## 2 Standard PCA and Recently Developed Robust PCAs  
In its standard formulation [2], PCA is designed to create uncorrelated components (\"orthogonal solutions\"). There are variants of PCA that create correlated components (\"oblique solutions\"), but we do not consider them here. Principal components are normally calculated using the singular value decomposition of the data matrix, an efficient and stable numerical procedure. Standard PCA can be summarized as follows:  
1. Calculate the mean of the point cloud and subtract it out of the data. 2. Set up the matrix \(X\) of the data that result from Step 1. The rows of \(X\) are the data vectors. 3. Calculate the singular values of \(X\), that is, the diagonal elements of the matrix \(\Sigma\) in the singular value decomposition \(X=U\Sigma V^{T}\). (These singular values are the eigenvalues of the covariance matrix of the data.) 4. Order the components (the columns of the matrix \(V\) in the singular value decomposition) in descending order of the singular values. 5. Select basis vectors to be the components corresponding to the largest singular values. 6. Conduct further processing (for example, project the data onto the basis consisting of the vectors selected in Step 5). Step 3 makes the limitations of standard PCA apparent. For the singular values to be meaningful, the covariance matrix of the continuum distribution from which the samples come needs to exist, that is, be finite. There are heavy-tailed distributions for which covariance matrices exist and others for which they do not. When there are a significant number of outliers in the data, an assertion that the covariance matrix of the continuum distribution exists can be dubious. A large number of outliers in the data is often an indication that the data come from a continuum heavy-tailed distribution for which the covariance matrix may not exist (has infinite entries). While the covariance matrix of a finite sample always exists, it is meaningful only if the covariance matrix of the underlying continuum distribution exists. In the remainder of this section, we discuss four robust variants of PCA that are currently in use. [PERSON] _et al_. [7] use the \"nuclear norm\" of a matrix in their robust PCA. The nuclear norm of a matrix \(P\), denoted by \(\|\,P\,\|_{\bullet}\), is the sum of the singular values of \(P\). Let \(\|\,E\,\|_{1}\) denote the sum of the absolute values of the entries of a matrix \(E\). Let \(D\) be the data matrix, \(P\) be the matrix of principal components, and \(E\) be the error matrix. [PERSON] _et al_. formulate the robust PCA problem as minimization of  
\[\|\,P\,\|_{\bullet}\,+\lambda\,\,\|\,E\,\|_{1}\quad\text{ subject to }P+E=D \tag{1}\]  
under assumptions that \(P\) is of low rank and \(E\) is sparse. Here, \(\lambda\) is a prescribed parameter based on the size of the data set [7]. Using an \(\ell^{1}\) norm to replace an \(\ell^{2}\) norm in a computational method is often a good procedure for robustifying the method. Minimization of expression (1) is seemingly based on the \(\ell^{1}\) norm, since both of the norms that occur in the objective-functional portion of expression (1) consist of sums of absolute values. However, the nuclear norm is not an \(\ell^{1}\) metric but only a \"pseudo-\(\ell^{1}\)\" metric because the singular values on which it is based are created by an \(\ell^{2}\)-based process. Moreover, the arithmetic mean used to center the data is also an \(\ell^{2}\), not an \(\ell^{1}\) quantity. It is our hypothesis that a procedure that avoids use of singular values, which are \(\ell^{2}\) quantities, and is based completely on themetric will provide more accurate output information about realistic outlier-rich data than robust PCAs that are based on the use of singular values. [PERSON] [10] proposes maximizing the \(\ell^{1}\) norm of the product of a vector and the data matrix and provides face recognition results that indicate success.