Reduced-Rank Estimation for Ill-Conditioned Stochastic Linear Model with High Signal-to-Noise Ratio
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Piotrowski, Tomasz Yamada, Isao |
| author_facet | Piotrowski, Tomasz Yamada, Isao |
| contents | Reduced-rank approach has been used for decades in robust linear estimation of both deterministic and random vector of parameters in linear model y=Hx+\sqrt{epsilon}n. In practical settings, estimation is frequently performed under incomplete or inexact model knowledge, which in the stochastic case significantly increases mean-square-error (MSE) of an estimate obtained by the linear minimum mean-square-error (MMSE) estimator, which is MSE-optimal among linear estimators in the theoretical case of perfect model knowledge. However, the improved performance of reduced-rank estimators over MMSE estimator in estimation under incomplete or inexact model knowledge has been established to date only by means of numerical simulations and arguments indicating that the reduced-rank approach may provide improved performance over MMSE estimator in certain settings. In this paper we focus on the high signal-to-noise ratio (SNR) case, which has not been previously considered as a natural area of application of reduced-rank estimators. We first show explicit sufficient conditions under which familiar reduced-rank MMSE and truncated SVD estimators achieve lower MSE than MMSE estimator if singular values of array response matrix H are perturbed. We then extend these results to the case of a generic perturbation of array response matrix H, and demonstrate why MMSE estimator frequently attains higher MSE than reduced-rank MMSE and truncated SVD estimators if H is ill-conditioned. The main results of this paper are verified in numerical simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01117 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reduced-Rank Estimation for Ill-Conditioned Stochastic Linear Model with High Signal-to-Noise Ratio Piotrowski, Tomasz Yamada, Isao Optimization and Control Methodology Reduced-rank approach has been used for decades in robust linear estimation of both deterministic and random vector of parameters in linear model y=Hx+\sqrt{epsilon}n. In practical settings, estimation is frequently performed under incomplete or inexact model knowledge, which in the stochastic case significantly increases mean-square-error (MSE) of an estimate obtained by the linear minimum mean-square-error (MMSE) estimator, which is MSE-optimal among linear estimators in the theoretical case of perfect model knowledge. However, the improved performance of reduced-rank estimators over MMSE estimator in estimation under incomplete or inexact model knowledge has been established to date only by means of numerical simulations and arguments indicating that the reduced-rank approach may provide improved performance over MMSE estimator in certain settings. In this paper we focus on the high signal-to-noise ratio (SNR) case, which has not been previously considered as a natural area of application of reduced-rank estimators. We first show explicit sufficient conditions under which familiar reduced-rank MMSE and truncated SVD estimators achieve lower MSE than MMSE estimator if singular values of array response matrix H are perturbed. We then extend these results to the case of a generic perturbation of array response matrix H, and demonstrate why MMSE estimator frequently attains higher MSE than reduced-rank MMSE and truncated SVD estimators if H is ill-conditioned. The main results of this paper are verified in numerical simulations. |
| title | Reduced-Rank Estimation for Ill-Conditioned Stochastic Linear Model with High Signal-to-Noise Ratio |
| topic | Optimization and Control Methodology |
| url | https://arxiv.org/abs/2408.01117 |