## 2 Standard PCA and Recently Developed Robust PCAs
This method uses inner products (a matrix or a vector times a matrix), which are \(\ell^{2}\) operations that do not exist in an \(\ell^{1}\) space. Here again, we surmise that a procedure that avoids use of all \(\ell^{2}\) quantities and is based completely on the \(\ell^{1}\) metric will have better performance. [PERSON] and [PERSON] [4] calculate robust estimates of the eigenvalues and eigenvectors of the covariance matrix without estimating the covariance matrix itself. Their method is based on a projection-pursuit approach developed by [PERSON] and [PERSON] [20]. In this approach, one searches for directions for which the data, projected onto these directions, have maximal dispersion. This dispersion is measured not by the variance but by a robust scale estimator \(S_{n}\). For data \(\{\mathbf{x}_{m}\}_{m=0}^{M-1}\), the estimate of the first eigenvector is defined to be  
\[v_{S_{n},1}=\text{argmax}_{\|a\|=1}S_{n}(a^{T}\mathbf{x}_{0},a^{T}\mathbf{x}_ {1},\ldots,a^{T}\mathbf{x}_{M-1}) \tag{2}\]  
and the associated eigenvalue is  
\[\lambda_{S_{n},1}=S_{n}^{2}((v_{S_{n},1})^{T}\mathbf{x}_{0},(v_{S_{n},1})^{T} \mathbf{x}_{1},\ldots,(v_{S_{n},1})^{T}\mathbf{x}_{M-1}) \tag{3}\]  
Subsequent eigenvectors are determined by searching in an analogous manner over subspaces orthogonal to all subspaces already found. [PERSON] and [PERSON] [4] made the projection-pursuit process of [PERSON] and [PERSON] more efficient by approximating on each step a full-space search for the vector \(a\) by a search over a finite set while retaining a high finite-sample breakdown point. Finally, [PERSON] and [PERSON] [8,9] seek a \(UV\) factorization of the data matrix \(X=(\mathbf{x}_{0},\mathbf{x}_{1},\ldots,\mathbf{x}_{M-1})\) such that  
\[\|X-UV^{T}\|_{1} \tag{4}\]  
is minimized. This method uses the \(\ell^{1}\) norm but also involves \(\ell^{2}\) inner products (matrix multiplication). The minimization of expression (4) is non-convex, so [PERSON] and [PERSON] calculate \(U\) and \(V\) iteratively with a random initialization of \(U\), optimizing one matrix while keeping the other one fixed, in the following manner. Given \(U=U^{(k)}\), \(V^{(k)}\) is defined to be  
\[V^{(k)}=\text{argmin}_{V}\|X-U^{(k)}V^{T}\|_{1} \tag{5}\]  
The next \(U\), that is, \(U^{(k+1)}\) is defined to be  
\[U^{(k+1)}=\text{argmin}_{U}\|X-UV^{(k)^{T}}\|_{1} \tag{6}\]  
These two \(\ell^{1}\) minimization problems can be decomposed into independent minimization problems that [PERSON] and [PERSON] solve by linear programming or, approximately, by quadratic programming. In addition to the issues mentioned above, currently available robust PCAs are not able to meaningfully handle data from distributions with multiple, irregularly spaced main directions or \"spokes,\" that is, directions in which the level surfaces of the probability density function extend further out from the central point than in neighboring directions, a situation that is increasingly common for some physical and a lot of non-physical (sociological, cognitive, human-behavior, _etc._) data. These PCAs provide only one main direction that accounts for the largest variability in the data and additional orthogonal directions that account for the remaining variability. But these directions may provide very little information about the data. In the literature, one extension of PCA that detects orthogonal spokes using a union-of-subspaces model is available [21]. However, this extension does not involve the \(\ell^{1}\) norm and is therefore not robust when a significant number of outliers is present. It is possible that \"superstructure\" could be added to standard PCA and to robust PCAs to allow them to calculate multiple non-orthogonal spokes, but such generalizations of these methods are not yet available in the literature. There is nevertheless a need to develop a method that can calculate the directions and spreads of multiple major components.