BigDataStatMeth (version 1.0.3)

bdpseudoinv: Compute Matrix Pseudoinverse (In-Memory)

Description

Computes the Moore-Penrose pseudoinverse of a matrix using SVD decomposition. This implementation handles both square and rectangular matrices, and provides numerically stable results even for singular or near-singular matrices.

Usage

bdpseudoinv(X, threads = NULL)

Value

The pseudoinverse matrix of X.

Arguments

X

Numeric matrix or vector to be pseudoinverted.

threads

Optional integer. Number of threads for parallel computation. If NULL, uses maximum available threads.

Mathematical Details

• SVD decomposition: \(A = U \Sigma V^T\)

• Pseudoinverse: \(A^+ = V \Sigma^+ U^T\)

• \(\Sigma^+_{ii} = 1/\Sigma_{ii}\) if \(\Sigma_{ii} > \text{tolerance}\)

• \(\Sigma^+_{ii} = 0\) otherwise

Key features:

• Robust computation:

  • Handles singular and near-singular matrices

  • Automatic threshold for small singular values

  • Numerically stable implementation

• Implementation details:

  • Uses efficient SVD algorithms

  • Parallel processing support

  • Memory-efficient computation

  • Handles both dense and sparse inputs

The pseudoinverse satisfies the Moore-Penrose conditions:

• \(AA^+A = A\)

• \(A^+AA^+ = A^+\)

• \((AA^+)^* = AA^+\)

• \((A^+A)^* = A^+A\)

Details

The Moore-Penrose pseudoinverse (denoted A+) of a matrix A is computed using Singular Value Decomposition (SVD).

For a matrix A = USigmaV^T (where ^T denotes transpose), the pseudoinverse is computed as:

$$A^+ = V \Sigma^+ U^T$$

where Sigma+ is obtained by taking the reciprocal of non-zero singular values.

References

• Golub, G. H., & Van Loan, C. F. (2013). Matrix Computations, 4th Edition. Johns Hopkins University Press.

• Ben-Israel, A., & Greville, T. N. E. (2003). Generalized Inverses: Theory and Applications, 2nd Edition. Springer.

See Also

• bdpseudoinv_hdf5 for HDF5-stored matrices

• bdSVD_hdf5 for singular value decomposition

Examples

library(BigDataStatMeth)

# Create a singular matrix
X <- matrix(c(1,2,3,2,4,6), 2, 3)  # rank-deficient matrix

# Compute pseudoinverse
X_pinv <- bdpseudoinv(X)

# Verify Moore-Penrose conditions
# 1. X %*% X_pinv %*% X = X
all.equal(X %*% X_pinv %*% X, X)

# 2. X_pinv %*% X %*% X_pinv = X_pinv
all.equal(X_pinv %*% X %*% X_pinv, X_pinv)