L1 Norm Based Principal Component Analysis and Its Multiresolution Analysis

The L
₁
norm based principal component analysis (PCA) is used for reducing the dimensions of the signals in many science and engineering disciplines recently. This is because the errors between the reconstructed vectors and the original training vectors are sparse. Different from the conventional L
₂
norm based PCA, as both the projection matrix and the reconstruction matrix are not necessary to be unitary, these two matrices are required to be determined. Another approach is to employ the same matrix for both the projection and the reconstruction. In this case, the total L
₁
norm error between the reconstructed vectors and the original training vectors is minimized subject to the exact unitary condition on the transform matrix. However, the errors between the reconstructed vectors and the original training vectors are not sparse. To address this issue, this paper minimizes both the total L
₁
norm error between the reconstructed vectors and the original training vectors as well as the unitary error of the transform matrix. Also, this concept is applied to the multiresolution analysis. That is, the training vectors are reduced from the first dimension to the second dimension and then from the second dimension to the third dimension.