Package 'filling' reference manual

Title:	Matrix Completion, Imputation, and Inpainting Methods
Description:	Filling in the missing entries of a partially observed data is one of fundamental problems in various disciplines of mathematical science. For many cases, data at our interests have canonical form of matrix in that the problem is posed upon a matrix with missing values to fill in the entries under preset assumptions and models. We provide a collection of methods from multiple disciplines under Matrix Completion, Imputation, and Inpainting. See Davenport and Romberg (2016) <doi:10.1109/JSTSP.2016.2539100> for an overview of the topic.
Authors:	Kisung You [aut, cre]
Maintainer:	Kisung You <[email protected]>
License:	MIT + file LICENSE
Version:	0.2.3
Built:	2024-10-13 07:49:25 UTC
Source:	CRAN

Matrix Completion, Imputation, and Inpainting Methods

Description

Filling in the missing entries of a partially observed data is one of fundamental problems in various disciplines of mathematical science. For many cases, data at our interests have canonical form of matrix in that the problem is posed upon a matrix with missing values to fill in the entries under preset assumptions and models. We provide a collection of methods from multiple disciplines under Matrix Completion, Imputation, and Inpainting. Currently, we have following methods implemented,

Name of Function	Method
`fill.HardImpute`	Generalized Spectral Regularization
`fill.KNNimpute`	Weighted $K$ -nearest Neighbors
`fill.nuclear`	Nuclear Norm Optimization
`fill.OptSpace`	OptSpace
`fill.simple`	Simple Rules of Mean, Median, and Random
`fill.SoftImpute`	Spectral Regularization
`fill.SVDimpute`	Iterative Regression against Right Singular Vectors
`fill.SVT`	Singular Value Thresholding for Nuclear Norm Optimization
`fill.USVT`	Universal Singular Value Thresholding

Randomly assign NAs to the data matrix with probability `x`

Description

aux.rndmissing randomly selects $100\cdot x$ % of entries from a given data matrix and turns them into missing entries, i.e., their values become NA.

Usage

aux.rndmissing(A, x = 0.1)
aux.rndmissing(A, x = 0.1)

Arguments

`A`	an $(n\times p)$ data matrix.
`x`	percentage of turning current entries into missing (`NA`).

Value

an $(n\times p)$ data matrix with missing entries at proportion $x$ .

Examples

# load lena64 image matrix
data(lena64)

# generate 10% of missing values
lena64_miss <- aux.rndmissing(lena64)

# visualize
par(mfrow=c(1,2))
image(lena64, axes=FALSE, main="original image")
image(lena64_miss, axes=FALSE, main="10% missing entries")

# load lena64 image matrix
data(lena64)

# generate 10% of missing values
lena64_miss <- aux.rndmissing(lena64)

# visualize
par(mfrow=c(1,2))
image(lena64, axes=FALSE, main="original image")
image(lena64_miss, axes=FALSE, main="10% missing entries")

HardImpute : Generalized Spectral Regularization

Description

If the assumed underlying model has sufficiently many zeros, the LASSO type shrinkage estimator is known to overestimate the number of non-zero coefficients. fill.HardImpute aims at overcoming such difficulty via low-rank assumption and hard thresholding idea, well-known concept in conventional regression analysis. In algorithmic aspect, it takes output of SoftImpute as warm-start matrices for iterative estimation process.

Usage

fill.HardImpute(
  A,
  lambdas = c(10, 1, 0.1),
  maxiter = 100,
  tol = 0.001,
  rk = (min(dim(A)) - 1)
)
fill.HardImpute(
  A,
  lambdas = c(10, 1, 0.1),
  maxiter = 100,
  tol = 0.001,
  rk = (min(dim(A)) - 1)
)

Arguments

`A`	an $(n\times p)$ partially observed matrix.
`lambdas`	a length- $t$ vector regularization parameters.
`maxiter`	maximum number of iterations to be performed.
`tol`	stopping criterion for an incremental progress.
`rk`	assumed rank of the matrix.

Value

a named list containing

X: an $(n\times p\times t)$ cubic array after completion at each lambda value.

