<abstract><p>Let $ T:X\to Y be a bounded linear operator between Banach spaces X, $. A vector x_0\in {\mathsf{S}}_X in the unit sphere of X is called supporting T provided that \|T(x_0)\| = \sup\{\|T(x)\|:\|x\| 1\} \|T\| Since matrices induce operators finite-dimensional Hilbert spaces, we can consider their vectors. In this manuscript, unveil relationship principal components matri...

Let X be an n× p matrix whose rows are iid random vectors Xi· with mean μ ∈ R and covariance Σ ∈ Sp+— for example, they might be (Xi·) iid ∼ No(μ,Σ). For many problems (such as multivariate regression of some Y on X) we might wish to reduce the dimension p of these rows. For example, if we have a vector of p = 1000 possible explanatory variables about each individual, we may hope that a small s...

In ordinary least squares regression, dimensionality is a sensitive issue. As the number of independent variables approaches the sample size, the least squares algorithm could easily fail, i.e., estimates are not unique or very unstable, (Draper and Smith, 1981). There are several problems usually encountered in modeling high dimensional data, including the difficulty of visualizing the data, s...

In this paper, we address the problem of dimension reduction for sequentially observed functional data (X k : k ∈ Z). Such functional time series arise frequently, e.g., when a continuous time process is segmented into some smaller natural units, such as days. Then each X k represents one intraday curve. We argue that functional principal component analysis (FPCA), though a key technique in the...

Principal Component Analysis (PCA) is a very successful dimensionality reduction technique, widely used in predictive modeling. A key factor in its widespread use in this domain is the fact that the projection of a dataset onto its first K principal components minimizes the sum of squared errors between the original data and the projected data over all possible rank K projections. Thus, PCA pro...