Восстановление сигналов, полученных косвенными измерениями, на основе усеченного сингулярного разложения и случайного проецирования

Abstract

Рассмотрены методы устойчивого решения дискретных некорректных задач с применением случайного проецирования, усеченного сингулярного разложения, регуляризации Тихонова, дан их сравнительный анализ и результаты экспериментального исследования.

Розглянуто методи стійкого рішення дискретних некоректних задач із застосуванням випадкового проектування, усіченого сингулярного розкладання, регуляризації Тихонова, наведено їх порівняльний аналіз та результати експериментального дослідження.

Introduction. The solution of the ill-posed inverse problem by the least squares method is unstable with a large solution error. Tikhonov regularization, truncated singular value decomposition, and random projection were used to overcome the instability and to increase the accuracy of the solution. Purpose. We provide an experimental comparison of the solution accuracy for the ill-posed inverse problem by Tikhonov regularization, truncated singular value decomposition, and random projection. Methods. Tikhonov's regularization imposes some restrictions on the solution, i.e. penalty on its Euclidean norm, that improves stability. Another approach approximates the original data by a model linear with respect to parameters. Selection of the optimal number of components of the linear model minimizes the error of solution and ensures stability. To obtain the optimal number of model components, model selection criteria are used. Results and Conclusion. A comparative analysis of the accuracy shows that the truncated singular value decomposition method with the CRSVD criterion and the random projection method with the CRQ and AIC criteria ensured the accuracy at the level of Tikhonov regularization with the regularization parameter selected by the discrepancy method. The advantage of the random projection method is a lower computational complexity due to the dimensionality reduction. Perspective. The directions for further research include the decreasing of the computational complexity and averaging over the realizations of the random matrix.