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Radius of Information for Two Intersected Centered Hyperellipsoids and Implications in Optimal Recovery from Inaccurate Data

Simon Foucart, Chunyang Liao

Abstract

For objects belonging to a known model set and observed through a prescribed linear process, we aim at determining methods to recover linear quantities of these objects that are optimal from a worst-case perspective. Working in a Hilbert setting, we show that, if the model set is the intersection of two hyperellipsoids centered at the origin, then there is an optimal recovery method which is linear. It is specifically given by a constrained regularization procedure whose parameters, short of being explicit, can be precomputed by solving a semidefinite program. This general framework can be swiftly applied to several scenarios: the two-space problem, the problem of recovery from $\ell_2$-inaccurate data, and the problem of recovery from a mixture of accurate and $\ell_2$-inaccurate data. With more effort, it can also be applied to the problem of recovery from $\ell_1$-inaccurate data. For the latter, we reach the conclusion of existence of an optimal recovery method which is linear, again given by constrained regularization, under a computationally verifiable sufficient condition. Experimentally, this condition seems to hold whenever the level of $\ell_1$-inaccuracy is small enough. We also point out that, independently of the inaccuracy level, the minimal worst-case error of a linear recovery method can be found by semidefinite programming.

Radius of Information for Two Intersected Centered Hyperellipsoids and Implications in Optimal Recovery from Inaccurate Data

Abstract

For objects belonging to a known model set and observed through a prescribed linear process, we aim at determining methods to recover linear quantities of these objects that are optimal from a worst-case perspective. Working in a Hilbert setting, we show that, if the model set is the intersection of two hyperellipsoids centered at the origin, then there is an optimal recovery method which is linear. It is specifically given by a constrained regularization procedure whose parameters, short of being explicit, can be precomputed by solving a semidefinite program. This general framework can be swiftly applied to several scenarios: the two-space problem, the problem of recovery from -inaccurate data, and the problem of recovery from a mixture of accurate and -inaccurate data. With more effort, it can also be applied to the problem of recovery from -inaccurate data. For the latter, we reach the conclusion of existence of an optimal recovery method which is linear, again given by constrained regularization, under a computationally verifiable sufficient condition. Experimentally, this condition seems to hold whenever the level of -inaccuracy is small enough. We also point out that, independently of the inaccuracy level, the minimal worst-case error of a linear recovery method can be found by semidefinite programming.
Paper Structure (15 sections, 16 theorems, 97 equations)

This paper contains 15 sections, 16 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Solution for the two-hyperellipsoid-intersection model set
  3. Main result
  4. Justification
  5. Side results
  6. Three easy consequences
  7. The two-space problem
  8. Recovery from $\ell_2$-inaccurate data
  9. Recovery for mixed accurate and $\ell_2$-inaccurate data
  10. One intricate consequence: recovery from $\ell_1$-inaccurate data
  11. Main result
  12. Justification
  13. Side result
  14. Polyak's S-procedure.
  15. Constrained Regularization.

Key Result

Theorem 1

For the two-hyperellipsoid-intersection model set 2EllipsModSet, the square of the radius of information of the observation map $\Lambda : H \to \mathbb{R}^m$ for the estimation of $Q$ is given by the optimal value of the program Further, if $a_\sharp, b_\sharp \ge 0$ are minimizers of this program, then $Q \circ \Delta_{a_\sharp,b_\sharp}$ is an optimal recovery map. In short,

Theorems & Definitions (31)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • proof : Proof of Theorem \ref{['Thm2Space']}
  • Proposition 5
  • proof
  • ...and 21 more