Estimation of projection operators with Gaussian noise

TL;DR

This paper derives non-asymptotic error bounds for estimated projection operators under Gaussian noise, considering various estimator distributions and introducing regularized methods.

Contribution

It provides new non-asymptotic bounds for projection estimation errors, including regularized estimators, in noisy settings with applications to PLS.

Findings

Derived non-asymptotic upper bounds on projection error.

Bounds depend on noise level and subspace properties.

Regularized estimators improve estimation under structural assumptions.

Abstract

This paper focuses on random projection operators when the subspace of projection is estimated. We derive non-asymptotic upper bounds on the error between the projection onto the estimated subspace and the projection onto the underlying subspace. The provided upper bounds depend on the noise and on intrinsic properties of the estimated subspace. Several scenarios are considered according to the distribution of the estimator of the matrix spanning the subspace. The aforementioned bounds are attained under a structural assumption on the Gram matrix associated with the subspace. Regularized estimators are introduced to circumvent this assumption. An example is given in the partial least square (PLS) framework where the estimated subspace is spanned by the PLS weights.