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 info Set and Notation1.6 ConclusionsChapter 2. PCA studying Algorithms withRead 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 houses of Invariant Sets2.2.2 stipulations for Invariant Sets2

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9 If w(0) ∈ S and w(0) ∈ / Vσ⊥ , then there exist constants θ1 > 0, Π1 ≥ 0, and d > 0 such that n j=l+1 zj2 (k) ≤ Π1 · e−θ1 k , for all k > N , where θ1 = ln σ + λp + 2ησλp σ + λp + 2ηλm+1 λp 2 > 0. Proof: Since w(0) ∈ / Vσ⊥ , there must exist some i(1 ≤ i ≤ m) such that zi (0) = 0. Without loss of generality, assume that z1 (0) = 0. 4, it follows that w(k) ∈ S for all k ≥ 0. 7, it holds that 1 + η λi (2 − w(k) 2 ) − wT (k)Cw(k) > 0, for k ≥ 0. 5, then w(k + 1) 2 ≤ 2σ < 2, σ + λp 2 · zj (k) z1 (k) 2 .

N). Next, we analyze the convergence of w(k) by studying the convergence of zi (k)(i = 1, . . , m) and zi (k)(i = m + 1, . . , n), respectively. 8899, 2 if w(0) ∈ S and w(0) ∈ / Vσ⊥ , then there exist constants θ1 > 0 and Π1 ≥ 0 such that n j=m+1 zj2 (k) ≤ Π1 · e−θ1 k , for all k ≥ 0, where 1 + ησ 1 + ηλm+1 θ1 = ln 2 > 0. Proof : Since w(0) ∈ / Vσ⊥ , there must exist some i(1 ≤ i ≤ m) such that zi (0) = 0. Without loss of generality, assume that z1 (0) = 0. 5) and for k ≥ 0. 1, S is an invariant set, then w(k) ∈ S for all k ≥ 0.

M), k→+∞ where zi∗ (i = 1, . . , m) are constants. Proof: Given any ǫ > 0, there exists a K ≥ 1 such that ΠKe−θK 2 (1 − e−θ ) ≤ ǫ. For any k1 > k2 ≥ K, it follows that k1 −1 |zi (k1 ) − zi (k2 )| = r=k2 [zi (r + 1) − zi (r)] k1 −1 ≤ η r=k2 σ − wT (r)Cw(r) zi (r) k1 −1 ≤ Π re−θr r=k2 +∞ ≤ Π re−θr r=K +∞ ≤ ΠKe−θK · = r e−θ r−1 r=0 ΠKe−θK 2 (1 − e−θ ) ≤ ǫ, (i = 1, . . , m). This shows that each sequence {zi (k)} is a Cauchy sequence. By Cauchy Convergence Principle, there must exist constants zi∗ (i = 1, .