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In high-dimensional statistical inference in which the number of parametersto be estimated is larger than that of the holding data, regularized linearestimation techniques are widely used. These techniques have, however, somedrawbacks. First, estimators are biased in the sense that their absolute valuesare shrunk toward zero because of the regularization effect. Second, theirstatistical properties are difficult to characterize as they are given asnumerical solutions to certain optimization problems. In this manuscript, wetackle such problems concerning LASSO, which is a widely used method for sparselinear estimation, when the measurement matrix is regarded as a sample from arotationally invariant ensemble. We develop a new computationally feasiblescheme to construct a de-biased estimator with a confidence interval andconduct hypothesis testing for the null hypothesis that a certain parametervanishes. It is numerically confirmed that the proposed method successfullyde-biases the LASSO estimator and constructs confidence intervals and p-valuesby experiments for noisy linear measurements.

A statistical mechanics approach to de-biasing and uncertainty estimation in LASSO for random measurements