Abstract

It is possible to express a matrix as a weighed sum of outer products of column and row vectors. The set of column vectors forms a basis for the columns and the set of row vectors forms a basis for the rows. The number of column (row) vectors required to span the columns (rows) of the matrix is defined as the rank of the matrix. If the elements of the matrix have uncertainties due to measurement errors and/or computer representations, the determination of rank can be difficult. We define pseudorank as that number of columns (rows) vectors required to reproduce the matrix to within the uncertainties of measurement and reproduction. Matrices that occur in image analysis and pattern recognition are often of very high dimension but have low psuedorank. For such matrices, an outer product expansion provides a significant data compression. Below a technique for finding truncated outer product representation of such matrices is described. The determination of approximate singular value decompositions and pseudoinverses follows directly.

© 1985 Optical Society of America