Identity mapping

The network has three hidden layers, including a bottleneck layer which is of a smaller dimension than either the input layer or the output layer. The network is trained to perform an identity mapping by approximating the input information at the output layer. Since there are fewer nodes in the bottleneck layer than the input or output layers, the bottleneck nodes implement data compression and encode the essential information in the inputs for its reconstruction in subsequent layers. In the NLPCA framework and terminology, autoassociative neural networks seek to provide a mapping of the form... 

Nonlinear PCA To address the nonlinearity in the identity mapping of multivariate data, a nonlinear counterpart of the PCA can be used (see Section 3.6.1). As the versions of NLPCA make use of the neural network (NN) concept to address the nonlinearity, they suffer from the known overparameterization problem in the case of noise corrupted data. Data with small SNR will also give rise to extensive computations during the training of the network. Shao et al. [266] used wavelet filtering to pre-process the data followed by IT-net to detect the non-conforming trends in an industrial spray drier. 

Any graph has at least one such automorphism, namely the identity mapping, where any vertex or edge is mapped onto itself ... 

The occurrence of further automorphisms, in addition to the identity mapping, will depend on the symmetry of the graph. 

Most of the basic properties of standard triangles involve homotopy, and so are best stated in K( ). For example, the mapping cone C of the identity map A —> A is homotopically equivalent to zero, a homotopy between the identity map of C and the zero map being as indicated ... 

It is easily checked—either directly, or because C is isomorphic to the cone of the identity map of A —that = 0 for all n and then from... 

Suppose we are given a jtseudofimctor, and functors f X —> Y, one for each S-morphism f X —> Y, such that is an identity functor whenever / is an identity map, and a family of functorial isomorphisms a — / - It is left as an exercise to show that then there is a unique... 

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