Solving Systems of Quadratic Equations via Exponential-type Gradient Descent Algorithm
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We consider the rank minimization problem from quadratic measurements, i.e., recovering a rank r matrix X ∈R^n × r from m scalar measurements y_i=a_i^ XX^ a_i, a_i∈R^n, i=1,...,m. Such problem arises in a variety of applications such as quadratic regression and quantum state tomography. We present a novel algorithm, which is termed exponential-type gradient descent algorithm, to minimize a non-convex objective function f(U)=1/4m∑_i=1^m(y_i-a_i^ UU^ a_i)^2. This algorithm starts with a careful initialization, and then refines this initial guess by iteratively applying exponential-type gradient descent. Particularly, we can obtain a good initial guess of X as long as the number of Gaussian random measurements is O(nr), and our iteration algorithm can converge linearly to the true X (up to an orthogonal matrix) with m=O(nr (cr)) Gaussian random measurements.
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