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<p>Our focus is on robust recovery algorithms in statistical linear inverse problem. We consider two recovery routines—the much-studied linear estimate originating from Kuks and Olman (Iswestija Akademija Nauk Estonskoj SSR 20:480–482, 1971) and polyhedral estimate introduced in Juditsky and Nemirovski (Electron J Stat 14(1):458–502, 2020). It was shown in Juditsky and Nemirovski (Statistical inference via convex optimization, Princeton University Press, Princeton, 2020) that risk of these estimates can be tightly upper-bounded for a wide range of a priori information about the model through solving a convex optimization problem, leading to a computationally efficient implementation of nearly optimal estimates of these types. The subject of the present paper is design and analysis of linear and polyhedral estimates which are robust with respect to the stochastic uncertainty in the observation matrix. In this setting, we show how to bound the estimation risk by the optimal value of an efficiently solvable convex optimization problem; “presumably good” estimates are then obtained through optimization of the risk bounds with respect to estimate parameters.</p>

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Estimation from Indirect Observations Under Stochastic Uncertainty in Observation Matrix

  • 期刊论文

  • 发表日期：

    2025年4月23日

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  • Yannis Bekri,
  • Anatoli Juditsky,
  • Arkadi Nemirovski

摘要

Our focus is on robust recovery algorithms in statistical linear inverse problem. We consider two recovery routines—the much-studied linear estimate originating from Kuks and Olman (Iswestija Akademija Nauk Estonskoj SSR 20:480–482, 1971) and polyhedral estimate introduced in Juditsky and Nemirovski (Electron J Stat 14(1):458–502, 2020). It was shown in Juditsky and Nemirovski (Statistical inference via convex optimization, Princeton University Press, Princeton, 2020) that risk of these estimates can be tightly upper-bounded for a wide range of a priori information about the model through solving a convex optimization problem, leading to a computationally efficient implementation of nearly optimal estimates of these types. The subject of the present paper is design and analysis of linear and polyhedral estimates which are robust with respect to the stochastic uncertainty in the observation matrix. In this setting, we show how to bound the estimation risk by the optimal value of an efficiently solvable convex optimization problem; “presumably good” estimates are then obtained through optimization of the risk bounds with respect to estimate parameters.