Adaptive confidence sets for matrix completion

Carpentier, Alexandra; Klopp, Olga; Löffler, Matthias; Nickl, Richard

doi:10.3150/17-bej933

Cited by 17 publications

(20 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The opening question is to choose the parameters of interest for the investigation. In [11] and [13], the authors proposed confidence regions of the matrix M with respect to the matrix Frobenius norm. In [5] and [12], the confidence intervals for M 's entries are established.…”

Section: Background and Motivationmentioning

confidence: 99%

Confidence Region of Singular Subspaces for Low-Rank Matrix Regression

Dong

2019

IEEE Trans. Inform. Theory

View full text Add to dashboard Cite

Low-rank matrix regression refers to the instances of recovering a low-rank matrix based on specially designed measurements and the corresponding noisy outcomes. In the last decade, numerous statistical methodologies have been developed for efficiently recovering the unknown low-rank matrices. However, in some applications, the unknown singular subspace is scientifically more important than the low-rank matrix itself. In this article, we revisit the low-rank matrix regression model and introduce a two-step procedure to construct confidence regions of the singular subspace. The procedure involves the de-biasing for the typical low-rank estimators after which we calculate the empirical singular vectors. We investigate the distribution of the joint projection distance between the empirical singular subspace and the unknown true singular subspace. We specifically prove the asymptotical normality of the joint projection distance with data-dependent centering and normalization when r 3/2 (m 1 +m 2 ) 3/2 = o(n/ log n) where m 1 , m 2 denote the matrix row and column sizes, r is the rank and n is the number of independent random measurements. Consequently, we propose data-dependent confidence regions of the true singular subspace which attains any pre-determined confidence level asymptotically. In addition, non-asymptotical convergence rates are also established. Numerical results are presented to demonstrate the merits of our methods.

show abstract

Section: Background and Motivationmentioning

confidence: 99%

Confidence Region of Singular Subspaces for Low-Rank Matrix Regression

Dong

2019

IEEE Trans. Inform. Theory

View full text Add to dashboard Cite

show abstract

“…For example Davenport et al (2014) study the problem of one-bit matrix completion, where the response is binary and the entrywise loss is logistic, with an additional ∞ -norm on the entries of the matrix, and Lafond (2015) study prediction error bounds for matrix completion for exponential family models with a nuclear norm penalty. Carpentier et al (2016) discuss confidence sets for the low-rank matrix completion problem and Klopp et al (2015) consider a multinomial matrix completion problem where the observed entries are quantized with a few levels (in their framework the missingness need not be uniform). They study a regularized negative log-likelihood problem, where the latent variables are regularized by a nuclear norm penalty and an additional constraint on the maximal absolute entries of the matrix.…”

Section: Nuclear Norm Regularizationmentioning

confidence: 99%

Flexible Low-Rank Statistical Modeling with Missing Data and Side Information

Fithian¹,

Mazumder²

2018

Statist. Sci.

View full text Add to dashboard Cite

We propose a general framework for reduced-rank modeling of matrix-valued data. By applying a generalized nuclear norm penalty we can directly model low-dimensional latent variables associated with rows and columns. Our framework flexibly incorporates row and column features, smoothing kernels, and other sources of side information by penalizing deviations from the row and column models. Moreover, a large class of these models can be estimated scalably using convex optimization. The computational bottleneck in each case is one singular value decomposition per iteration of a large but easy-to-apply matrix. Our framework generalizes traditional convex matrix completion and multi-task learning methods as well as maximum a posteriori estimation under a large class of popular hierarchical Bayesian models.

show abstract

“…It is an important issue in applications to be able to say from the observations how well the recovery procedure has worked or, in the sequential sampling setting, to be able to give data-driven stopping rules that guarantee the recovery of the matrix M 0 at a given precision. This fundamental statistical question was recently studied in [7] where two statistical models for matrix completion are considered: the trace regression model and the Bernoulli model (for details see Section 1.1). In particular, in [7], the authors show that in the case of unknown noise variance, the information-theoretic structure of these two models is fundamentally different.…”

Section: Introductionmentioning

confidence: 99%

“…This fundamental statistical question was recently studied in [7] where two statistical models for matrix completion are considered: the trace regression model and the Bernoulli model (for details see Section 1.1). In particular, in [7], the authors show that in the case of unknown noise variance, the information-theoretic structure of these two models is fundamentally different.…”

Section: Introductionmentioning

confidence: 99%