Generalized Inverse of Matrices and Its Applications

Banerjee, K. S.

doi:10.1080/00401706.1973.10489026

Cited by 680 publications

(20 citation statements)

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“…There are several methods to calculate the Moore-Penrose generalized inverse of a matrix [38], [39]. For example, in the orthogonal projection method,…”

Section: A Heterogeneous Multilayer Generalized Operational Perceptrmentioning

confidence: 99%

Heterogeneous Multilayer Generalized Operational Perceptron

Tran

Kıranyaz

Gabbouj

et al. 2020

IEEE Trans. Neural Netw. Learning Syst.

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The traditional Multilayer Perceptron (MLP) using McCulloch-Pitts neuron model is inherently limited to a set of neuronal activities, i.e., linear weighted sum followed by nonlinear thresholding step. Previously, Generalized Operational Perceptron (GOP) was proposed to extend conventional perceptron model by defining a diverse set of neuronal activities to imitate a generalized model of biological neurons. Together with GOP, Progressive Operational Perceptron (POP) algorithm was proposed to optimize a pre-defined template of multiple homogeneous layers in a layerwise manner. In this paper, we propose an efficient algorithm to learn a compact, fully heterogeneous multilayer network that allows each individual neuron, regardless of the layer, to have distinct characteristics. Based on the complexity of the problem, the proposed algorithm operates in a progressive manner on a neuronal level, searching for a compact topology, not only in terms of depth but also width, i.e., the number of neurons in each layer. The proposed algorithm is shown to outperform other related learning methods in extensive experiments on several classification problems.

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“…There are several methods to calculate the Moore-Penrose generalized inverse of a matrix [38], [39]. For example, in the orthogonal projection method,…”

Section: A Heterogeneous Multilayer Generalized Operational Perceptrmentioning

confidence: 99%

Heterogeneous Multilayer Generalized Operational Perceptron

Tran

Kıranyaz

Gabbouj

et al. 2020

IEEE Trans. Neural Netw. Learning Syst.

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“…According to Equation (13), as long as the input weight w i and the offset b i are randomly determined, the output matrix H is uniquely determined. The Moore-Penrose generalized inverse matrix H † of H is used to analyze and determine the least-squares solution β of the smallest norm [25,26]…”

Section: As Shown Inmentioning

confidence: 99%

An Improved LDA-Based ELM Classification for Intrusion Detection Algorithm in IoT Application

Zheng

Zhou

Wang

et al. 2020

Sensors

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The Internet of Things (IoT) is widely applied in modern human life, e.g., smart home and intelligent transportation. However, it is vulnerable to malicious attacks, and the current existing security mechanisms cannot completely protect the IoT. As a security technology, intrusion detection can defend IoT devices from most malicious attacks. However, unfortunately the traditional intrusion detection models have defects in terms of time efficiency and detection efficiency. Therefore, in this paper, we propose an improved linear discriminant analysis (LDA)-based extreme learning machine (ELM) classification for the intrusion detection algorithm (ILECA). First, we improve the linear discriminant analysis (LDA) and then use it to reduce the feature dimensions. Moreover, we use a single hidden layer neural network extreme learning machine (ELM) algorithm to classify the dimensionality-reduced data. Considering the high requirement of IoT devices for detection efficiency, our scheme not only ensures the accuracy of intrusion detection, but also improves the execution efficiency, which can quickly identify the intrusion. Finally, we conduct experiments on the NSL-KDD dataset. The evaluation results show that the proposed ILECA has good generalization and real-time characteristics, and the detection accuracy is up to 92.35%, which is better than other typical algorithms.

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“…The solution of in Eq. (2) can be directly obtained by: is the Moore-Penrose generalized inverse of matrix [36]. That is…”

Section: Extreme Learning Machine (Elm)mentioning

confidence: 99%