Estimating the Jacobian of the Singular Value Decomposition: Theory and Applications
ECCV '00 Proceedings of the 6th European Conference on Computer Vision-Part I
Robust adaptive signal processing methods for heterogeneous radar clutter scenarios
Signal Processing - Special section: New trends and findings in antenna array processing for radar
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Let $A$ be a rank-deficient matrix and let $N$ be a matrix whose norm is small compared with that of $A$. The left singular vectors of $A$ can be grouped into two matrices $U_1$ and $U_2$ whose columns provide orthonormal bases for the $p$-dimensional column space of $A$ and for its $n - p$ dimensional orthogonal complement. The left singular vectors of $\tilde{A} = A + N$ can also be partitioned into the first $p$ columns, $\tilde{U}_1$, and the last $n - p$ columns $\tilde{U}_2$. When analyzing a variety of signal processing algorithms, it is useful to know how different the spaces spanned by $U_1$ and $\tilde{U}_1$ (or $U_2$ and $\tilde{U}_2$) are. This question can be answered by developing a perturbation expansion for the subspace spanned by a set of singular vectors. A first-order expansion of this type has recently been developed and used to analyze the performance of direction-finding algorithms in array signal processing. In this paper, a new second-order expansion is derived and the result is illustrated with two examples.