High Dimensional Multivariate Regression and Precision Matrix Estimation via Nonconvex Optimization
This addresses a bottleneck in high-dimensional statistical inference for researchers and practitioners, offering a computationally efficient method with strong theoretical guarantees, though it is incremental in building on existing nonconvex optimization approaches.
The paper tackles the problem of joint multivariate regression and precision matrix estimation in high-dimensional settings with sparsity constraints, proposing a nonconvex estimator solved via gradient descent with hard thresholding that achieves linear convergence and optimal statistical rates up to a logarithmic factor.
We propose a nonconvex estimator for joint multivariate regression and precision matrix estimation in the high dimensional regime, under sparsity constraints. A gradient descent algorithm with hard thresholding is developed to solve the nonconvex estimator, and it attains a linear rate of convergence to the true regression coefficients and precision matrix simultaneously, up to the statistical error. Compared with existing methods along this line of research, which have little theoretical guarantee, the proposed algorithm not only is computationally much more efficient with provable convergence guarantee, but also attains the optimal finite sample statistical rate up to a logarithmic factor. Thorough experiments on both synthetic and real datasets back up our theory.