rcur {rsvd}

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 NAs 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.

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.

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.

Author(s)

N. Benjamin Erichson, [e-mail]

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. doi: 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))

## End(Not run)