References

Mazumder R, Hastie T, Tibshirani R (2010). “Spectral Regularization Algorithms for Learning Large Incomplete Matrices.” J. Mach. Learn. Res., 11, 2287–2322. ISSN 1532-4435.

Examples

## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
A <- aux.rndmissing(lena128, x=0.05)

## apply the method with 3 rank conditions
fill1 <- fill.HardImpute(A, lambdas=c(500,100,50), rk=10)
fill2 <- fill.HardImpute(A, lambdas=c(500,100,50), rk=50)
fill3 <- fill.HardImpute(A, lambdas=c(500,100,50), rk=100)

## visualize only the last ones from each run
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(fill1$X[,,3], col=gray((0:100)/100), axes=FALSE, main="Rank 10")
image(fill2$X[,,3], col=gray((0:100)/100), axes=FALSE, main="Rank 50")
image(fill3$X[,,3], col=gray((0:100)/100), axes=FALSE, main="Rank 100")
par(opar)

## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
A <- aux.rndmissing(lena128, x=0.05)

## apply the method with 3 rank conditions
fill1 <- fill.HardImpute(A, lambdas=c(500,100,50), rk=10)
fill2 <- fill.HardImpute(A, lambdas=c(500,100,50), rk=50)
fill3 <- fill.HardImpute(A, lambdas=c(500,100,50), rk=100)

## visualize only the last ones from each run
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(fill1$X[,,3], col=gray((0:100)/100), axes=FALSE, main="Rank 10")
image(fill2$X[,,3], col=gray((0:100)/100), axes=FALSE, main="Rank 50")
image(fill3$X[,,3], col=gray((0:100)/100), axes=FALSE, main="Rank 100")
par(opar)

Imputation using Weighted K-nearest Neighbors

Description

One of the simplest idea to guess missing entry is to use portion of the data that has most similar characteristics across all covariates. fill.KNNimpute follows such reasoning in that it finds $K$ -nearest neighbors based on observed variables and uses weighted average of nearest elements to fill in the missing entry. Note that when there are many missing entries, it's possible that there are no surrogates to be computed upon. Therefore, if there exists an entire row or column full of missing entries, the algorithm stops.

Usage

fill.KNNimpute(A, k = ceiling(nrow(A)/2))
fill.KNNimpute(A, k = ceiling(nrow(A)/2))

Arguments

`A`	an $(n\times p)$ partially observed matrix.
`k`	the number of neighbors to use.

Value

a named list containing

X: an $(n\times p)$ matrix after completion.

References

Troyanskaya O, Cantor M, Sherlock G, Brown P, Hastie T, Tibshirani R, Botstein D, Altman RB (2001). “Missing value estimation methods for DNA microarrays.” Bioinformatics, 17(6), 520–525. ISSN 1367-4803.

Examples

## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
set.seed(5)
A <- aux.rndmissing(lena128, x=0.05)

## apply the method with 3 different neighborhood size
fill1 <- fill.KNNimpute(A, k=5)
fill2 <- fill.KNNimpute(A, k=25)
fill3 <- fill.KNNimpute(A, k=50)

## visualize only the last ones from each run
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(fill1$X, col=gray((0:100)/100), axes=FALSE, main="5-neighbor")
image(fill2$X, col=gray((0:100)/100), axes=FALSE, main="25-neighbor")
image(fill3$X, col=gray((0:100)/100), axes=FALSE, main="50-neighbor")
par(opar)

## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
set.seed(5)
A <- aux.rndmissing(lena128, x=0.05)

## apply the method with 3 different neighborhood size
fill1 <- fill.KNNimpute(A, k=5)
fill2 <- fill.KNNimpute(A, k=25)
fill3 <- fill.KNNimpute(A, k=50)

## visualize only the last ones from each run
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(fill1$X, col=gray((0:100)/100), axes=FALSE, main="5-neighbor")
image(fill2$X, col=gray((0:100)/100), axes=FALSE, main="25-neighbor")
image(fill3$X, col=gray((0:100)/100), axes=FALSE, main="50-neighbor")
par(opar)

Low-Rank Completion with Nuclear Norm Optimization

Description

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

$\|X\|_* = \sum \sigma_i(X)$

where $\sigma_i (X)$ 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.

