Table 1.

Correspondences between L₁-PCA* and L₂-PCA for estimating the k-dimensional best-fit subspace

Concept	Formula
1 kth principal component loadings vector αk, k = 2, … ,m  (Set α1 orthogonal to α2, … , αm)	$\left({\prod }_{\ell =m+1}^{k+1}{V}^{\ell }\right)\frac{{\beta }^k}{{\mid \mid {\beta }^k\mid \mid }_2}$
2 Score of observation i  (from Step 4 of Algorithm L1-PCA*)	$x_i^k$
3 Projection of point xi for observation i  (in terms of original coordinates)	$\left({\prod }_{\ell =m+1}^{k+1}{V}^{\ell }\right)x_i^k$
4 Score of a new point xn+1	$\left({\prod }_{\ell =m+1}^{m+1}{\left(V^{\ell }\right)}^T{\left(I^{j\ast }\right)}^{\ell }\right)x_{n+1}$
5 Projection of a new point xn+1  (in terms of original coordinates)	$\left({\prod }_{r=m+1}^{k+1}{V}^r\right)\left({\prod }_{\ell =k+1}^{m+1}{\left(V^{\ell }\right)}^T{\left(I^{j\ast }\right)}^{\ell }\right)x_{n+1}$