1-Bit Matrix Completion under Exact Low-Rank Constraint

We consider the problem of noisy 1-bit matrix completion under an exact rank constraint on the true underlying matrix $M^*$. Instead of observing a subset of the noisy continuous-valued entries of a matrix $M^*$, we observe a subset of noisy 1-bit (or binary) measurements generated according to a probabilistic model. We consider constrained maximum likelihood estimation of $M^*$, under a constraint on the entry-wise infinity-norm of $M^*$ and an exact rank constraint. This is in contrast to previous work which has used convex relaxations for the rank. We provide an upper bound on the matrix estimation error under this model. Compared to the existing results, our bound has faster convergence rate with matrix dimensions when the fraction of revealed 1-bit observations is fixed, independent of the matrix dimensions. We also propose an iterative algorithm for solving our nonconvex optimization with a certificate of global optimality of the limiting point. This algorithm is based on low rank factorization of $M^*$. We validate the method on synthetic and real data with improved performance over existing methods.Comment: 6 pages, 3 figures, to appear in CISS 201

Nearly Optimal Sample Size in Hypothesis Testing for High-Dimensional Regression

We consider the problem of fitting the parameters of a high-dimensional linear regression model. In the regime where the number of parameters $p$ is comparable to or exceeds the sample size $n$, a successful approach uses an $\ell_1$-penalized least squares estimator, known as Lasso. Unfortunately, unlike for linear estimators (e.g., ordinary least squares), no well-established method exists to compute confidence intervals or p-values on the basis of the Lasso estimator. Very recently, a line of work \cite{javanmard2013hypothesis, confidenceJM, GBR-hypothesis} has addressed this problem by constructing a debiased version of the Lasso estimator. In this paper, we study this approach for random design model, under the assumption that a good estimator exists for the precision matrix of the design. Our analysis improves over the state of the art in that it establishes nearly optimal \emph{average} testing power if the sample size $n$ asymptotically dominates $s_0 (\log p)^2$, with $s_0$ being the sparsity level (number of non-zero coefficients). Earlier work obtains provable guarantees only for much larger sample size, namely it requires $n$ to asymptotically dominate $(s_0 \log p)^2$. In particular, for random designs with a sparse precision matrix we show that an estimator thereof having the required properties can be computed efficiently. Finally, we evaluate this approach on synthetic data and compare it with earlier proposals.Comment: 21 pages, short version appears in Annual Allerton Conference on Communication, Control and Computing, 201