Usage

fill.nuclear(A, tolerance = 0.001)
fill.nuclear(A, tolerance = 0.001)

Arguments

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

Value

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.

References

Candès EJ, Recht B (2009). “Exact Matrix Completion via Convex Optimization.” Foundations of Computational Mathematics, 9(6), 717–772. ISSN 1615-3375, 1615-3383.

Examples

## 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)

## 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)

OptSpace

Description

OptSpace is an algorithm for matrix completion when a matrix is partially observed. It performs what authors called trimming and projection repeatedly based on singular value decompositions. Original implementation is borrowed from ROptSpace package, which was independently developed by the maintainer. See OptSpace for more details.

Usage

fill.OptSpace(A, ropt = NA, niter = 50, tol = 1e-06)
fill.OptSpace(A, ropt = NA, niter = 50, tol = 1e-06)

Arguments

`A`	an $(n\times p)$ partially observed matrix.
`ropt`	`NA` to guess the rank, or a positive integer as a pre-defined rank.
`niter`	maximum number of iterations allowed.
`tol`	stopping criterion for reconstruction in Frobenius norm.

Value

a named list containing

X: an $(n\times p)$ matrix after completion.
error: a vector of reconstruction errors for each successive iteration.

References

Keshavan RH, Montanari A, Oh S (2010). “Matrix Completion From a Few Entries.” IEEE Transactions on Information Theory, 56(6), 2980–2998. ISSN 0018-9448.

Examples

## 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 with different rank assumptions
filled10 <- fill.OptSpace(A, ropt=10)
filled20 <- fill.OptSpace(A, ropt=20)

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(filled10$X, col=gray((0:100)/100), axes=FALSE, main="rank 10")
image(filled20$X, col=gray((0:100)/100), axes=FALSE, main="rank 20")
par(opar)

## End(Not run)


## 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 with different rank assumptions
filled10 <- fill.OptSpace(A, ropt=10)
filled20 <- fill.OptSpace(A, ropt=20)

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(filled10$X, col=gray((0:100)/100), axes=FALSE, main="rank 10")
image(filled20$X, col=gray((0:100)/100), axes=FALSE, main="rank 20")
par(opar)

## End(Not run)

Imputation by Simple Rules

Description

One of the most simplest ways to fill in the missing entries is to apply any simple rule for each variable. In this example, we provide 3 options, "mean", "median", and "random". It assumes that every column has at least one non-missing entries in that for each column, the rule is applied from the subset of non-missing values.

Usage

fill.simple(A, method = c("mean", "median", "random"))
fill.simple(A, method = c("mean", "median", "random"))

Arguments

`A`	an $(n\times p)$ partially observed matrix.
`method`	simple rule to fill in the missing entries in a columnwise manner.

Value

a named list containing

X: an $(n\times p)$ matrix after completion.

References

Gelman A, Hill J (2007). Data analysis using regression and multilevel/hierarchical models, Analytical methods for social research. Cambridge University Press, Cambridge ; New York. ISBN 978-0-521-86706-1 978-0-521-68689-1, OCLC: ocm67375137.

Examples

## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
A <- aux.rndmissing(lena128, x=0.05)

## apply all three methods#'
fill1 <- fill.simple(A, method="mean")
fill2 <- fill.simple(A, method="median")
fill3 <- fill.simple(A, method="random")

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="original")
image(fill1$X, col=gray((0:100)/100), axes=FALSE, main="method:mean")
image(fill2$X, col=gray((0:100)/100), axes=FALSE, main="method:median")
image(fill3$X, col=gray((0:100)/100), axes=FALSE, main="method:random")
par(opar)

## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
A <- aux.rndmissing(lena128, x=0.05)

## apply all three methods#'
fill1 <- fill.simple(A, method="mean")
fill2 <- fill.simple(A, method="median")
fill3 <- fill.simple(A, method="random")

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="original")
image(fill1$X, col=gray((0:100)/100), axes=FALSE, main="method:mean")
image(fill2$X, col=gray((0:100)/100), axes=FALSE, main="method:median")
image(fill3$X, col=gray((0:100)/100), axes=FALSE, main="method:random")
par(opar)

SoftImpute : Spectral Regularization

Description

fill.SoftImpute implements convex relaxation techniques to generate a sequence of regularized low-rank solutions for matrix completion problems. For the nuclear norm optimization problem, it uses soft thresholding technique iteratively in that the algorithm returns several matrices in accordance with the provided vector of regularization parameters $\lambda$ (lambdas).

