Randomized core reduction for discrete ill-posed problem

Zhang, Li Ping; Wei, Yimin

doi:10.1016/j.cam.2020.112797

“…The approach [15,8,16,10,7] based on finding the minimum error of solving a discrete ill-posed problem using random projection, ensures the stability of the solution and allows one to reduce the computational complexity. Random projections and other randomizations are also used for various versions of DIPs in [29][30][31][32].…”

Section: Regularization Methods For Solving Dipmentioning

confidence: 99%

Random Projection and Truncated SVD for Estimating Direction of Arrival in Antenna Array

Revunova¹,

Rachkovskij²

2018

Kibern. vyčisl. teh.

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Introduction. The need to solve inverse problems arises in many areas of science and technology in connection with the recovery of the object signal based on the results of indirect remote measurements. In the case where the transformation matrix has a high conditional number, the sequence of its singular numbers falls to zero, and the output of the measuring system contains noise, the problem of estimating the input vector is called discrete ill-posed problem (DIP). It is known that the DIP solution using pseudoinverse of the input-output transformation matrix is unstable. To overcome the instability and to improve the accuracy of the solution, regularization methods are used. Our approaches to ensuring the stability of the DIP solution (truncated singular decomposition (TSVD) and random projection (RP)) use the integer regularization parameter, which is the number of terms of the linear model. Regularization with an integer parameter makes it possible to provide a model close to the best in terms of the accuracy of the input vector recovery, and also to reduce the computational complexity by reducing the dimensionality of the problem. The purpose of the article is to develop an approach to estimating the direction of arrival of signals in the antenna array using the DIP solution, to compare the results with the well-known MUSIC method, to reveal the advantages and disadvantages of the methods. Results. Comparison of TSVD and MUSIC (implemented in real numbers) when working with correlated sources and five snapshots showed the advantage of TSVD in terms of the power of the useful signal P ratio by 2.2 times with the number of antenna elements K = 15 and by 4.7 times with K = 90. The advantage of TSVD in P ratio is by 3.7 times for K = 15 and by 4.2 times for K = 90. Comparison of RP and MUSIC (implemented in real numbers), when working with correlated sources and five snapshots, showed the advantage of RP in P ratio by 3

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“…The approach [15,8,16,10,7] based on finding the minimum error of solving a discrete ill-posed problem using random projection, ensures the stability of the solution and allows one to reduce the computational complexity. Random projections and other randomizations are also used for various versions of DIPs in [29][30][31][32].…”

Section: Regularization Methods For Solving Dipmentioning

confidence: 99%

Untitled

2018

Kibern. vyčisl. teh.

0

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Introduction. The need to solve inverse problems arises in many areas of science and technology in connection with the recovery of the object signal based on the results of indirect remote measurements. In the case where the transformation matrix has a high conditional number, the sequence of its singular numbers falls to zero, and the output of the measuring system contains noise, the problem of estimating the input vector is called discrete ill-posed problem (DIP). It is known that the DIP solution using pseudoinverse of the input-output transformation matrix is unstable. To overcome the instability and to improve the accuracy of the solution, regularization methods are used. Our approaches to ensuring the stability of the DIP solution (truncated singular decomposition (TSVD) and random projection (RP)) use the integer regularization parameter, which is the number of terms of the linear model. Regularization with an integer parameter makes it possible to provide a model close to the best in terms of the accuracy of the input vector recovery, and also to reduce the computational complexity by reducing the dimensionality of the problem. The purpose of the article is to develop an approach to estimating the direction of arrival of signals in the antenna array using the DIP solution, to compare the results with the well-known MUSIC method, to reveal the advantages and disadvantages of the methods. Results. Comparison of TSVD and MUSIC (implemented in real numbers) when working with correlated sources and five snapshots showed the advantage of TSVD in terms of the power of the useful signal P ratio by 2.2 times with the number of antenna elements K = 15 and by 4.7 times with K = 90. The advantage of TSVD in P ratio is by 3.7 times for K = 15 and by 4.2 times for K = 90. Comparison of RP and MUSIC (implemented in real numbers), when working with correlated sources and five snapshots, showed the advantage of RP in P ratio by 3

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“…A productive approach to overcome these problems is the randomization approach. It allows not only to reduce computational costs when searching for solutions, but also, as it turned out, to give stability to numerical methods, see also [58][59][60][61].…”

Section: Input Data Transformationsmentioning

confidence: 99%

Neural Distributed Representations for Artificial Intelligence and Modeling of Thinking

Rachkovskij¹,

Gritsenko²,

Volkov³

et al. 2022

Kibern. vyčisl. teh.

0

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coming these shortcomings is a necessary condition for advancing from specialized Artificial Intelligence to general one, which requires the development of alternative approaches.The purpose of the paper is to present an overview of research in this direction, which has been carried out at the International Center for 25 years. The approach being developed stems from the ideas of N. M. Amosov and his scientific school. Connections to the Hyperdimensional Computing (HDC) and Vector Symbolic Architectures (VSA) field as well as to current brain research are also provided.Results. The concept of distributed data representation is outlined, including HDC/VSA that are capable of representing various data structures. The developed paradigm of Associative-Projective Neural Networks is considered: codevector representation of data, superposition and binding operations, general architecture, transformation of data of various types into codevectors, methods for solving problems and applications. Conclusion. An adequate representation of data is one of the key issues within the Artificial Intelligence. The main area of research reviewed in this article is the problem of representing heterogeneous data in Artificial Intelligence systems in a unified format based on modeling the neural organization of the brain and the mechanisms of thinking. The approach under development is based on the hypothesis of distributed representation of information in the brain and allows representing various types of data, from numeric values to graphs, as vectors of large but fixed dimensionality.The most important advantages of the developed approach are the possibility of natural integration and efficient processing of various types of data and knowledge, a high degree of parallel computing, reliability and resistance to noise, the possibility of hardware implementation with high performance and energy efficiency, data processing based on associative similarity search -similar to how human memory works. This allows one to unify the methods, algorithms, and software and hardware for Artificial Intelligence systems, increase their scalability in terms of speed and memory with an increase in data volume and complexity.The research creates the basis for overcoming the shortcomings of current approaches to the specialized Artificial Intelligence based on Deep Neural Networks and paves the way for the creation of Artificial General Intelligence.

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Randomized core reduction for discrete ill-posed problem

Cited by 13 publications

References 41 publications

Random Projection and Truncated SVD for Estimating Direction of Arrival in Antenna Array

Random Projection and Truncated SVD for Estimating Direction of Arrival in Antenna Array

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Neural Distributed Representations for Artificial Intelligence and Modeling of Thinking

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