I am planning to do my PhD in machine learning. From my personal experience, I found that mathematics is a primary way to make a good research, but I'm doubting about the depth of mathematics required. Most of the impressive inventions in most of the computer science domains comes from mathematics only. But if I spend most of my time in learning mathematics, then I may end up with wasting my time. So, how much depth of mathematics is recommended to carry my PhD? Shall I study only books on machine learning or core mathematics also?
Here is a somewhat different bit of advice. If you are about to apply, and your background is strong enough to get accepted and find a suitable advisor, then just do that now. Let the rest follow.
There is a lot of mathematics that can be applied to machine learning but until you have a specific problem it is hard to say which is best.
But, mathematics is a way of thinking as much as, and perhaps more than, any given topic. So any mathematics that you have time to study now, and for which you have any lack, or any hunger, will probably aid you in that way of thinking.
There are a lot of ways to think, math is just one. For someone in machine learning or any aspect of AI, having lots of ways to approach a problem is a big win.
But get yourself connected first and then look at the specifics of what you need to do.