Solving Quadratic Systems with Full-Rank Matrices Using Sparse or Generative Priors
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The problem of recovering a signal x∈ℝ^n from a quadratic system {y_i=x^⊤A_ix,i=1,…,m} with full-rank matrices A_i frequently arises in applications such as unassigned distance geometry and sub-wavelength imaging. With i.i.d. standard Gaussian matrices A_i, this paper addresses the high-dimensional case where m≪ n by incorporating prior knowledge of x. First, we consider a k-sparse x and introduce the thresholded Wirtinger flow (TWF) algorithm that does not require the sparsity level k. TWF comprises two steps: the spectral initialization that identifies a point sufficiently close to x (up to a sign flip) when m=O(k^2log n), and the thresholded gradient descent (with a good initialization) that produces a sequence linearly converging to x with m=O(klog n) measurements. Second, we explore the generative prior, assuming that x lies in the range of an L-Lipschitz continuous generative model with k-dimensional inputs in an ℓ_2-ball of radius r. We develop the projected gradient descent (PGD) algorithm that also comprises two steps: the projected power method that provides an initial vector with O(√(k log L/m)) ℓ_2-error given m=O(klog(Lnr)) measurements, and the projected gradient descent that refines the ℓ_2-error to O(δ) at a geometric rate when m=O(klogLrn/δ^2). Experimental results corroborate our theoretical findings and show that: (i) our approach for the sparse case notably outperforms the existing provable algorithm sparse power factorization; (ii) leveraging the generative prior allows for precise image recovery in the MNIST dataset from a small number of quadratic measurements.
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