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Well-Behaved Landscape of Phase Retrieval with Optimal Sampling Complexity

Abstract: The problem of finding a vector x which obeys a set of quadratic equations |<a_k,x>|^2=y_k

k = 1, · · · , m, plays an important role in many applications. In this talk, we construct a new loss function for this problem, which combines the smooth quadratic loss function with an activation function. Under the Gaussian measurement model of a_k, we establish that the proposed loss function has a well-behaved landscape of critical points. In particalar, with high probability, the target solution x is the unique local minimizer (up to a global phase factor) of the new loss function provided m=O(n), and the loss function always has a negative directional curvature around its saddle points. Therefore, any algorithm finding a local minimum will solve the phase retrieval problem with theoretical guarantee in the optimal sampling complexity.