1 Preliminaries • p. 28 l. -2 : Insert ‘provided the automaton is complete’ after ‘The matrix M/k is stochastic’. • p. 30 l. Replace lines 3–17 by : Applying by induction the theorem to U and W , we obtain nonnegative eigenvectors u and w for the eigenvalues ρU and ρW of U and W . We prove that max(ρU , ρW ) is an eigenvalue of M with some nonnegative eigenvector. If ρU ≥ ρW , then ρU is an eig...

Principal Component Analysis (PCA) is a standard tool for dimensional reduction of a set of n observations (samples), each with p variables. In this paper, using a matrix perturbation approach, we study the non-asymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite sample of size n, to those of the limiting population PCA as n → ∞. As in machine learning, we pr...

Kernel method is a very powerful tool in machine learning. The trick of kernel has been effectively and extensively applied in many areas of machine learning, such as support vector machine (SVM) and kernel principal component analysis (kernel PCA). Kernel trick is to define a kernel function which relies on the inner-product of data in the feature space without knowing these feature space data...

In this paper we provide faster algorithms and improved sample complexities for approximating the top eigenvector of a matrix A>A. In particular we give the following results for computing an approximate eigenvector i.e. some x such that x>A>Ax ≥ (1− )λ1(AA): • Offline Eigenvector Estimation: Given an explicit matrix A ∈ Rn×d, we show how to compute an approximate top eigenvector in time Õ ([ n...