There's not really anything about direct injection that inherently makes more power. In Mazda's case, it allows them to run an Atkinson cycle without having to manage fuel in the intake tract. In cases where it's used with forced induction, it helps cut pre-ignition.
It used to be that you set up an engine to run at stoichiometric in the interest of the catalytic converter being able to do its job, and to continue to do it for years. Direct injection (theoretically) gave engine designers an alternate method to cut NOx emissions by getting lower peak temperatures in the combustion chamber. The problem is that cats still don't like dealing with an excess of oxygen flowing through them.
It used to be that you set up an engine to run at stoichiometric in the interest of the catalytic converter being able to do its job, and to continue to do it for years. Direct injection (theoretically) gave engine designers an alternate method to cut NOx emissions by getting lower peak temperatures in the combustion chamber. The problem is that cats still don't like dealing with an excess of oxygen flowing through them.