Matching schemes using the steepest-ascent/descent methods

A tool is presented for solving some types of pattern matching problems using the method of steepest-descent/ascent. These matching problems can be recast as function maximization problems with the functions to be maximized being constrained over the orthogonal group. The optimization of these functions is then performed using the steepest-ascent algorithm. This is made possible by representing the orthogonal group as a Lie group and then investigating the gradient vector field associated with the function to be maximized or minimized. Since the orthogonal group includes the group of permutations as a subgroup, the proposed procedure works not only for the continuous optimization problem, but also for the combinatorial problem. The conditions for the convergence of the steepest-ascent algorithms are also shown and simulations are performed