It is my understanding that statistical machine learning uses functional analysis and probability theory

Amazon.com: High-Dimensional Statistics: A Non-Asymptotic Viewpoint (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 48): 9781108498029: Wainwright, Martin J.: Books

www.amazon.com

A computer science approach into machine learning with solid undergrad courses in algorithms, probability and stats is an efficient way into the field for masters level with an applied focus, unless one is seeking a purely "black box" approach following a Python, tensor flow and/or R tutorial.

eg. Introduction to Algorithms Cormen… and Statistical Inference George Casella, Roger L. Berger

see

CS 170
CMU is a leader in machine learning and seem to follow a balanced Bayesian and classical approach to statistics

All of Statistics: A Concise Course in Statistical Inference by Larry Wasserman

Probability and Statistics 4th Edition by Morris H. DeGroot, Mark J. Schervish

Stanford course requirements for 315a in statistical learning, as taught by the authors, requires graduate probability, statistics and regression theory, per their course catalog. On the other hand, Berkeley's intro to machine learning, taught in their computer science department uses ISL and ESL texts, but supplements it with a lot of course notes and articles, they also seems to present this material after a semester in AI.