rNMF (version 0.5.0)

rnmf: Robust Penalized Nonnegative Matrix Factorization (rNMF).

Description

rNMF performs robust penalized nonnegative matrix factorizaion on a nonnegative matrix X to obtain low dimensional nonnegative matrices W and H, such that X ~ WH, while detecting and trimming different types of outliers in X.

Usage

rnmf(X, k = 5, alpha = 0, beta = 0, maxit = 20, tol = 0.001, gamma = FALSE, ini.W = NULL, ini.zeta = NULL, my.seed = NULL, variation = "cell", quiet = FALSE, nreg = 1, showprogress = TRUE)

Arguments

X

a 'p by n' nonnegative numerical matrix to be decomposed into X ~ WH.

k

the integer dimension to which X is projected (equals the number of columns of W and the number of rows of H, and smaller than min(p,n)). Default = 5.

alpha

a nonnegative number that controls the magnitude of W. The larger the alpha is, the higher penalty on ||W|| is, or ||W|| is forced to be smaller. Default = 0, representing no penalty on the magnitude ||W||.

beta

a nonnegative number that controls the sparsity of H. Default= 0. The larger the beta is, the higher penalty on sum_i ||H_j|| is, where H_j is the j-th column of H. Default = 0, representing no penalty on the sparsity of H.

maxit

the maximum number of iterations. Default = 20. The algorithm is done as follows: 1) fits W given H, then trims outliers in X, i.e. ones with large residuals from X - WH, then refits W without the outliers [this step can be repeated 'nreg' times, currently nreg = 1], then 2) repeat as in 1) with the roles of W and H swapped, and then iterate through 1) and 2) until convergence. Default = 20, allowing a maximum of 20 pairs of 1) and 2).

tol

the convergence tolerance. We suggest it to be a positive number smaller than 0.01. Default = 0.001.

gamma

a trimming percentage in (0, 1) of X. Default = 0.05 will trim 5% of elements of X (measured by cell, row or column as specified in 'variation'). If trim=0, there is no trim; so rNMF then performs the regular NMF.

ini.W

the initialization of the left matrix W, which is a "p by k" nonnegative numeric matrix. Default = NULL directs the algorithm to randomly generate an ini.W.

ini.zeta

a "p by n" logical matrix of True or False, indicating the initial locations of the outliers. The number of "TRUE"s in ini.zeta must be less than or equal to m = the rounded integer of (gamma * p * n). Default = NULL, initializes the cells as TRUE with the m largest residuals after the first iteration. Required only for "cell" trimming option.

my.seed

the random seed for initialization of W. If left to be NULL(default) a random seed is used. Required only if ini.W is not NULL.

variation

a character string indicating which trimming variation is used. The options are: 'cell' (default), 'col', 'row' and 'smooth'.

quiet

default = FALSE indicating a report would be given at the end of execution. If quiet = TRUE, no report is provide at the end.

nreg

the number of inner loop iterations [see 'maxit' above] to find outliers in X, given either H or W. Default = 1, is currently only implemented option in the "cell" variation.

showprogress

default = TRUE, shows a progress bar during iterations. If showprogress = FALSE, no progress bar is shown.

Value

An object of class 'rnmf', which is a list of the following items:

• W: left matrix of the decomposition X ~ WH, columns of which (i.e. W) are basis vectors of the low dimension projection.
• H: right matrix of the decomposition X ~ WH, columns of which (i.e. W) are low dimensional encoding of the data.
• fit: the fitted matrix W %*% H.
• trimmed: a list of locations of trimmed cells in each iteration.
• niter: the number of iterations performed.

Details

rNMF decomposes a nonnegative p by n data matrix X as X ~ WH and detect and trims outliers. Here W and H are p by k and k by n nonnegative matrices, respectively; and k <= min{p,="" n}="" is="" the="" dimension="" of="" subspace="" to="" which="" x="" projected.="" objective="" function="" is<="" p="">

||X - WH||_gamma + alpha * ||W||_2^2 + beta * sum(|H_.j|)^2

where alpha controls the magnitude of W, and beta controls the sparsity of H. The algorithm iteratively updates W, H and the outlier set zeta with alternating conditional nonnegative least square fittings until convergence.

Four variations of trimming are included in the algorithm: "cell", "row", "column" and "smooth". Specifically, the "cell" variation trims individual cell-wise outliers while "row" and "column" variations trim entire row or column outliers. The fourth variation "smooth" fills the cells that are declared outliers in each iteration by the average of the surrounding cells.

Examples

## Load a clean single simulated tumor image.
data("tumor")
image(tumor) # look at the tumor image
dim(tumor) # it is a '70 by 70' matrix
## Add 5\% corruptions.
tumor.corrupted <- tumor
set.seed(1)
tumor.corrupted[sample(1:4900, round(0.05 * 4900), replace = FALSE)] <- 1
## Run rnmf with different settings
# No trimming
res.rnmf1 <- rnmf(X = tumor.corrupted, gamma = FALSE, my.seed = 1)
# 6 percent trimming, low dimension k = 5 (default)
res.rnmf2 <- rnmf(X = tumor.corrupted, tol = 0.001, gamma = 0.06, my.seed = 1)
# add sparsity constraint of H (beta = 0.1) with k = 10, and the "smooth" variation.
res.rnmf3 <- rnmf(X = tumor.corrupted, k = 10, beta = 0.1,
                  tol = 0.001, gamma = 0.06, my.seed = 1,
                  variation = "smooth", maxit = 30)

## Show results:
par(mfrow = c(2,2), mar = c(2,2,2,2))
image(tumor.corrupted, main = "Corrupted tumor image", xaxt = "n", yaxt = "n")
image(res.rnmf1$fit, main = "rnmf (no trimming) fit", xaxt = "n", yaxt = "n")
image(res.rnmf2$fit, main = "rnmf (cell) fit 2", xaxt = "n", yaxt = "n")
image(res.rnmf3$fit, main = "rnmf (smooth) fit 3", xaxt = "n", yaxt = "n")