R: Randomized interpolative decomposition (ID).

rid {rsvd} | R Documentation

## 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 NAs 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.

### 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.

### 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.

### Author(s)

N. Benjamin Erichson, erichson@uw.edu

### References

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

- [2] N. B. Erichson, S. Voronin, S. Brunton, J. N. Kutz. "Randomized matrix decompositions using R" (2016). (available at 'arXiv http://arxiv.org/abs/1608.02148).

### 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))

    ## End(Not run)

[Package rsvd version 0.9 Index]