Usage

fill.SoftImpute(A, lambdas = c(10, 1, 0.1), maxiter = 100, tol = 0.001)
fill.SoftImpute(A, lambdas = c(10, 1, 0.1), maxiter = 100, tol = 0.001)

Arguments

`A`	an $(n\times p)$ partially observed matrix.
`lambdas`	a length- $t$ vector regularization parameters.
`maxiter`	maximum number of iterations to be performed.
`tol`	stopping criterion for an incremental progress.

Value

a named list containing

X: an $(n\times p\times t)$ cubic array after completion at each lambda value.

References

Mazumder R, Hastie T, Tibshirani R (2010). “Spectral Regularization Algorithms for Learning Large Incomplete Matrices.” J. Mach. Learn. Res., 11, 2287–2322. ISSN 1532-4435.

Examples

## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
A <- aux.rndmissing(lena128, x=0.05)

## apply the method with 3 lambda values
fill <- fill.SoftImpute(A, lambdas=c(500,100,50))

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(fill$X[,,1], col=gray((0:100)/100), axes=FALSE, main="lambda=500")
image(fill$X[,,2], col=gray((0:100)/100), axes=FALSE, main="lambda=100")
image(fill$X[,,3], col=gray((0:100)/100), axes=FALSE, main="lambda=50")
par(opar)

## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
A <- aux.rndmissing(lena128, x=0.05)

## apply the method with 3 lambda values
fill <- fill.SoftImpute(A, lambdas=c(500,100,50))

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(fill$X[,,1], col=gray((0:100)/100), axes=FALSE, main="lambda=500")
image(fill$X[,,2], col=gray((0:100)/100), axes=FALSE, main="lambda=100")
image(fill$X[,,3], col=gray((0:100)/100), axes=FALSE, main="lambda=50")
par(opar)

Iterative Regression against Right Singular Vectors

Description

Singular Value Decomposition (SVD) is the best low-rank approximation of a given matrix. fill.SVDimpute exploits such idea. First, it starts with simple filling using column mean values for filling. Second, it finds SVD of a current matrix. Then, each row vector is regressed upon top- $k$ right singular vectors. Missing entries are then filled with predicted estimates.

Usage

fill.SVDimpute(A, k = ceiling(ncol(A)/2), maxiter = 100, tol = 0.01)
fill.SVDimpute(A, k = ceiling(ncol(A)/2), maxiter = 100, tol = 0.01)

Arguments

`A`	an $(n\times p)$ partially observed matrix.
`k`	the number of regressors to be used.
`maxiter`	maximum number of iterations to be performed.
`tol`	stopping criterion for an incremental progress.

Value

a named list containing

X: an $(n\times p)$ matrix after completion.

References

Examples

## Not run: 
## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
set.seed(5)
A <- aux.rndmissing(lena128, x=0.05)

## apply the method with 3 different number of regressors
fill1 <- fill.SVDimpute(A, k=5)
fill2 <- fill.SVDimpute(A, k=25)
fill3 <- fill.SVDimpute(A, k=50)

## visualize only the last ones from each run
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(fill1$X, col=gray((0:100)/100), axes=FALSE, main="5 regressors")
image(fill2$X, col=gray((0:100)/100), axes=FALSE, main="25 regressors")
image(fill3$X, col=gray((0:100)/100), axes=FALSE, main="50 regressors")
par(opar)

## End(Not run)


## Not run: 
## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
set.seed(5)
A <- aux.rndmissing(lena128, x=0.05)

## apply the method with 3 different number of regressors
fill1 <- fill.SVDimpute(A, k=5)
fill2 <- fill.SVDimpute(A, k=25)
fill3 <- fill.SVDimpute(A, k=50)

## visualize only the last ones from each run
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(fill1$X, col=gray((0:100)/100), axes=FALSE, main="5 regressors")
image(fill2$X, col=gray((0:100)/100), axes=FALSE, main="25 regressors")
image(fill3$X, col=gray((0:100)/100), axes=FALSE, main="50 regressors")
par(opar)

