The forward and backward probabilities , p0j and pkn , can be computed using single-source algebraic path for the simpler semiring ( R , + , x , â ) -- or equivalently , by solving a sparse linear system of equations over R , a much-studied problem at O ( n ) space , O ( nm ) time , and faster approximations ( Greenbaum , 1997 ) .
iterative methods for solving linear systems
Anne Greenbaum
Write wjk as ( pjk , vjk ) , and let w1jk = ( p1jk , v1 jk ) denote the weight of the edge from j to k. 19 Then it can be shown that w0n = ( p0n , Ej , k p0jv1jkpkn ) . The forward and backward probabilities , p0j and pkn , can be computed using single-source algebraic path for the simpler semiring ( R , + , x , â ) -- or equivalently , by solving a sparse linear system of equations over R , a much-studied problem at O ( n ) space , O ( nm ) time , and faster approximations ( Greenbaum , 1997 ) . â¢ A Viterbi variant of the expectation semiring exists : replace ( 3 ) with if ( p1 > p2 , ( p1 , v1 ) , ( p2 , v2 ) ) .
The forward and backward probabilities , p0j and pkn , can be computed using single-source algebraic path for the simpler semiring ( R , + , x , â ) -- or equivalently , by solving a sparse linear system of equations over R , a much-studied problem at O ( n ) space , O ( nm ) time , and faster approximations ( @@CITATION ) .