Projection-Based Partial Correlation Inference Beyond Sparsity

Valid inference for partial correlation in high-dimensional Gaussian models is challenging when the conditioning variables exhibit dense dependence and conventional sparsity assumptions are inappropriate. We develop a target-specific projection framework for estimating and testing a population partial correlation without requiring coordinate sparsity. A population quadratic criterion characterizes the information retained by a linear reduction and is maximized by a one-dimensional projection. Under a rank-one alignment condition, this optimal projection preserves the original partial correlation. To obtain a feasible procedure, we estimate the optimal direction using diagonal ridge regularization on an independent training sample. Under conditions on regularized effective dimension and target-specific ridge bias, we establish a quantitative direction-estimation rate in the covariance-induced prediction norm. We further show that, under alignment, the induced partial-correlation error is second order in the direction error. These results yield oracle equivalence and asymptotic normality for the feasible split-sample estimator, as well as asymptotically valid testing of the original zero-partial-correlation null. Exact finite-sample \(F\)-calibration holds under the projected null. An explicit coordinate-dense Gaussian model verifies the required rates even when the dimension exceeds the sample size. Numerical experiments illustrate the finite-sample performance of the proposed procedures.