## End(Not run)

Singular Value Thresholding for Nuclear Norm Optimization

Description

fill.SVT is an iterative updating scheme for Nuclear Norm Minimization problem. An unconstrained parahrase of the problem introduced in fill.nuclear is

$\mathrm{minimize}\quad \frac{1}{2}\|P_{\Omega}(X-A) \|_F^2 + \lambda \| X \|_*$

where $P_{\Omega}(X)=X_{ij}$ if it is observed, or $0$ otherwise. It performs iterative shrinkage on newly computed singular values.

Usage

fill.SVT(A, lambda = 1, maxiter = 100, tol = 0.001)
fill.SVT(A, lambda = 1, maxiter = 100, tol = 0.001)

Arguments

`A`	an $(n\times p)$ partially observed matrix.
`lambda`	a regularization parameter.
`maxiter`	maximum number of iterations to be performed.
`tol`	stopping criterion for an incremental progress.

Value

a named list containing

X: an $(n\times p)$ matrix after completion.

References

Cai J, Candès EJ, Shen Z (2010). “A Singular Value Thresholding Algorithm for Matrix Completion.” SIAM Journal on Optimization, 20(4), 1956–1982. ISSN 1052-6234, 1095-7189.

Examples

## Not run: 
## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
A <- aux.rndmissing(lena128, x=0.05)

## apply the method
fill1 <- fill.SVT(A, lambda=0.1)
fill2 <- fill.SVT(A, lambda=1.0)
fill3 <- fill.SVT(A, lambda=20)

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(fill1$X, col=gray((0:100)/100), axes=FALSE, main="lbd=0.1")
image(fill2$X, col=gray((0:100)/100), axes=FALSE, main="lbd=1")
image(fill3$X, col=gray((0:100)/100), axes=FALSE, main="lbd=10")
par(opar)

## End(Not run)

## Not run: 
## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
A <- aux.rndmissing(lena128, x=0.05)

## apply the method
fill1 <- fill.SVT(A, lambda=0.1)
fill2 <- fill.SVT(A, lambda=1.0)
fill3 <- fill.SVT(A, lambda=20)

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(fill1$X, col=gray((0:100)/100), axes=FALSE, main="lbd=0.1")
image(fill2$X, col=gray((0:100)/100), axes=FALSE, main="lbd=1")
image(fill3$X, col=gray((0:100)/100), axes=FALSE, main="lbd=10")
par(opar)

## End(Not run)

Matrix Completion by Universal Singular Value Thresholding

Description

fill.USVT is a matrix estimation method suitable for low-rank structure. In the context of our package, we provide this method under the matrix completion problem category. It aims at exploiting the idea of thresholding the singular values to minimize the mean-squared error, defined as

$\mathrm{MSE}(\hat{A}):={E} \left\{ \frac{1}{np} \sum_{i=1}^{n} \sum_{j=1}^{p} (\hat{a}_{ij} - a_{ij})^2 \right\}$

where $A$ is an $(n\times p)$ matrix with some missing values and $\hat{A}$ is an estimate.

Usage

fill.USVT(A, eta = 0.01)
fill.USVT(A, eta = 0.01)

Arguments

`A`	an $(n\times p)$ partially observed matrix.
`eta`	control for thresholding $\in (0,1)$ .

Value

a named list containing

X: an $(n\times p)$ estimated matrix after completion, which is $\hat{A}$ in the equation above.

References

Chatterjee S (2015). “Matrix estimation by Universal Singular Value Thresholding.” Ann. Statist., 43(1), 177–214.

