Matrix permanent inequalities for approximating joint assignment matrices in tracking systems

This paper examines some of the combinatorial issues associated with the batch data association problem arising in tracking and correlation applications. A procedure is developed that addresses a large class of data association problems involving the calculation of permanents of submatrices of the original association matrix. This procedure yields what is termed the Joint Assignment Matrix (JAM), which can be used to optimally rank associations for hypothesis selection. Because the computational cost of the permanent scales exponentially with the size of the matrix, improved algorithms are developed both for calculating the exact JAM and for generating approximations to it. Empirical results suggest that at least one of the approximations is suitable for real-time hypothesis generation in large scale tracking and correlation applications. Novel theoretical results include an improved upper bound on the calculation of the JAM and new upper bound inequalities for the permanent of general nonnegative matrices. One of these inequalities is an improvement over the best previously known inequality.