Complex-Valued Boltzmann Manifold

2009 ◽  
pp. 1-26 ◽  
Author(s):  
Masaki Kobayashi
Keyword(s):  
Learning Algorithm ◽  
Information Geometry ◽  
Exponential Families ◽  
Learning Models ◽  
Boltzmann Machines ◽  
Effective Learning ◽  
Stochastic Learning ◽  
Leibler Divergence ◽  
Hidden Neurons ◽  
Complex Valued

Information geometry is one of the most effective tools to investigate stochastic learning models. In it, stochastic learning models are regarded as manifolds in the view of differential geometry. Amari applied it to Boltzmann Machines, which is one of the stochastic learning models. The purpose of this chapter is to apply information geometry to complex-valued Boltzmann Machines. First, we construct the complex-valued Boltzmann Machines. Next, the author describes information geometry. The author will know some important notions of information geometry, exponential families, mixture families, Kullback-Leibler divergence, connections, geodesics, Fisher metrics, potential functions and so on. Finally, they apply information geometry to complex-valued Boltzmann Machines. They will investigate the structure of complex-valued Boltzmann manifold and know the notions of the connections and Fisher metric. Moreover we will get an effective learning algorithm, what is called em algorithm, for complex-valued Boltzmann machines with hidden neurons.

COMPLEX-VALUED MINIMAL RESOURCE ALLOCATION NETWORK FOR NONLINEAR SIGNAL PROCESSING

2000 ◽  
Vol 10 (02) ◽  
pp. 95-106 ◽  
Author(s):  
JIANPING DENG ◽  
N. SUNDARARAJAN ◽  
P. SARATCHANDRAN
Keyword(s):  
Resource Allocation ◽  
Signal Processing ◽  
Communication Systems ◽  
Learning Algorithm ◽  
Nonlinear Signal Processing ◽  
Hidden Neurons ◽  
Nonlinear Signal ◽  
Complex Valued ◽  
Complex Channel ◽  
Minimal Resource

This paper presents a sequential learning algorithm and evaluates its performance on complex valued signal processing problems. The algorithm is referred to as Complex Minimal Resource Allocation Network (CMRAN) algorithm and it is an extension of the MRAN algorithm originally developed for online learning in real valued RBF networks. CMRAN has the ability to grow and prune the (complex) RBF network's hidden neurons to ensure a parsimonious network structure. The performance of the learning algorithm is illustrated using two applications from signal processing of communication systems. The first application considers identification of a nonlinear complex channel. The second application considers the use of CMRAN to QAM digital channel equalization problems. Simulation results presented clearly show that CMRAN is very effective in modeling and equalization with performance achieved often being superior to that of some of the well known methods.

scholarly journals An Investigation into the Application of Deep Learning in the Detection and Mitigation of DDOS Attack on SDN Controllers

Technologies ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 14
Author(s):  
James Dzisi Gadze ◽  
Akua Acheampomaa Bamfo-Asante ◽  
Justice Owusu Agyemang ◽  
Henry Nunoo-Mensah ◽  
Kwasi Adu-Boahen Opare
Keyword(s):  
Deep Learning ◽  
Network Architecture ◽  
Learning Algorithm ◽  
Single Point ◽  
Denial Of Service ◽  
Specific Work ◽  
Learning Models ◽  
Ddos Attack ◽  
Deep Learning Algorithm ◽  
Proposed Model

Software-Defined Networking (SDN) is a new paradigm that revolutionizes the idea of a software-driven network through the separation of control and data planes. It addresses the problems of traditional network architecture. Nevertheless, this brilliant architecture is exposed to several security threats, e.g., the distributed denial of service (DDoS) attack, which is hard to contain in such software-based networks. The concept of a centralized controller in SDN makes it a single point of attack as well as a single point of failure. In this paper, deep learning-based models, long-short term memory (LSTM) and convolutional neural network (CNN), are investigated. It illustrates their possibility and efficiency in being used in detecting and mitigating DDoS attack. The paper focuses on TCP, UDP, and ICMP flood attacks that target the controller. The performance of the models was evaluated based on the accuracy, recall, and true negative rate. We compared the performance of the deep learning models with classical machine learning models. We further provide details on the time taken to detect and mitigate the attack. Our results show that RNN LSTM is a viable deep learning algorithm that can be applied in the detection and mitigation of DDoS in the SDN controller. Our proposed model produced an accuracy of 89.63%, which outperformed linear-based models such as SVM (86.85%) and Naive Bayes (82.61%). Although KNN, which is a linear-based model, outperformed our proposed model (achieving an accuracy of 99.4%), our proposed model provides a good trade-off between precision and recall, which makes it suitable for DDoS classification. In addition, it was realized that the split ratio of the training and testing datasets can give different results in the performance of a deep learning algorithm used in a specific work. The model achieved the best performance when a split of 70/30 was used in comparison to 80/20 and 60/40 split ratios.

scholarly journals A MLMVN with Arbitrary Complex-Valued Inputs and a Hybrid Testability Approach for the Extraction of Lumped Models Using FRA

2019 ◽  
Vol 9 (1) ◽  
pp. 5-19 ◽  
Author(s):  
Igor Aizenberg ◽  
Antonio Luchetta ◽  
Stefano Manetti ◽  
Maria Cristina Piccirilli
Keyword(s):  
Neural Network ◽  
Frequency Response ◽  
Learning Algorithm ◽  
Repeated Measurements ◽  
Distributed Parameter ◽  
Response Analysis ◽  
Frequency Response Analysis ◽  
Arbitrary Complex ◽  
Lumped Models ◽  
Complex Valued

Abstract A procedure for the identification of lumped models of distributed parameter electromagnetic systems is presented in this paper. A Frequency Response Analysis (FRA) of the device to be modeled is performed, executing repeated measurements or intensive simulations. The method can be used to extract the values of the components. The fundamental brick of this architecture is a multi-valued neuron (MVN), used in a multilayer neural network (MLMVN); the neuron is modified in order to use arbitrary complex-valued inputs, which represent the frequency response of the device. It is shown that this modification requires just a slight change in the MLMVN learning algorithm. The method is tested over three completely different examples to clearly explain its generality.

A new learning algorithm with logarithmic performance index for complex-valued neural networks

2009 ◽  
Vol 72 (16-18) ◽  
pp. 3771-3781 ◽  
Author(s):  
R. Savitha ◽  
S. Suresh ◽  
N. Sundararajan ◽  
P. Saratchandran
Keyword(s):  
Neural Networks ◽  
Performance Index ◽  
Learning Algorithm ◽  
New Learning ◽  
Complex Valued

Learning in Boltzmann Trees

1994 ◽  
Vol 6 (6) ◽  
pp. 1174-1184 ◽  
Author(s):  
Lawrence Saul ◽  
Michael I. Jordan
Keyword(s):  
Supervised Learning ◽  
Gradient Descent ◽  
Learning Algorithm ◽  
Mean Field ◽  
Large Family ◽  
Weight Space ◽  
Hidden Symmetries ◽  
Boltzmann Machines ◽  
Field Annealing

We introduce a large family of Boltzmann machines that can be trained by standard gradient descent. The networks can have one or more layers of hidden units, with tree-like connectivity. We show how to implement the supervised learning algorithm for these Boltzmann machines exactly, without resort to simulated or mean-field annealing. The stochastic averages that yield the gradients in weight space are computed by the technique of decimation. We present results on the problems of N-bit parity and the detection of hidden symmetries.

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