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...

Multivariate Gaussian data is completely characterized by its mean and covariance, yet modern non-Gaussian data makes higher-order statistics such as cumulants inevitable. For univariate data, the third and fourth scalar-valued cumulants are relatively well-studied as skewness and kurtosis. For multivariate data, these cumulants are tensor-valued, higher-order analogs of the covariance matrix c...

Principal Component Analysis (PCA) is a canonical and well-studied tool for dimensionality reduction. However, when few data are available, the poor quality of the covariance estimator at its core may compromise its performance. We leverage this issue by casting the PCA into a multitask framework, and doing so, we show how to solve simultaneously several related PCA problems. Hence, we propose ...

A new method for performing a nonlinear form of Principal Component Analysis is proposed. By the use of integral operator kernel functions, one can e ciently compute principal components in high{ dimensional feature spaces, related to input space by some nonlinear map; for instance the space of all possible d{pixel products in images. We give the derivation of the method and present experimenta...

Many tasks involving high-dimensional data, such as face recognition, suffer from the curse of dimensionality: the number of training samples required to accurately learn a classifier increases exponentially with the dimensionality of the data. Structured Principal Component Analysis (SPCA) reduces the dimensionality of the data by choosing a small number of features to represent larger sets of...

We present a new technique called contrastive principal component analysis (cPCA) that is designed to discover low-dimensional structure that is unique to a dataset, or enriched in one dataset relative to other data. The technique is a generalization of standard PCA, for the setting where multiple datasets are available – e.g. a treatment and a control group, or a mixed versus a homogeneous pop...