization of the Split Covariance Intersection Filter
Hao Li
TL;DR
The paper addresses the w-optimization step in the split covariance intersection filter (Split CIF) by proving the convexity of the objective $\det(\mathbf{P}(w))$ with respect to $w$. It formalizes $\mathbf{P}(w)=(\mathbf{P}_1(w)^{-1}+\mathbf{P}_2(w)^{-1})^{-1}$, where $\mathbf{P}_1(w)$ and $\mathbf{P}_2(w)$ depend on $w$ through $\mathbf{D}_1=P_{1d}/w$ and $\mathbf{D}_2=P_{2d}/(1-w)$, under the assumption that the base matrices are PSD and that $\mathbf{P}_1(w)$, $\mathbf{P}_2(w)$ remain positive definite for $w\in(0,1)$. The core contribution is a detailed derivation showing that $\frac{d^2}{dw^2} \ln \det \mathbf{P}(w) \ge 0$ on $(0,1)$ via Jacobi's formula and matrix calculus, which implies convexity of the optimization problem in $w$. This result enables the use of convex optimization techniques to efficiently compute the w-parameter, enhancing both performance and implementation efficiency of Split CIF in general data fusion tasks. The paper also provides a demo code appendix to facilitate practical adoption.
Abstract
The split covariance intersection filter (split CIF) is a useful tool for general data fusion and has the potential to be applied in a variety of engineering tasks. An indispensable optimization step (referred to as w-optimization) involved in the split CIF concerns the performance and implementation efficiency of the Split CIF, but explanation on w-optimization is neglected in the paper [1] that provides a theoretical foundation for the Split CIF. This note complements [1] by providing a theoretical proof for the convexity of the w-optimization problem involved in the split CIF (convexity is always a desired property for optimization problems as it facilitates optimization considerably).