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We can tighten (or improve) the relaxation (6.37) by adding other constraints that cut away parts of the feasible set, without excluding rank 1 solutions. By tightening the relaxation, we reduce the gap between the optimal values of the original problem and the relaxation. For example, we can add the constraints

0 X i j 1 , i = 1 , , n , j = 1 , , n , X i i X i j , i = 1 , , n , j = 1 , , n ,

and so on. This will usually have a dramatic impact on solution times and memory requirements. Already constraining a semidefinite matrix to be doubly nonnegative ( X i j 0 ) introduces additional n 2 linear inequality constraints.

Similarly to NIKE Mens Air Force 1 07 LV8 Basketball Shoe Medium Olive/Baroque Brown/Sequoia/Black luYAh4w3YE
we can use semidefinite relaxations of boolean constraints x { 1 , + 1 } n . To that end, we note that

(6.38) x { 1 , + 1 } n X = x x T , d i a g ( X ) = e ,

with a semidefinite relaxation X x x T of the rank-1 constraint.

As a (standard) example of a combinatorial problem with boolean constraints, we consider an undirected graph G = ( V , E ) described by a set of vertices V = { v 1 , , v n } and a set of edges E = { ( v i , v j ) v i , v j V , i j } , and we wish to find the cut of maximum capacity. A cut C partitions the nodes V into two disjoint sets S and T = V S , and the cut-set I is the set of edges with one node in S and another in T , i.e.,

Published by: Kowsar
Human Informatics Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan

Nitta T. Complex-Valued Neurocomputing and Singular Points, Arch Neurosci. 2015 ;2(4):e27461. doi: 10.5812/archneurosci.27461 .

Context: Recently, the singular points of neural networks have attracted attention from the artificial intelligence community, and their interesting properties have been demonstrated. The objective of this study is to provide an overview of studies on the singularities of complex-valued neural networks.

Evidence Acquisition: This review is based on the relevant literature on complex-valued neural networks and singular points.

Results: Review of the studies and available literature on the subject area shows that the singular points of complex-valued neural networks have negative effects on learning, as do those of real-valued neural networks. However, the nature of the singular points in complex-valued neural networks is superior in quality, and the methods for improving the learning performance have been proposed.

Conclusions: A complex-valued neural network could be a promising learning method from the viewpoint of a singularity.

Keywords: Neural Network; Learning; Non-linear Dynamics

Copyright © 2015, Tehran University of Medical Sciences. This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/) which permits copy and redistribute the material just in noncommercial usages, provided the original work is properly cited.

An artificial neural network is a machine learning model and has recently attracted attention owing to its human-like intelligence such as learning and generalization ( Stuart Weitzman Womens Metro SlipOn Loafer Brandy Mirror GNQxn

A complex-valued neural network extends (real-valued) parameters (weights and threshold values) in a real-valued neural network to complex numbers. It is advantageous because the good-natured behavior of a complex number to rotate is assured. Thus, it is suitable for the information processing of complex-valued data and two-dimensional data and has been applied to various fields such as communications, image processing, biological information processing, land-mine detection, wind prediction, and independent component analysis (ICA) ( 2 ). Recently, a complex-valued firing-rate model was presented ( EnllerviiD Women Multicolor Strappy Flat Jelly Sandals Open Toe Slip On Garden Shoes Rose F2U93
), which is an attempt to implement a neural network of an actual brain with complex numbers.

Generally, a hierarchical structure causes singular points. For example, consider a three-layered real-valued neural network. If a weight between a hidden neuron and an output neuron is equal to zero, then no value of the weight vector between the hidden neuron and the input neurons affects the output value of the real-valued neural network. Then, the weight vector is called an unidentifiable parameter, and is a singular point. It has been shown that singular points affect the learning dynamics of learning models and that they can cause a standstill in learning ( CHFSO Womens Stylish Solid Round Toe Lace Up With Buckle Mid Chunky Heel Knee High Boots Black 80TsU
- 6 ).


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