## 1 Introduction  
Discerning the major components of a point cloud is of importance for finding patterns in the cloud and for compressing it. Principal Component Analysis (PCA) is a widely used successful tool to determine the direction, spread, and dimensionality of point clouds [1; 2]). In its standard formulation, PCA is based on analysis in the square of the \(\ell^{2}\) norm applicable to data from distributions with finite second-order moments. This results in excellent performance when the point cloud has Gaussian structure or the structure of a distribution close to Gaussian. However, point clouds obtained under conditions other than benign laboratory conditions often contain significant numbers of outliers and the points may follow a heavy-tailed distribution that does not have finite second-order moments, which strongly limits the accuracy of standard PCA or prevents it from being applicable.  
To remedy this situation, robust PCAs [3; 4; 5; 6] and especially robust PCAs involving the \(\ell^{1}\) norm [7; 8; 9; 10] have been investigated over the past few years. In most of the \(\ell^{1}\) reformulations of PCA that have been proposed, the \(\ell^{1}\) norm is applied only to parts of the PCA process. An analytical connection with heavy-tailed statistics is not present in any of these reformulations. Much of the previous work on \(\ell^{1}\) methods for PCA and in other areas has been carried out under the assumption of sparsity of the principal components or of the error. But the principal components and the error are often not sparse. While an assumption of sparsity is common in many areas (for example, in compressed sensing) and can lead to meaningful analytical and computational results in those areas, it restricts the set of situations that a reformulated PCA can address. Finally, neither standard PCA nor any of the reformulated robust PCAs are able to provide reliable information for distributions with two or more irregularly spaced main directions (spokes, such as from superimposing several classical distributions).  
We hypothesize that \(\ell^{2}\)-based concepts such as singular values, inner products, orthogonal projection, averaging and second-order moments (variances and covariances) in the reformulated PCAs that have been investigated in the recent literature are limiting factors in the applicability of these PCAs to realistic outlier-rich point clouds that occur in geometric modeling, image analysis, object and face recognition, data mining, network analysis and many other areas. To remedy this situation, we carry out here a fundamental reformulation of the PCA approach in a framework based not just in part but in total on the \(\ell^{1}\) norm. The \(\ell^{1}\) norm is chosen because it is an appropriate norm when the data cloud has a significant number of outliers, either artificial outliers or outliers of a heavy-tailed statistical distribution. While analysis of multivariate heavy-tailed distributions is still underdeveloped, there has been progress [11; 12; 13; 14]). Our \(\ell^{1}\) approach here is related to the approach by which \(L_{1}\) splines, a new class of splines that preserve shape for highly irregular data, have been created over the past 12 years [15; 16; 17; 18; 19].