Low-Rank Matrix Recovery with Composite Optimization: Good Conditioning and Rapid Convergence

A superlinearly convergent subgradient method for sharp semismooth problems

“…In particular, [10] shows that f satisfies (A1) when (i) i , r i are i.i.d. standard Gaussian vectors, (ii) m rd, and (iii) at most a small constant fraction of entries of ξ are nonzero.…”

“…Sharp growth is known to hold in a range of problems, most classically in feasibility formulations of linear programs [34] (see also the survey [57]). Several contemporary problems also exhibit sharp growth, for example, nonconvex formulations of low-rank matrix sensing and completion problems [10].…”

“…To close this section, we mention that the sharp growth property (A1) is well-studied in the settings of these propositions. For example, the setting of Proposition 2.1 arises in low-rank matrix estimation problems [10], where regularity property (A1) is a consequence of the restricted isometry property [9] of the "measurement operator." Next, for f defined in Proposition 2.2, regularity property (A1) is simply the classical metric subregularity assumption.…”

“…Tensorflow [1]) and in robust low-rank matrix estimation problems [10]. For the latter problem class, recent work has highlighted the prevalence and benefits of the so-called sharp growth property, which stipulates that f grows at least linearly away from its minimizers:…”

Convex and Nonconvex Optimization Are Both Minimax-Optimal for Noisy Blind Deconvolution Under Random Designs

“…Another line of works has focused on the development of fast nonconvex algorithms , Lee et al, 2018, Ma et al, 2018, Huang and Hand, 2018, Charisopoulos et al, 2019, which was largely motivated by recent advances in efficient nonconvex optimization for tackling statistical estimation problems [ Candes et al, 2015, Chen and Candès, 2017, Charisopoulos et al, 2021, Keshavan et al, 2009, Jain et al, 2013, Zhang et al, 2016, Chen and Wainwright, 2015, Sun and Luo, 2016, Zheng and Lafferty, 2016, Wang et al, 2017a, Cai et al, 2021b, Wang et al, 2017b, Qu et al, 2017, Duchi and Ruan, 2019, Ma et al, 2019 (see Chi et al [2019] for an overview). proposed a feasible nonconvex recipe by attempting to optimize a regularized squared loss (which includes extra penalty term to promote incoherence), and showed that in conjunction with proper initialization, nonconvex gradient descent converges to the ground truth in the absence of noise.…”