A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations|Ax|^(2)=*** algorithms are developed by exploiting the inherent low rank structure of the problem b...
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A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations|Ax|^(2)=*** algorithms are developed by exploiting the inherent low rank structure of the problem based on the embedded manifold of rank-1 positive semidefinite *** recovery guarantee has been established for the truncated variant,showing that the algorithm is able to achieve successful recovery when the number of equations is proportional to the number of *** key ingredients in the analysis are the restricted well conditioned property and the restricted weak correlation property of the associated truncated linear *** evaluations show that our algorithms are competitive with other state-of-the-art first order nonconvex approaches with provable guarantees.
The problem of finding a vector boldsymbol {x} which obeys a set of quadratic equations | boldsymbol {a}_{text {k}}top boldsymbol {x} |{2}=text {y}_{text {k}} , text {k}=1,cdots,text {m} , plays an important role in m...
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The problem of finding a vector boldsymbol {x} which obeys a set of quadratic equations | boldsymbol {a}_{text {k}}<^>top boldsymbol {x} |<^>{2}=text {y}_{text {k}} , text {k}=1,cdots,text {m} , plays an important role in many applications. In this paper we consider the case when both boldsymbol {x} and boldsymbol {a}_{k} are real-valued vectors of length n. A new loss function is constructed for this problem, which combines the smooth quadratic loss function with an activation function. Under the Gaussian measurement model, we establish that with high probability the target solution boldsymbol {x} is the only minimizer (up to a global sign) of the new loss function provided text {m}gtrsim text {n} . Moreover, the loss function always has a negative directional curvature around its saddle points.
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