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Abstract: Inverse problems determine the inner mechanisms or causes of an observed phenomenon. Network inversion solves inverse problems to estimate causes from results by using a multilayer neural network. The original network inversion method has been applied to a usual multilayer neural network with real-valued inputs and outputs. Recently, a complex-valued neural network that is an extension of a neural network to the complex region has been studied. A complex-valued network inversion method has been proposed to solve inverse problems that include complex numbers. In general, this is a problem attributed to the ill-posedness, which implies that the existence, uniqueness, and stability of the solution are not guaranteed, related to inverse problems. To solve this ill-posedness, the concept of regularization is used for adding some conditions to the solution. In this study, the author applies complex-valued network inversion with regularization to an ill-posed inverse mapping problem and show the effectiveness of the proposed method in solving ill-posed inverse problems.
Key words: Inverse problems, complex-valued neural networks, network inversion, ill-posedness, regularization.
1. Introduction
It is necessary to solve inverse problems for estimating causes from observed results in various fields [1-2]. An inverse problem determines the inner mechanisms or causes of an observed phenomenon. As a solution to inverse problems, a neural-network-based method has been proposed, while other methods such as the statistical method and the parametric method have also been studied. Network inversion solves inverse problems using a multilayer neural network [3]. In this method, inverse problems are solved by the inverse use of the input-output relation of trained multilayer neural networks. Network inversion has been applied to actual problems [4-5]. The original concept of network inversion solves an inverse problem by using a usual multilayer neural network that handles the relation between a real-valued input and output. On the other hand, there exists an extension of the multilayer neural network to the complex domain [6-7]. The complex-valued neural network has been studied in various application fields. Such research has led to the development of the concept of complex-valued network inversion for solving general inverse problems whose causes and results extend to the complex domain [8-9].
In general, there is a problem attributable to the ill-posedness on the inverse problems. The term“ill-posedness” implies that the existence, uniqueness, and stability of a solution are not guaranteed. It is often difficult to obtain a solution because of ill-posedness. Regularization imposes specific conditions on an ill-posed inverse problem in order to convert it into a well-posed problem [10-11]. In the case of a complex-valued network inversion, ill-posedness becomes an important problem that needs to be solved. By solving the mapping problem on a complex plane, the author can solve various ill-posed problems of inverse mapping. It is possible to apply the obtained result to various problems by examining a complex-valued neural network using a mapping problem.
In this study, the author examines the application of complex-valued network inversion with regularization to ill-posed inverse problems. Further, the author aims to confirm the effect of the regularization on the ill-posed inverse problems. He considers inverse mapping problems that include ill-posedness related to the existence, uniqueness, and stability of a solution. For example, the relation of the absolute value becomes an ill-posed inverse problem related to uniqueness as both positive and negative values are mapped to a positive value, that is, a two-to-one relation. Furthermore, the inverse relation of the absolute value to a negative value is an ill-posed problem related to existence. Moreover, the relation of reduction becomes an ill-posed problem related to stability because the noise is enlarged in the inverse estimation from the output containing noise. In this study, the author extends the above-mentioned issue to a complex mapping problem and examine a method to improve the ill-posedness of inverse problems. With respect to uniqueness and stability, the author introduces a complex regularization method for providing a restriction to a solution and examines the effect. From the viewpoint of existence, how to distinguish the existence of a solution from the provided output is examined. Thus, the effectiveness of the complex-valued network inversion in solving ill-posed problems is shown.
The rest of this paper is organized as follows: Section 2 discusses the inverse problems and neural networks; section 3 introduces the complex-valued network inversion; section 4 introduces the regularization for complex-valued network inversion; in section 5, complex inverse mapping problems are discussed; section 6 presents simulation; section 7 describes results and discussion; section 8 gives conclusions.
References
[1] C.W. Groetsch, Inverse Problems in the Mathematical Sciences, Friedr. Vieweg and Sohn Verlags GmbH, 1993.
[2] J. Kaipio, E. Somersalo, Statistical and Computational Inverse Problems, Springer, 2005.
[3] A. Linden, J. Kindermann, Inversion of multilayer nets, in: Proc. IJCNN, 1989, pp. 425-430.
[4] W.R. Murray, C.T. Heg, C.M. Pohlhammer, Iterative inversion of a neural network for estimating the location of a planar object, in: Proc. World Congress on Neural Networks, 1993, Vol. 3, pp. 188-193.
[5] I. Valova, K. Kameyama, Y. Kosugi, Image decomposition by answer-in-weights neural network, IEICE Trans. on Information and Systems E78-D-9 (1995) 1221-1224.
[6] T. Nitta, An extension of the back-propagation algorithm to complex numbers, Neural Networks 10 (8) (1997) 1392-1415.
[7] A. Hirose, Complex-Valued Neural Networks, Springer, 2005.
[8] T. Ogawa, H. Kanada, Network inversion for complex-valued neural networks, in: Proc. of ISSPIT 2005, 2005, pp. 850-855.
[9] T. Ogawa, Complex-valued neural network and inverse problems, in: T. Nitta (Ed.), Complex-Valued Neural Networks: Utilizing High-Dimensional Parameters, IGI-Global, 2009, pp. 27-55.
