Exactly or Approximately Wasserstein Distributionally Robust Estimation According to Wasserstein Radii Being Small or Large
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912625207541760 |
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| author | Ding, Xiao Song, Enbin Niu, Dunbiao Cao, Zhujun Shi, Qingjiang |
| author_facet | Ding, Xiao Song, Enbin Niu, Dunbiao Cao, Zhujun Shi, Qingjiang |
| contents | This paper primarily considers the robust estimation problem under Wasserstein distance constraints on the parameter and noise distributions in the linear measurement model with additive noise, which can be formulated as an infinite-dimensional nonconvex minimax problem. We prove that the existence of a saddle point for this problem is equivalent to that for a finite-dimensional minimax problem, and give a counterexample demonstrating that the saddle point may not exist. Motivated by this observation, we present a verifiable necessary and sufficient condition whose parameters can be derived from a convex problem and its dual. Additionally, we also introduce a simplified sufficient condition, which intuitively indicates that when the Wasserstein radii are small enough, the saddle point always exists. In the absence of the saddle point, we solve an finite-dimensional nonconvex minimax problem, obtained by restricting the estimator to be linear. Its optimal value establishes an upper bound on the robust estimation problem, while its optimal solution yields a robust linear estimator. Numerical experiments are also provided to validate our theoretical results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_01763 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exactly or Approximately Wasserstein Distributionally Robust Estimation According to Wasserstein Radii Being Small or Large Ding, Xiao Song, Enbin Niu, Dunbiao Cao, Zhujun Shi, Qingjiang Signal Processing Optimization and Control Statistics Theory This paper primarily considers the robust estimation problem under Wasserstein distance constraints on the parameter and noise distributions in the linear measurement model with additive noise, which can be formulated as an infinite-dimensional nonconvex minimax problem. We prove that the existence of a saddle point for this problem is equivalent to that for a finite-dimensional minimax problem, and give a counterexample demonstrating that the saddle point may not exist. Motivated by this observation, we present a verifiable necessary and sufficient condition whose parameters can be derived from a convex problem and its dual. Additionally, we also introduce a simplified sufficient condition, which intuitively indicates that when the Wasserstein radii are small enough, the saddle point always exists. In the absence of the saddle point, we solve an finite-dimensional nonconvex minimax problem, obtained by restricting the estimator to be linear. Its optimal value establishes an upper bound on the robust estimation problem, while its optimal solution yields a robust linear estimator. Numerical experiments are also provided to validate our theoretical results. |
| title | Exactly or Approximately Wasserstein Distributionally Robust Estimation According to Wasserstein Radii Being Small or Large |
| topic | Signal Processing Optimization and Control Statistics Theory |
| url | https://arxiv.org/abs/2510.01763 |