  Developing efficient and guaranteed nonconvex algorithms has been an
important challenge in modern machine learning. Algorithms with good empirical
performance such as stochastic gradient descent often lack theoretical
guarantees. In this paper, we analyze the class of homotopy or continuation
methods for global optimization of nonconvex functions. These methods start
from an objective function that is efficient to optimize (e.g. convex), and
progressively modify it to obtain the required objective, and the solutions are
passed along the homotopy path. For the challenging problem of tensor PCA, we
prove global convergence of the homotopy method in the "high noise" regime. The
signal-to-noise requirement for our algorithm is tight in the sense that it
matches the recovery guarantee for the best degree-4 sum-of-squares algorithm.
In addition, we prove a phase transition along the homotopy path for tensor
PCA. This allows to simplify the homotopy method to a local search algorithm,
viz., tensor power iterations, with a specific initialization and a noise
injection procedure, while retaining the theoretical guarantees.
