## Control Systems and Computers, N1, 2018, Article 2

Upr. sist. maš., 2018, Issue 1 (273), pp. 16-27.

UDC 004.942+6232.454.862

Revunova Elena G., PhD in Technical Sciences,  Deputy Director on Research, International Research and Training Center for Information Technologies and Systems of the NAS and MES of Ukraine, Glushkov ave., 40, Kyiv, 03187, Ukraine, E-mail: helab@i.com.ua

### Studying the Accuracy for the Solution of Discrete Ill-Posed Problems Using the Method of Random Projection

Introduction. Discrete ill-posed problems (DIP) often arise when restoring signals obtained as a result of indirect measurements. A stable solution of DIP can be obtained, for example, by the method of truncated singular value decomposition and by the method of random projection (RP). The accuracy of solving DIP by the RP method is determined by the influence of two independent random variables: additive noise in the output vector and a random variable that forms a random matrix. The change in the number of rows of a random matrix leads to a change in the accuracy of the DIP solution. Noisiness of the output vector leads to the appearance of a component of the error, the value of which increases with the number of rows of the random matrix and is scaled by the noise level. Therefore, the dependence of the error of DIP solution on the number of rows of the random matrix at certain noise levels has a minimum at  (row number of the random matrix less then row number of the DIP matrix). Averaging over random matrices leads to a smoothing of the dependence of the solution error on k and to a decrease in the number of local minima. The study of bias and variance of the error, arising from averaging over the realizations of the random matrix, made it possible to obtain a much more accurate estimate of the input vector.

Purpose. The aim is develop a method for determining the optimal size of a random matrix for the method of DIP solving based on the refined evaluation of the input vector.

Results and conclusions. The method of DIP solving based on the analytical averaging of random projection has been proposed. For this method, we have developed the criterion  for determining the number of rows of a random matrix which provides the error of DIP solution close to the minimum. We conducted an experimental investigation of the accuracy of DIP solution by a deterministic method based on the analytical averaging of random projection with the search for the optimal solution by the model selection criteria of Mallows, Akaike, Minimum description length, and . The experiments showed that  and Akaike criteria provide the error value close to the minimum.

Perspective. The direction of further work is the use of a deterministic method for DIP solution based on analytical averaging of random projection in applied problems.

Keywords: discrete ill-posed problem, regularization, random projection.

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