Let (t, d) be all the parameters of the HMM.
as a first step in getting a convex HMM, one could follow the path developed in (Simion et al., 2015a) and directly replace the HMM's objective terms the geometric mean function is convex ( (Boyd and Vandenberghe, 2004)) and, for a given sentence pair (e (k) , f (k) ) with alignment a (k) we can find a projection matrix P so that j=1 are exactly the parameters used in the term above (in particular, t, d are the set of all parameters whilet,d are the set of parameters for the specific training pair k; P projects from the full space onto only the parameters used for training pair k).