Multiple Testing of Linear Forms for Noisy Matrix Completion

See, e.g., Resnick and Varian (1997); Schafer et al. (2007); Koren et al. (2009); Davidson et al. (2010); McAuley and Leskovec (2013); Das et al. (2017). Consider, more specifically, representing the ratings of d 1 users on d 2 products/items by a d 1 d 2 matrix. For all practical purposes, both d 1 and d 2 can be very large yet only a rather small number of the entries can be observed. The idea is that if the interaction between users and products can be approximately captured by a handful of latent user-specific and product-specific characteristics, then it is possible to infer the whole user-item rating matrix from these sparsely observed entries, and hence recommend products to users who may be genuinely interested in them. Since the pioneering works of Cand` es and Tao (2009); Candes and Plan (2010); Candes and Recht (2012), a lot of impressive progress has been made to make these techniques more accurate and scalable, and to better understand the statistical and computational underpinnings of the problem.