Large Covariance Matrix Estimation by Penalized Log‐Det Heuristics

We study covariance estimation in high‐dimensional approximate factor models with a sparse residual covariance. The estimator combines a log‐det loss with nuclear‐norm and off‐diagonal penalties on a sample‐normalized covariance scale. Under light‐tailed sampling, pervasive factors and factor‐space incoherence, we establish local curvature and characterize a local fixed point of the associated threshold iteration. Its low‐rank component consistently recovers the latent rank; exact signed residual‐support recovery additionally requires a minimum‐signal condition. We derive componentwise estimation rates and sufficient conditions for positive definiteness. The analysis separates the spectral requirements for rank recovery from the entrywise requirements for residual‐support recovery. A boundary‐adaptive search identifies the two thresholds by extending an initial grid when its selected solution lies on a boundary. Simulations examine structural recovery and covariance accuracy, and compare log‐det and Frobenius losses with covariance shrinkage methods. An application to European bank supervisory indicators illustrates the resulting decomposition into latent factors and sparse residual relationships.