Spatial covariance estimation (shrinkage)

Every Riemannian feature is only as good as the covariance estimate it starts from, and the sample covariance matrix is a poor estimator when the number of channels approaches or exceeds the number of samples: its small eigenvalues are systematically underestimated (pushed toward zero), making the matrix ill-conditioned or singular and destroying SPD geometry. Shrinkage estimators (Ledoit-Wolf, OAS) fix this by convex-combining the sample covariance with a scaled identity target, (1-lambda) S + lambda (trace(S)/C) I, with an analytically chosen lambda that minimizes expected error.

Shrinkage guarantees a well-conditioned, strictly positive-definite matrix, a prerequisite for matrix logarithms, geodesic distances, and stable Karcher means, and it often improves downstream decoding more than the choice of Riemannian classifier does. Robust and block or Toeplitz-structured estimators are used when artifacts or known spatial structure warrant them.