In this paper we present closed-form solutions for efficiently updating the prin-cipal components of a set of n points, when m points are added or deleted fromthe point set. For both operations performed on a discrete point set in R, we cancompute the new principal components in O(m) time for fixed d. This is a signifi-cant improvement over the commonly used approach of reco...

Derivation of PCA I: For a set of d-dimensional data vectors {x}i=1, the principal axes {e}qj=1 are those orthonormal axes onto which the retained variance under projection is maximal. It can be shown that the vectors ej are given by the q dominant eigenvectors of the sample covariance matrix S, such that Sej = λjej . The q principal components of the observed vector xi are given by the vector ...

Principal components analysis is an important and well-studied subject in statistics and signal processing. The literature has an abundance of algorithms for solving this problem, where most of these algorithms could be grouped into one of the following three approaches: adaptation based on Hebbian updates and deflation, optimization of a second order statistical criterion (like reconstruction ...

Consider the standard setting where we are given n points in d dimensions. Call these ~ x1, ~ x2, . . . , ~ xn. As before, our goal is to reduce the number of dimensions to a small number k. In principal component analysis (or PCA), we will model the data by a k-dimensional subspace, and find the subspace for which the error in this representation is smallest. Suppose k = 1. Then we want to app...

Principal component analysis (PCA) is a well-established tool for making sense of high dimensional data by reducing it to a smaller dimension. Its extension to sparse principal component analysisprincipal component analysis!sparce, which provides a sparse low-dimensional representation of the data, has attracted alot of interest in recent years (see, e.g., [1, 2, 3, 5, 6, 7, 8, 9]). In many app...

One of the central issues in the use of principal component analysis (PCA) for data modelling is that of choosing the appropriate number of retained components. This problem was recently addressed through the formulation of a Bayesian treatment of PCA (Bishop, 1999a) in terms of a probabilistic latent variable model. A central feature of this approach is that the effective dimensionality of the...