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... AltMinSense converge to X? linearly under a RIP condition. However, the least squares problems are often ill-conditioned, it is difficult to observe AltMinSense converging to X? in practice. As described above, considerable progress has been made on algorithms for rank minimization and certain semidefinite programming problems. Yet truly efficient, scalable and provably convergent 3 algorithms have not yet been obtained. In the specific setting that X? is positive semidefinite, our algorithm exploits this structure to achieve these goals. We note that recent and independent work of Tu et al. [21] proposes a hybrid algorithm called Procrustes Flow (PF), which uses a few iterations of SVP as initialization, and then applies gradient descent. 4 A Gradient Descent Algorithm for Rank Minimization Our method is described in Algorithm 1. It is parallel to the Wirtinger Flow (WF) algorithm for phase retrieval [6], to recover a complex vector x ∈ Cn given the squared magnitudes of its linear measurements bi = |〈ai, x〉|2, i ∈ [m], where a1, . . . , am ∈ Cn. Candes et al. [6] propose a first-order method to minimize the sum of squared residuals fWF(z) = n∑ i=1 ( |〈ai, z〉|2 − bi )2 . (5) The aut...

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... d. Work in [21] considers the deterministic corruption model, and improves this running time without sacrificing the robustness guarantee on α. They propose an alternating projection (AltProj) method to estimate the low rank and sparse structures iteratively and simultaneously, and show their algorithm has complexity O(r2d2 log(1/ε)), which is faster than the convex approach but still slower than SVD. Non-convex approaches have recently seen numerous developments for applications in low-rank estimation, including alternating minimization (see e.g. [19, 17, 16]) and gradient descent (see e.g. [4, 12, 23, 24, 29, 30]). These works have fast running times, yet do not provide robustness guarantees. One exception is [12], where the authors analyze a row-wise `1 projection method for recovering S∗. Their analysis hinges on positive semidefinite M∗, and the algorithm requires prior knowledge of the `1 norm of every row of S∗ and is thus prohibitive in practice. Another exception is work [16], which analyzes alternating minimization plus an overall sparse projection. Their algorithm is shown to tolerate at most a fraction of α = O(1/(µ2/3r2/3d)) corruptions. As we discuss in Section 1.2, we can allow S∗ to have...

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...rieval problem has a natural generalization to recovering low-rank positive semidefinite matrices. Consider the problem of recovering an unknown rank-r matrix M 0 in Rn×n from linear measurement of the form zk = tr(AkM) with symmetricAk for k = 1, . . . ,m. One can solve the problem by considering the “factorized” version: recoveringX ∈ Rn×r (up to right invertible transform) from measurements zk = tr(X ∗AkX). This is a natural generalization of GPR, as one can write the GPR measurements as y2k = |a∗kx| 2 = x∗(aka ∗ k)x. This generalization and related problems have recently been studied in [SRO15, ZL15, TBSR15, CW15]. 1.5 Notations, Organization, and Reproducible Research Basic notations and facts. Throughout the paper, we define complex inner product as: 〈a, b〉 .= a∗b for any a, b ∈ Cn. We use CSn−1 for the complex unit sphere in Cn. CSn−1(λ) with λ > 0 denotes the centered complex sphere with radius λ in Cn. Similarly, we use CBn(λ) to denote the centered complex ball of radius λ. We use CN (k) for a standard complex Gaussian vector of length k defined in (1.2). We reserve C and c, and their indexed versions to denote absolute constants. Their value vary with the context. Let < (z) ∈ Rn and =(z) ∈ Rn de...