  In high-dimensional statistical inference in which the number of parameters
to be estimated is larger than that of the holding data, regularized linear
estimation techniques are widely used. These techniques have, however, some
drawbacks. First, estimators are biased in the sense that their absolute values
are shrunk toward zero because of the regularization effect. Second, their
statistical properties are difficult to characterize as they are given as
numerical solutions to certain optimization problems. In this manuscript, we
tackle such problems concerning LASSO, which is a widely used method for sparse
linear estimation, when the measurement matrix is regarded as a sample from a
rotationally invariant ensemble. We develop a new computationally feasible
scheme to construct a de-biased estimator with a confidence interval and
conduct hypothesis testing for the null hypothesis that a certain parameter
vanishes. It is numerically confirmed that the proposed method successfully
de-biases the LASSO estimator and constructs confidence intervals and p-values
by experiments for noisy linear measurements.
