We consider a problem involving estimation of a high-dimensional covariance matrix that is the sum of a diagonal matrix and a low-rank matrix, and making a decision based on the resulting estimate. Such problems arise, for example, in portfolio management, where a common approach employs principal component analysis (PCA) to estimate factors used in constructing the low-rank term of the covaria...

Given a set of signals, a classical construction of an optimal truncatable basis for optimally representing the signals, is the principal component analysis (PCA for short) approach. When the information about the signals one would like to represent is a more general property, like smoothness, a different basis should be considered. One example is the Fourier basis which is optimal for represen...

We consider the online version of the well known Principal Component Analysis (PCA) problem. In standard PCA, the input to the problem is a set of vectors X = [x1, . . . , xn] in Rd×n and a target dimension k < d; the output is a set of vectors Y = [y1, . . . , yn] in Rk×n that minimize minΦ ‖X − ΦY ‖F where Φ is restricted to be an isometry. The global minimum of this quantity, OPTk, is obtain...

Efficient and compact representation of images is a fundamental problem in computer vision. Principal Component Analysis (PCA) has been widely used for image representation and has been successfully applied to many computer vision algorithms. In this paper, we propose a method that uses Haar-like binary box functions to span a subspace which approximates the PCA subspace. The proposed method ca...

Conventional blind signal separation algorithms do not adopt any asymmetric information of the input sources, thus the convergence point of a single output is always unpredictable. However, in most of the applications, we are usually interested in only one or two of the source signals and prior information is almost always available. In this paper, a principal independent component analysis (PI...

where each column is a data sample. Analyzing—and hopefully understanding— the result of an experiment often consists in uncovering regularity or structure in the data matrix. Unfortunately, measured data is often “messy” in the sense that it is too high-dimensional for us to detect structure by direct inspection and in the sense that noise and/or redundancy often impairs data visualization. Pr...

This article presents a unified theory for analysis of components in discrete data, and compares the methods with techniques such as independent component analysis (ICA), non-negative matrix factorisation (NMF) and latent Dirichlet allocation (LDA). The main families of algorithms discussed are mean field, Gibbs sampling, and Rao-Blackwellised Gibbs sampling. Applications are presented for voti...

We study the extraction of nonlinear data models in high dimensional spaces with modi ed self organizing maps We present a general algorithm which maps low dimensional lattices into high dimensional data manifolds without violation of topology The approach is based on a new principle exploiting the speci c dynamical properties of the rst order phase tran sition induced by the noise of the data ...

Principal component analysis (PCA) is a multivariate technique that analyzes a data table in which observations are described by several inter-correlated quantitative dependent variables. Its goal is to extract the important information from the table, to represent it as a set of new orthogonal variables called principal components, and to display the pattern of similarity of the observations a...