By Jian Cheng Lv, Zhang Yi, Jiliu Zhou
PrefaceChapter 1. Introduction1.1 Introduction1.1.1 Linear Neural Networks1.1.2 Subspace Learning1.2 Subspace studying Algorithms1.2.1 PCA studying Algorithms1.2.2 MCA studying Algorithms1.2.3 ICA studying Algorithms1.3 tools for Convergence Analysis1.3.1 SDT Method1.3.2 DCT Method1.3.3 DDT Method1.4 Block Algorithms1.5 Simulation information Set and Notation1.6 ConclusionsChapter 2. PCA studying Algorithms with Constants studying Rates2.1 Oja's PCA studying Algorithms2.1.1 The Algorithms2.1.2 Convergence Issue2.2 Invariant Sets2.2.1 houses of Invariant Sets2.2.2 stipulations for Invariant Sets2. Read more...
summary: PrefaceChapter 1. Introduction1.1 Introduction1.1.1 Linear Neural Networks1.1.2 Subspace Learning1.2 Subspace studying Algorithms1.2.1 PCA studying Algorithms1.2.2 MCA studying Algorithms1.2.3 ICA studying Algorithms1.3 tools for Convergence Analysis1.3.1 SDT Method1.3.2 DCT Method1.3.3 DDT Method1.4 Block Algorithms1.5 Simulation information Set and Notation1.6 ConclusionsChapter 2. PCA studying Algorithms with Constants studying Rates2.1 Oja's PCA studying Algorithms2.1.1 The Algorithms2.1.2 Convergence Issue2.2 Invariant Sets2.2.1 homes of Invariant Sets2.2.2 stipulations for Invariant Sets2
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3 Outline of This Book The rest of this chapter is organized as follows. In Section 2, the methods for convergence analysis is introduced and the relationship between SDT algorithm and the corresponding DDT algorithm is given in Section 3. Section 4 presents some notations and preliminaries. In Chapter 2, the convergence of Oja’s and Xu’s algorithms with constant learning rates is studied in detail. Some invariant sets are obtained and local convergence is proven rigorously. The most important contribution is the convergence analysis framework of deterministic discrete time (DDT) method is established in this chapter.
Moreover, some trajectories may diverge and even become chaotic. Thus, it is quite interesting to explore whether there exists invariant sets to retain the trajectories within the sets and based on these invariant sets to explore the bigger issue of convergence. This section will study these problems in details. The DDT algorithm has many equilibria and thus the study of convergence belongs to a multistability problem. Multistability analysis has recently received attractive attentions, see, for examples, , , , .
Then, it holds that, f (s) ≤ f (ξ) = 4 · (1 + ησ)3 , 27η for all 0 ≤ s ≤ σ + 1/η. The proof is completed. 1 to extract the first principal component from the input data . Based on the well-known Hebbian learning rule, Oja proposed a principal component analysis (PCA) learning algorithm to update the weights of the network. This network, under Oja’s algorithm, is able to extract a principal component from input data adaptively. The results have been useful in online data processing applications.