Robust PCA Based on Incoherence With Geometrical Interpretation

Robust principal component analysis, which extracts low-dimensional data from high-dimensional data, can also be regarded as a source separation problem of the sparse error matrix and the low-rank matrix. Until recently, various methods have attempted to precisely predict the discrete rank function by assigning a weight to the nuclear norm. However, if the weights are not in ascending order, the algorithms will diverge and exhibit high computational complexity. Moreover, from the viewpoint of source separation, these methods overlook the fact that two components must be sufficiently different for accurate demixing. In this paper, we employ the incoherence term with convex shape, which considers that components must appear different from one another for boosting separability. Since it is intractable to directly exploit mutual incoherence defined in linear algebra, we guarantee the incoherence by indirectly making the sparse matrix lack the low-rank property by using the duality norm principle. This approach can also be associated with the null space. To analyze the results of the proposed algorithm geometrically, we measure the geodesic distance between the tangent spaces of the manifolds of two separate components. As this distance increases, the degree of dissimilarity of the two components is adequately assured; thus, separation succeeds. Furthermore, this paper is the first to provide insights into the relationship between source separation conditions and the derivatives of the nuclear norm and L₁ norm. Experiments are conducted on still image separation and background subtraction to confirm the superiority of the proposed methods both qualitatively and quantitatively.