Oja在线主成分分析的扩散近似
Diffusion approximations of Oja's online principal component analysis
摘要：
Oja的主成分分析（PCA）算法是一种在实际中应用的降维方法。本文主要研究离散算法的收敛性。为了实现这一点，我们将算法看作是参数空间和半群上的一个随机过程。我们用随机微分方程逼近它，并证明了随机微分方程的大时间收敛性，以保证其性能。这个过程分三步完成。首先，离散算法可以看作是一个半群：$S^kφ=\mathbb{E}[φ(\mathbf W(k))]$。其次，我们构造了Stiefel流形上的随机微分方程，即扩散近似，来逼近半群。通过证明算法的弱收敛性，我们证明了算法是“接近”的SDES。最后，我们使用的SDES的可逆性来证明长时间收敛。
英文摘要：
Oja's algorithm of principal component analysis (PCA) has been one of the methods utilized in practice to reduce dimension. In this paper, we focus on the convergence property of the discrete algorithm. To realize that, we view the algorithm as a stochastic process on the parameter space and semi-group. We approximate it by SDEs, and prove large time convergence of the SDEs to ensure its performance. This process is completed in three steps. First, the discrete algorithm can be viewed as a semigroup: $S^kφ=\mathbb{E}[φ(\mathbf W(k))]$. Second, we construct stochastic differential equations (SDEs) on the Stiefel manifold, i.e. the diffusion approximation, to approximate the semigroup. By proving the weak convergence, we verify that the algorithm is 'close to' the SDEs. Finally, we use the reversibility of the SDEs to prove long-time convergence.
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