Well-Conditioned Linear Minimum Mean Square Error Estimation

Linear minimum mean square error (LMMSE) estimation is often ill-conditioned, suggesting that unconstrained minimization of the an inadequate approach to filter design. To address this, we first develop a unifying framework for studying constrained LMMSE problems. Using this framework, explore important structural property filters involving certain prefilter. Optimality invariant under invertible linear transformations This parameterizes all optimal by equivalence classes prefilters. We then clarify merely constraining rank does not suitably problem ill-conditioning. Instead, adopt constraint explicitly requires solutions be well-conditioned in specific sense. introduce two and show they converge as their truncation-power loss goes zero, at same rate low-rank Wiener filter. also extensions case weighted trace determinant covariance objective functions. Finally, our quantitative results with historical VIX data demonstrate have stable performance while standard deteriorates increasing condition number.