  The landscape of empirical risk has been widely studied in a series of
machine learning problems, including low-rank matrix factorization, matrix
sensing, matrix completion, and phase retrieval. In this work, we focus on the
situation where the corresponding population risk is a degenerate non-convex
loss function, namely, the Hessian of the population risk can have zero
eigenvalues. Instead of analyzing the non-convex empirical risk directly, we
first study the landscape of the corresponding population risk, which is
usually easier to characterize, and then build a connection between the
landscape of the empirical risk and its population risk. In particular, we
establish a correspondence between the critical points of the empirical risk
and its population risk without the strongly Morse assumption, which is
required in existing literature but not satisfied in degenerate scenarios. We
also apply the theory to matrix sensing and phase retrieval to demonstrate how
to infer the landscape of empirical risk from that of the corresponding
population risk.
