rsvd (version 1.0.5)

rcur: Randomized CUR matrix decomposition.

Description

Randomized CUR matrix decomposition.

Usage

rcur(A, k = NULL, 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; target rank of the low-rank approximation, i.e., the number of columns/rows to be selected. It is required that \(k\) is smaller or equal to \(min(m,n)\).

p

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

q

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

idx_only

bool, optional; if (\(TRUE\)), only the index set C.idx and R.idx is returned, but not the matrices C and 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

rcur returns a list with class \(id\) containing the following components:

C

array_like; column subset \(C = A[,C.idx]\); \((m, k)\) dimensional array.

R

Array_like. row subset \(R = A[R.idx, ]\); \((k, n)\) dimensional array.

U

array_like; connector matrix; \((k,k)\) dimensional array.

C.idx

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

R.idx

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

C.scores

array_like; scores of the selected columns.

R.scores

array_like; scores of the selected rows.

Details

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

$$A = C * U * R$$

using the rid decomposition. The factor matrix \(C\) is formed using actual columns of \(A\), also called the partial column skeleton. The factor matrix \(R\) is formed using actual rows of \(A\), also called the partial row skeleton.

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

References

• [1] 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.

• [2] 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).

See Also

rid

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 <- rcur(tiger, k = 150)

# Reconstruct image
tiger.re <- out$C %*% out$U %*% out$R

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

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