## 3 Computational Experiments
: _In Tables 1-4, the radius information for the robust PCA methods [5, 17] was generated by Step 6 of the algorithm described in this paper with the major directions returned by their own. Some results are not available because the computational time exceeded the limit of 3600 s._  
**Remark 4**. : _In Tables 6-9, no results for either [PERSON] and [PERSON]'s robust PCA or [PERSON] et al. 's \(L_{1}\) PCA are presented, because these two methods provide only one major direction for the superimposed distributions and do not yield any meaningful information about the individual major direction._  
The results in Tables 1 and 2 indicate that, when there is only one major direction in the data, \(\ell_{1}\) MCDA obtains comparable results to [PERSON] and [PERSON]'s robust PCA in accuracy. Although [PERSON] _et al. 's \(L_{1}\)_ PCA outperforms the proposed \(\ell_{1}\) MCDA, it requires much longer computational time as shown in Table 5. The results in Tables 3 and 4 indicate that \(\ell_{1}\) MCDA outperforms [PERSON] and [PERSON]'s robust PCA in all cases, especially when the dimension is larger than 30. [PERSON] _et al. 's \(L_{1}\)_ PCA obtained similar results as \(\ell_{1}\) MCDA but consumed much more CPU time as shown in Table 5. It is noticeable that the accuracy of the proposed method decreases as the dimension increases. The reason is that for a given data size, the number of points falling into a fixed neighbor of the major direction decreases and the two-level median estimation deteriorates as the dimension grows higher. In contrast, the accuracy of the robust PCAs increases as the dimension increases because both robust PCAs used project-pursuit method, whose performance degrades when the underlying dimension decreases [17]. The results in Tables 6-9 indicate that our method can obtain good accuracy by using a small neighbor size (up to 2% of the size of given data) for the four-major-directions case. It is worth to point out that the four major directions are not orthogonal to each other, and the proposed \(\ell_{1}\) MCDA is very robust by noting that the accuracy of \(\ell_{1}\) MCDA for data with the artificial outliers is as good as the one of \(\ell_{1}\) MCDA without  
(1) it allows use of various distance functions in the data space (although we used \(\ell_{2}\) norm in this paper); (2) it works well for both heavy-tailed and light-tailed distributions; (3) it is applicable to data that have multiple spokes, contain patterned outliers, and do not necessarily have mutually orthogonal major directions; (4) it does not require the assumption of sparsity of the major components or of the error, and (5) it utilizes local information, instead of global information, in each iteration to improve the efficiency. These advantages allow \(\ell_{1}\) MCDA to be used in many circumstances in which robust PCAs cannot be used. For many geometric problems and for, perhaps, most \"soft\" problems (face and pattern recognition, image analysis, social network analysis, _etc._), irregularly positioned \"spokes\" with many outliers are likely to be commonly encountered. Identifying these spokes is part of a process of generating semantics from raw data. With the ability to recover complex spoke structures in spite of the presence of heavy-tailed statistical noise and of clutter, \(\ell_{1}\) MCDA shows its potential to generate better semantics from data than other methods. Topics that require further investigation include:  
* Use of alternative principles for calculation of the central point (for example, for V-shaped distributions or more complicated asymmetric distributions with spokes for which the median of the data is not a meaningful central point);
* Properties of \(\ell_{1}\) MCDA for high dimensions (\(10^{4}\) to \(10^{6}\));
* Ability of \(\ell_{1}\) MCDA to deal with missing data. Irregular, high-dimensional heavy-tailed distributions are likely to describe a large number of financial, economic, social, networking, and physical phenomena.