  We propose a general theory for studying the \xl{landscape} of nonconvex
\xl{optimization} with underlying symmetric structures \tz{for a class of
machine learning problems (e.g., low-rank matrix factorization, phase
retrieval, and deep linear neural networks)}. In specific, we characterize the
locations of stationary points and the null space of Hessian matrices \xl{of
the objective function} via the lens of invariant groups\removed{for associated
optimization problems, including low-rank matrix factorization, phase
retrieval, and deep linear neural networks}. As a major motivating example, we
apply the proposed general theory to characterize the global \xl{landscape} of
the \xl{nonconvex optimization in} low-rank matrix factorization problem. In
particular, we illustrate how the rotational symmetry group gives rise to
infinitely many nonisolated strict saddle points and equivalent global minima
of the objective function. By explicitly identifying all stationary points, we
divide the entire parameter space into three regions: ($\cR_1$) the region
containing the neighborhoods of all strict saddle points, where the objective
has negative curvatures; ($\cR_2$) the region containing neighborhoods of all
global minima, where the objective enjoys strong convexity along certain
directions; and ($\cR_3$) the complement of the above regions, where the
gradient has sufficiently large magnitudes. We further extend our result to the
matrix sensing problem. Such global landscape implies strong global convergence
guarantees for popular iterative algorithms with arbitrary initial solutions.
