## 1 Introduction  
In the analysis of human-based activities, including those in social media, one increasingly encounters \"irregular\" data, that is, data that follows a star-shaped statistical distribution with many irregularly positioned \"spokes\" and \"clutters\". The \"spokes\" are the data from distributions whose level surfaces of the probability density functions extend much further out in certain directions from the central point than other directions; and the \"clutters\" often consist of patterned outliers. The spokes and clutters often have special meaning but are currently difficult to recover. For example, the spokes in point clouds representing the intersection of roads and fences encountered in urban modeling contain the information about the roads and fences, while clutters in such data may represent obstructions between the sensing mechanism and the roads or fences. Analyzing the actual irregular structure in the point cloud is important for recovering semantic information out of the data. Classical principal component analysis (PCA) [1], which is based on the \(\ell_{2}\) norm, assumes few outliers and assumes a light-tailed statistical distribution similar to the elliptically shaped Gaussian, is not a meaningful approach for determining the major components of such data. However, point clouds obtained from reality often contain a large number of outliers and the points may be incomplete or follow a heavy-tailed distribution. To deal with this case, various types of \"robust principal component analysis\" (robust PCA) involving the \(\ell_{1}\) norm have been proposed and successfully developed [2; 3; 4; 5; 6; 7; 8; 9]. This shift from the \(\ell_{2}\) norm to the \(\ell_{1}\) norm is part of a larger tendency in recent research that has taken place also in signal and image processing [10; 11], compressive sensing [12; 13], shape-preserving geometric modeling [14; 15] and other areas.  
The robust PCAs that have been developed so far have been successful for many classes of problems but are not able to handle the data that (1) follows a star-shaped statistical distribution with multiple irregularly positioned spokes and (2) includes many statistical and patterned outliers. Our goal in this paper is to create a method for identifying the major components of such data in three and higher dimensions. In [16], [PERSON] _et al._ created an \(\ell_{1}\) major component detection and analysis (\(\ell_{1}\) MCDA) for 2D data that can handle data from distributions with irregularly positioned spokes. The \(\ell_{1}\) MCDA in [16] consists of the following steps:  
1. Calculate the central point of the data and subtract it out of the data.
2. Find a local neighbor of a direction.
3. Calculate the quadratic surface that best fits the data in the current local neighbor in \(\ell_{1}\) norm.
4. Determine the location of the maximum of the quadratic surface over the local neighbor. If the location of the maximum is strictly inside the local neighbor, go to Step 5; otherwise move to the direction on the boundary of the local neighbor and return to Step 2.
5. Calculate a new best-fit quadratic surface on a larger neighbor of the direction identified in Step 4. Output the maximum of the new quadratic surface.  
In this paper, we extend the 2D \(\ell_{1}\) MCDA of [16] to higher dimensions. Just like the 2D \(\ell_{1}\) MCDA of [16], the \(\ell_{1}\) MCDA for higher dimensions that we propose here involves a fundamental reformulation of all steps of PCA in a framework based on the assumption that the data points follow a heavy-tailed statistical distribution. The rest of this paper is arranged as follows. In Section 2, \(\ell_{1}\) MCDA in the \(n\)-dimensional space is derived. Computational results for different types of light-tailed and heavy-tailed distributions are presented in Section 3. Section 4 gives conclusions and future work.