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Estimation of projection operators with Gaussian noise

Luca Castelli

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.

Estimation of projection operators with Gaussian noise

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.

Paper Structure

This paper contains 46 sections, 25 theorems, 230 equations, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Linear model and projection
  4. Estimation of Projection Operators
  5. Other related methods and questions
  6. General framework
  7. Main results
  8. Projection operator estimation
  9. Regularized versions
  10. The PLS case
  11. Conclusion
  12. Precise results
  13. Detailed results in each Scenario
  14. Detailed results with Ridge regularization
  15. Preliminaries
  16. ...and 31 more sections

Key Result

Theorem 4.1

Let $\delta\in(0,1)$. Consider Scenario 2 and assume that with $d_{\delta}$ with depending only on $\delta$. Then with probability higher than $1-\delta$, then exists a constant $D_{\delta}^{(1)}>0$, depending only on $\delta$, such that

Theorems & Definitions (34)

  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Corollary 4.5
  • Theorem A.1
  • Theorem A.2
  • Theorem A.3
  • Corollary A.4
  • Theorem A.5
  • ...and 24 more