[10] A.N. Tikhonov, V.Y. Arsenin, Solutions of Ill-Posed Problems, Winstion and Sons, 1977.
[11] Y.P. Petrov, V.S. Sizikov, Well-Posed, Ill-Posed, and Intermediate Problems with Application, Koninklijke Brill NV, 2005.
Key words: Inverse problems, complex-valued neural networks, network inversion, ill-posedness, regularization.
1. Introduction
It is necessary to solve inverse problems for estimating causes from observed results in various fields [1-2]. An inverse problem determines the inner mechanisms or causes of an observed phenomenon. As a solution to inverse problems, a neural-network-based method has been proposed, while other methods such as the statistical method and the parametric method have also been studied. Network inversion solves inverse problems using a multilayer neural network [3]. In this method, inverse problems are solved by the inverse use of the input-output relation of trained multilayer neural networks. Network inversion has been applied to actual problems [4-5]. The original concept of network inversion solves an inverse problem by using a usual multilayer neural network that handles the relation between a real-valued input and output. On the other hand, there exists an extension of the multilayer neural network to the complex domain [6-7]. The complex-valued neural network has been studied in various application fields. Such research has led to the development of the concept of complex-valued network inversion for solving general inverse problems whose causes and results extend to the complex domain [8-9].
In general, there is a problem attributable to the ill-posedness on the inverse problems. The term“ill-posedness” implies that the existence, uniqueness, and stability of a solution are not guaranteed. It is often difficult to obtain a solution because of ill-posedness. Regularization imposes specific conditions on an ill-posed inverse problem in order to convert it into a well-posed problem [10-11]. In the case of a complex-valued network inversion, ill-posedness becomes an important problem that needs to be solved. By solving the mapping problem on a complex plane, the author can solve various ill-posed problems of inverse mapping. It is possible to apply the obtained result to various problems by examining a complex-valued neural network using a mapping problem.
In this study, the author examines the application of complex-valued network inversion with regularization to ill-posed inverse problems. Further, the author aims to confirm the effect of the regularization on the ill-posed inverse problems. He considers inverse mapping problems that include ill-posedness related to the existence, uniqueness, and stability of a solution. For example, the relation of the absolute value becomes an ill-posed inverse problem related to uniqueness as both positive and negative values are mapped to a positive value, that is, a two-to-one relation. Furthermore, the inverse relation of the absolute value to a negative value is an ill-posed problem related to existence. Moreover, the relation of reduction becomes an ill-posed problem related to stability because the noise is enlarged in the inverse estimation from the output containing noise. In this study, the author extends the above-mentioned issue to a complex mapping problem and examine a method to improve the ill-posedness of inverse problems. With respect to uniqueness and stability, the author introduces a complex regularization method for providing a restriction to a solution and examines the effect. From the viewpoint of existence, how to distinguish the existence of a solution from the provided output is examined. Thus, the effectiveness of the complex-valued network inversion in solving ill-posed problems is shown.
The rest of this paper is organized as follows: Section 2 discusses the inverse problems and neural networks; section 3 introduces the complex-valued network inversion; section 4 introduces the regularization for complex-valued network inversion; in section 5, complex inverse mapping problems are discussed; section 6 presents simulation; section 7 describes results and discussion; section 8 gives conclusions.
References
[1] C.W. Groetsch, Inverse Problems in the Mathematical Sciences, Friedr. Vieweg and Sohn Verlags GmbH, 1993.
[2] J. Kaipio, E. Somersalo, Statistical and Computational Inverse Problems, Springer, 2005.
[3] A. Linden, J. Kindermann, Inversion of multilayer nets, in: Proc. IJCNN, 1989, pp. 425-430.
[4] W.R. Murray, C.T. Heg, C.M. Pohlhammer, Iterative inversion of a neural network for estimating the location of a planar object, in: Proc. World Congress on Neural Networks, 1993, Vol. 3, pp. 188-193.
[5] I. Valova, K. Kameyama, Y. Kosugi, Image decomposition by answer-in-weights neural network, IEICE Trans. on Information and Systems E78-D-9 (1995) 1221-1224.
[6] T. Nitta, An extension of the back-propagation algorithm to complex numbers, Neural Networks 10 (8) (1997) 1392-1415.
[7] A. Hirose, Complex-Valued Neural Networks, Springer, 2005.
[8] T. Ogawa, H. Kanada, Network inversion for complex-valued neural networks, in: Proc. of ISSPIT 2005, 2005, pp. 850-855.
[9] T. Ogawa, Complex-valued neural network and inverse problems, in: T. Nitta (Ed.), Complex-Valued Neural Networks: Utilizing High-Dimensional Parameters, IGI-Global, 2009, pp. 27-55.
[10] A.N. Tikhonov, V.Y. Arsenin, Solutions of Ill-Posed Problems, Winstion and Sons, 1977.
[11] Y.P. Petrov, V.S. Sizikov, Well-Posed, Ill-Posed, and Intermediate Problems with Application, Koninklijke Brill NV, 2005.