rsvd (version 1.0.5)

rid: Randomized interpolative decomposition (ID).

Description

Randomized interpolative decomposition.

Usage

rid(A, k = NULL, mode = "column", p = 10, q = 0, idx_only = FALSE, rand = TRUE)

Arguments

A

array_like; numeric \((m, n)\) input matrix (or data frame). If the data contain \(NA\)s na.omit is applied.

k

integer, optional; number of rows/columns to be selected. It is required that \(k\) is smaller or equal to \(min(m,n)\).

mode

string c('column', 'row'), optional; columns or rows ID.

p

integer, optional; oversampling parameter (default \(p=10\)).

q

integer, optional. number of additional power iterations (default \(q=0\)).

idx_only

bool, optional; if (\(TRUE\)), the index set idx is returned, but not the matrix C or R. This is more memory efficient, when dealing with large-scale data.

rand

bool, optional; if (\(TRUE\)), a probabilistic strategy is used, otherwise a deterministic algorithm is used.

Value

rid returns a list containing the following components:

C

array_like; column subset \(C = A[,idx]\), if mode='column'; array with dimensions \((m, k)\).

R

array_like; row subset \(R = A[idx, ]\), if mode='row'; array with dimensions \((k, n)\).

Z

array_like; well conditioned matrix; Depending on the selected mode, this is an array with dimensions \((k,n)\) or \((m,k)\).

idx

array_like; index set of the \(k\) selected columns or rows used to form \(C\) or \(R\).

pivot

array_like; information on the pivoting strategy used during the decomposition.

scores

array_like; scores of the columns or rows of the input matrix \(A\).

scores.idx

array_like; scores of the \(k\) selected columns or rows in \(C\) or \(R\).

Details

Algorithm for computing the ID of a rectangular \((m, n)\) matrix \(A\), with target rank \(k << min(m,n)\). The input matrix is factored as

$$A = C * Z$$

using the column pivoted QR decomposition. The factor matrix \(C\) is formed as a subset of columns of \(A\), also called the partial column skeleton. If mode='row', then the input matrix is factored as

$$A = Z * R$$

using the row pivoted QR decomposition. The factor matrix \(R\) is now formed as a subset of rows of \(A\), also called the partial row skeleton. The factor matrix \(Z\) contains a \((k, k)\) identity matrix as a submatrix, and is well-conditioned.

If \(rand='TRUE'\) a probabilistic strategy is used to compute the decomposition, otherwise a deterministic algorithm is used.

References

• [1] N. Halko, P. Martinsson, and J. Tropp. "Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions" (2009). (available at arXiv https://arxiv.org/abs/0909.4061).

• [2] N. B. Erichson, S. Voronin, S. L. Brunton and J. N. Kutz. 2019. Randomized Matrix Decompositions Using R. Journal of Statistical Software, 89(11), 1-48. 10.18637/jss.v089.i11.

See Also

rcur,

Examples

# NOT RUN {
# Load test image
data("tiger")

# Compute (column) randomized interpolative decompsition
# Note that the image needs to be transposed for correct plotting
out <- rid(t(tiger), k = 150)

# Show selected columns 
tiger.partial <- matrix(0, 1200, 1600)
tiger.partial[,out$idx] <- t(tiger)[,out$idx]
image(t(tiger.partial), col = gray((0:255)/255), useRaster = TRUE)

# Reconstruct image
tiger.re <- t(out$C %*% out$Z)

# Compute relative error
print(norm(tiger-tiger.re, 'F') / norm(tiger, 'F')) 

# Plot approximated image
image(tiger.re, col = gray((0:255)/255))
# }