Assume that we have -dimensional data vectors , which form the data matrix = . The matrix is decomposed into

where is a matrix, is a matrix and . Principal subspace methods [6,4] find and such that the reconstruction error

is minimized. There denotes the Frobenius norm, and , , and elements of the matrices , , and , respectively. Typically the row-wise mean is removed from as a preprocessing step.

Without any further constraints, there exist infinitely many ways to perform
such a decomposition. However, the subspace spanned by the column vectors of
the matrix
, called the *principal subspace*, is unique.
In PCA, these vectors are mutually orthogonal and have unit length.
Further, for each
, the first vectors form the -dimensional
principal subspace. This makes the solution practically
unique, see [4,2,5] for details.

There are many ways to determine the principal subspace and components [6,4,2]. We will discuss three common methods that can be adapted for the case of missing values.