51 research outputs found
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 . Instead of observing a subset
of the noisy continuous-valued entries of a matrix , we observe a subset
of noisy 1-bit (or binary) measurements generated according to a probabilistic
model. We consider constrained maximum likelihood estimation of , under a
constraint on the entry-wise infinity-norm of 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 . 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 is
comparable to or exceeds the sample size , a successful approach uses an
-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 asymptotically
dominates , with being the sparsity level (number of
non-zero coefficients). Earlier work obtains provable guarantees only for much
larger sample size, namely it requires to asymptotically dominate .
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
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