DOCUMENTATION
Sign In

Get access to My SAS, trials, communities and more.

Machine Learning Procedures
 

RPCA Procedure

Details: RPCA Procedure

Robust principal component analysis (RPCA) is a matrix decomposition algorithm that decomposes an input matrix M into a low-rank matrix upper L 0 and a sparse matrix upper S 0, where upper M equals upper L 0 plus upper S 0. This decomposition is obtained by solving a convex programming problem called principal component pursuit (PCP).

Let StartAbsoluteValue EndAbsoluteValue upper L StartAbsoluteValue EndAbsoluteValue Subscript asterisk colon equals sigma-summation Underscript i Endscripts rho Subscript i Baseline left-parenthesis upper L right-parenthesis denote the nuclear norm of the matrix L (that is, the sum of the singular values of L), and let StartAbsoluteValue EndAbsoluteValue upper S StartAbsoluteValue EndAbsoluteValue Subscript 1 Baseline equals sigma-summation Underscript i j Endscripts StartAbsoluteValue upper S Subscript i j Baseline EndAbsoluteValue denote the l 1 norm of S. In this case, PCP can be formulated as

StartLayout 1st Row 1st Column minimize 2nd Column StartAbsoluteValue EndAbsoluteValue upper L StartAbsoluteValue EndAbsoluteValue Subscript asterisk plus lamda StartAbsoluteValue EndAbsoluteValue upper S StartAbsoluteValue EndAbsoluteValue Subscript 1 2nd Row 1st Column subject to 2nd Column upper L plus upper S equals upper M EndLayout

For more information about this formulation, see Candès et al. (2011).

RPCA is robust because the principal components are computed from the low-rank matrix. The sparse matrix includes the outlier values.[10] As a result, many applications of RPCA focus on the sparse matrix.

Two algorithms are implemented in the RPCA procedure: the augmented Lagrange multiplier algorithm and the accelerated proximal gradient algorithm.


[10] The sparse matrix can also include noise. The sparsity of the sparse matrix depends highly on the settings of the lamda and lamda Subscript normal w normal e normal i normal g normal h normal t parameters.

Last updated: August 06, 2026