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...g from the same neighborhood of U0 with only Ω(nr) measurements. Moreover, the theory of restricted isometries in our work considerably simplifies the analysis. Finally, we would also like to mention =-=[SOR14]-=- for guarantees using stochastic gradient algorithms. The results of [SOR14] are applicable to a variety of models; focusing on the Gaussian ensemble, the authors require Ω ((nr log n)/) samples to r...

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...completion: Another variant of online matrix completion studied in the literature is where observations are made on a column by column basis e.g., [16, 26]. These models can give improved offline performance in terms of space and could potentially work under relaxed regularity conditions. However, they do not tackle the version where only entries (as opposed to columns) are observed. Non-convex optimization: Over the last few years, there has also been a significant amount of work in designing other efficient algorithms for solving non-convex problems. Examples include eigenvector computation [6, 11], sparse coding [20, 1] etc. For general non-convex optimization, an interesting line of recent work is that of [7], which proves gradient descent with noise can also escape saddle point, but they only provide polynomial rate without explicit dependence. Later [17, 21] show that without noise, the space of points from where gradient descent converges to a saddle point is a measure zero set. However, they do not provide a rate of convergence. Another related piece of work to ours is [10], proves global convergence along with rates of convergence, for the special case of computing matrix squarer...

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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...