Examples

## Not run: 
## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
set.seed(5)
A <- aux.rndmissing(lena128, x=0.05)

## apply the method with 3 different control 'eta'
fill1 <- fill.USVT(A, eta=0.01)
fill2 <- fill.USVT(A, eta=0.5)
fill3 <- fill.USVT(A, eta=0.99)

## visualize only the last ones from each run
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(fill1$X, col=gray((0:100)/100), axes=FALSE, main="eta=0.01")
image(fill2$X, col=gray((0:100)/100), axes=FALSE, main="eta=0.5")
image(fill3$X, col=gray((0:100)/100), axes=FALSE, main="eta=0.99")
par(opar)

## End(Not run)

## Not run: 
## load image data of 'lena128'
data(lena128)

## transform 5% of entries into missing
set.seed(5)
A <- aux.rndmissing(lena128, x=0.05)

## apply the method with 3 different control 'eta'
fill1 <- fill.USVT(A, eta=0.01)
fill2 <- fill.USVT(A, eta=0.5)
fill3 <- fill.USVT(A, eta=0.99)

## visualize only the last ones from each run
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(A, col=gray((0:100)/100), axes=FALSE, main="5% missing")
image(fill1$X, col=gray((0:100)/100), axes=FALSE, main="eta=0.01")
image(fill2$X, col=gray((0:100)/100), axes=FALSE, main="eta=0.5")
image(fill3$X, col=gray((0:100)/100), axes=FALSE, main="eta=0.99")
par(opar)

## End(Not run)

lena image at size of $(128 \times 128)$

Description

Lena is probably one of the most well-known example in image processing and computer vision.

Usage

data(lena128)
data(lena128)

Format

matrix of size $(128\times 128)$

Source

USC SIPI Image Database.

References

Gonzalez, Rafael C. and Woods, Richard E. (2017) Digital Image Processing (4th ed.). ISBN 0133356728.

Examples

data(lena128)
image(lena128, col=gray((0:100)/100), axes=FALSE, main="lena128")
data(lena128)
image(lena128, col=gray((0:100)/100), axes=FALSE, main="lena128")

lena image at size of $(256 \times 256)$

Description

Lena is probably one of the most well-known example in image processing and computer vision.

Usage

data(lena256)
data(lena256)

Format

matrix of size $(256\times 256)$

Source

USC SIPI Image Database.

References

Gonzalez, Rafael C. and Woods, Richard E. (2017) Digital Image Processing (4th ed.). ISBN 0133356728.

Examples

data(lena256)
image(lena256, col=gray((0:100)/100), axes=FALSE, main="lena256")
data(lena256)
image(lena256, col=gray((0:100)/100), axes=FALSE, main="lena256")

lena image at size of $(64 \times 64)$

Description

Lena is probably one of the most well-known example in image processing and computer vision.

Usage

data(lena64)
data(lena64)

Format

matrix of size $(64\times 64)$

Source

USC SIPI Image Database.

References

Gonzalez, Rafael C. and Woods, Richard E. (2017) Digital Image Processing (4th ed.). ISBN 0133356728.

Examples

data(lena64)
image(lena64, col=gray((0:100)/100), axes=FALSE, main="lena64")
data(lena64)
image(lena64, col=gray((0:100)/100), axes=FALSE, main="lena64")

Package 'filling'

Help Index

Matrix Completion, Imputation, and Inpainting Methods

Description

Randomly assign NAs to the data matrix with probability x

Description

Usage

Arguments

Value

Examples

HardImpute : Generalized Spectral Regularization

Description

Usage

Arguments

Value

References

See Also

Examples

Imputation using Weighted K-nearest Neighbors

Description

Usage

Arguments

Value

References

See Also

Examples

Low-Rank Completion with Nuclear Norm Optimization

Description

Usage

Arguments

Value

References

Examples

OptSpace

Description

Usage

Arguments

Value

References

Examples

Imputation by Simple Rules

Description

Usage

Arguments

Value

References

Examples

SoftImpute : Spectral Regularization

Description

Usage

Arguments

Value

References

See Also

Examples

Iterative Regression against Right Singular Vectors

Description

Usage

Arguments

Value

References

See Also

Examples

Singular Value Thresholding for Nuclear Norm Optimization

Description

Usage

Arguments

Value

References

See Also

Examples

Matrix Completion by Universal Singular Value Thresholding

Description

Usage

Arguments

Value

References

Examples

lena image at size of (128×128)(128 \times 128)(128×128)

Description

Randomly assign NAs to the data matrix with probability `x`

lena image at size of $(128 \times 128)$

lena image at size of $(256 \times 256)$

lena image at size of $(64 \times 64)$