fill_nuclear: Low-Rank Completion with Nuclear Norm Optimization
In filling: Matrix Completion, Imputation, and Inpainting Methods

Description Usage Arguments Value References Examples

In many circumstances, it is natural to assume that there exists an underlying low-rank structure. The assumption of low-rank property leads to an optimization problem for matrix completion problem,

\mathrm{minimize}\quad rank(X)

\mathrm{s.t}~~ X_{ij}=A_{ij} ~~\mathrm{for}~~ A_{ij} \in E

where A_{ij}\in E means the (i,j)-th entry of data matrix A is not missing. The objective function can be further relaxed by nuclear norm

\|X\|_* = ∑ σ_i(X)

where σ_i (X) is i-th singular value of the matrix X. Note that for modeling purpose, we adopted closeness parameter tolerance for equality constraint. CVXR package was used in implementation. Computational efficiency may not be guaranteed for large data matrix.

1	fill.nuclear(A, tolerance = 0.001)

`A`	an (n\times p) partially observed matrix.
`tolerance`	level of tolerance for entrywise equality condition.

a named list containing

X: an (n\times p) matrix after completion.
norm: solution of the minimization problem; approximate rank.
cvxr.status: “optimal” denotes the problem was solved. See psolve for more details on solvability.
cvxr.niters: the number of iterations taken.
cvxr.solver: type of solver used by CVXR.

\insertRef

candes_exact_2009filling

## Not run: 
## load image data of 'lena64'
data(lena64)

## transform 5% of entries into missing
A <- aux.rndmissing(lena64, x=0.05)

## apply the method
filled <- fill.nuclear(A)

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(filled$X, col=gray((0:100)/100), axes=FALSE, main="processed")
par(opar)

## End(Not run)