Adaptive strategies for solving parameterized systems using homotopy continuation
For researchers using homotopy continuation on parameterized systems, the paper offers incremental improvements to numerical stability and efficiency, but lacks empirical validation.
The paper investigates three adaptive strategies for solving parameterized polynomial systems via homotopy continuation: adaptive affine patches for homogeneous systems, adaptive subsystem selection for overdetermined systems, and heuristic truncation of nonreal solution paths. These are demonstrated on two computer vision problems, but no quantitative results are provided.
Three aspects of applying homotopy continuation, which is commonly used to solve parameterized systems of polynomial equations, are investigated. First, for parameterized systems which are homogeneous, we investigate options for performing computations on an adaptively chosen affine coordinate patch. Second, for parameterized systems which are overdetermined, we investigate options for adaptively selecting a well-constrained subsystem to restore numerical stability. Finally, since one is typically interested in only computing real solutions for parameterized problems which arise from applications, we investigate a scheme for heuristically identifying solution paths which appear to be ending at nonreal solutions and truncating them. We demonstrate these three aspects on two problems arising in computer vision.