Three Cuts for Accelerated Interval Propagation

This paper addresses the problem of nonlinear multivariate root finding. In an earlier paper we described a system called Newton which finds roots of systems of nonlinear equations using refinements of interval methods. The refinements are inspired by AI constraint propagation techniques. Newton is competative with continuation methods on most benchmarks and can handle a variety of cases that are infeasible for continuation methods. This paper presents three "cuts" which we believe capture the essential theoretical ideas behind the success of Newton. This paper describes the cuts in a concise and abstract manner which, we believe, makes the theoretical content of our work more apparent. Any implementation will need to adopt some heuristic control mechanism. Heuristic control of the cuts is only briefly discussed here.

Belief Propagation and Revision in Networks with Loops

Local belief propagation rules of the sort proposed by Pearl(1988) are guaranteed to converge to the optimal beliefs for singly connected networks. Recently, a number of researchers have empirically demonstrated good performance of these same algorithms on networks with loops, but a theoretical understanding of this performance has yet to be achieved. Here we lay the foundation for an understanding of belief propagation in networks with loops. For networks with a single loop, we derive ananalytical relationship between the steady state beliefs in the loopy network and the true posterior probability. Using this relationship we show a category of networks for which the MAP estimate obtained by belief update and by belief revision can be proven to be optimal (although the beliefs will be incorrect). We show how nodes can use local information in the messages they receive in order to correct the steady state beliefs. Furthermore we prove that for all networks with a single loop, the MAP estimate obtained by belief revisionat convergence is guaranteed to give the globally optimal sequence of states. The result is independent of the length of the cycle and the size of the statespace. For networks with multiple loops, we introduce the concept of a "balanced network" and show simulati.

Uncertainty Propagation in Model-Based Recognition

Building robust recognition systems requires a careful understanding of the effects of error in sensed features. Error in these image features results in a region of uncertainty in the possible image location of each additional model feature. We present an accurate, analytic approximation for this uncertainty region when model poses are based on matching three image and model points, for both Gaussian and bounded error in the detection of image points, and for both scaled-orthographic and perspective projection models. This result applies to objects that are fully three- dimensional, where past results considered only two-dimensional objects. Further, we introduce a linear programming algorithm to compute the uncertainty region when poses are based on any number of initial matches. Finally, we use these results to extend, from two-dimensional to three- dimensional objects, robust implementations of alignmentt interpretation- tree search, and ransformation clustering.