The possibilities include excess wear of front-end components such as ball joints and tie rod ends. This would be a significant safety hazard.
It is also possible that your tires may need a more effective type of balancing, known as Road Force Balancing. Only a few tire dealers are likely to have the Hunter GPS9700 machine that is needed for this job.
I would suggest that you call a few tire dealers and perhaps even a couple of Mazda dealers to see if they have this type of equipment. If Road Force Balancing does not resolve the issue, then you need to have a qualified front-end shop check for excess wear of front-end components.
I should also mention that, if the shaking only takes place when braking, then the probable reason is warped brake rotors. If braking has nothing to do with the situation, then you should focus on the previously-mentioned items.