Package 'softImpute'

Standardize a matrix to have optionally row means zero and variances one, and/or column means zero and variances one.

Description

A function for standardizing a matrix in a symmetric fashion. Generalizes the scale function in R. Works with matrices with NAs, matrices of class "Incomplete", and matrix in "sparseMatrix" format.

Usage

biScale(
  x,
  maxit = 20,
  thresh = 1e-09,
  row.center = TRUE,
  row.scale = TRUE,
  col.center = TRUE,
  col.scale = TRUE,
  trace = FALSE
)

Arguments

x	matrix, possibly with NAs, also of class "Incomplete" or "sparseMatrix" format.
maxit	When both row and column centering/scaling is requested, iteration is may be necessary
thresh	Convergence threshold
row.center	if row.center==TRUE (the default), row centering will be performed resulting in a matrix with row means zero. If row.center is a vector, it will be used to center the rows. If row.center=FALSE nothing is done. See details for more info.
row.scale	if row.scale==TRUE, the rows are scaled (after possibly centering, to have variance one. Alternatively, if a positive vector is supplied, it is used for row centering. See details for more info.
col.center	Similar to row.center
col.scale	Similar to row.scale
trace	with trace=TRUE, convergence progress is reported, when iteration is needed.

Details

This function computes a transformation

$\frac{X_{ij}-\alpha_i-\beta_j}{\gamma_i\tau_j}$

to transform the matrix $X$. It uses an iterative algorithm based on "method-of-moments". At each step, all but one of the parameter vectors is fixed, and the remaining vector is computed to solve the required condition. Although in genereal this is not guaranteed to converge, it mostly does, and quite rapidly. When there are convergence problems, remove some of the required constraints. When any of the row/column centers or scales are provided, they are used rather than estimated in the above model.

Value

A matrix like x is returned, with attributes:

biScale:row	a list with elements "center" and "scale" (the $alpha$ and $gamma$ above. If no centering was done, the center component will be a vector of zeros. Likewise, of no row scaling was done, the scale component will be a vector of ones.
biScale:column	Same details as biScale:row

For matrices with missing values, the constraints apply to the non-missing entries. If x is of class "sparseMatrix", then the sparsity is maintained, and an object of class "SparseplusLowRank" is returned, such that the low-rank part does the centering.

Note

This function will be described in detail in a forthcoming paper

Author(s)

Trevor Hastie, with help from Andreas Buja and Steven Boyd
, Maintainer: Trevor Hastie [email protected]

References

Trevor Hastie, Rahul Mazumder, Jason Lee, Reza Zadeh (2015) Matrix Completion and Low-rank SVD via Fast Alternating Least Squares, https://arxiv.org/abs/1410.2596
Journal of Machine Learning Research, 16, 3367-3402

See Also

softImpute,Incomplete,lambda0,impute,complete, and class "SparseplusLowRank"

Examples

set.seed(101)
n=200
p=100
J=50
np=n*p
missfrac=0.3
x=matrix(rnorm(n*J),n,J)%*%matrix(rnorm(J*p),J,p)+matrix(rnorm(np),n,p)/5
xc=biScale(x)
ix=seq(np)
imiss=sample(ix,np*missfrac,replace=FALSE)
xna=x
xna[imiss]=NA
xnab=biScale(xna,row.scale=FALSE,trace=TRUE)
xnaC=as(xna,"Incomplete")
xnaCb=biScale(xnaC)
nnz=trunc(np*.3)
inz=sample(seq(np),nnz,replace=FALSE)
i=row(x)[inz]
j=col(x)[inz]
x=rnorm(nnz)
xS=sparseMatrix(x=x,i=i,j=j)
xSb=biScale(xS)
class(xSb)

---

rdname softImpute-internal

Description

rdname softImpute-internal

Usage

clean.warm.start(a)

Arguments

a	an svd object with components u, d and v or NULL

---

Recompute the $d component of a "softImpute" object through regression.

Description

softImpute uses shrinkage when completing a matrix with missing values. This function debiases the singular values using ordinary least squares.

Usage

deBias(x, svdObject)

Arguments

x	matrix with missing entries, or a matrix of class "Incomplete"
svdObject	an SVD object, the output of softImpute

Details

Treating the "d" values as parameters, this function recomputes them by linear regression.

Value

An svd object is returned, with components "u", "d", and "v".

Author(s)

Trevor Hastie
Maintainer: Trevor Hastie [email protected]

Examples

set.seed(101)
n=200
p=100
J=50
np=n*p
missfrac=0.3
x=matrix(rnorm(n*J),n,J)%*%matrix(rnorm(J*p),J,p)+matrix(rnorm(np),n,p)/5
ix=seq(np)
imiss=sample(ix,np*missfrac,replace=FALSE)
xna=x
xna[imiss]=NA
fit1=softImpute(xna,rank=50,lambda=30)
fit1d=deBias(xna,fit1)

---

make predictions from an svd object

Description

These functions produce predictions from the low-rank solution of softImpute

Usage

impute(object, i, j, unscale = TRUE)

Arguments

object	an svd object with components u, d and v
i	vector of row indices for the locations to be predicted
j	vector of column indices for the locations to be predicted
unscale	if object has biScale attributes, and unscale=TRUE, the imputations reversed the centering and scaling on the predictions.

Details

impute returns a vector of predictions, using the reconstructed low-rank matrix representation represented by object. It is used by complete, which returns a complete matrix with all the missing values imputed.

Value

Either a vector of predictions or a complete matrix. WARNING: if x has large dimensions, the matrix returned by complete might be too large.

Author(s)

Trevor Hastie

See Also

softImpute, biScale and Incomplete

Examples

set.seed(101)
n=200
p=100
J=50
np=n*p
missfrac=0.3
x=matrix(rnorm(n*J),n,J)%*%matrix(rnorm(J*p),J,p)+matrix(rnorm(np),n,p)/5
ix=seq(np)
imiss=sample(ix,np*missfrac,replace=FALSE)
xna=x
xna[imiss]=NA
fit1=softImpute(xna,rank=50,lambda=30)
complete(xna,fit1)

---

create a matrix of class Incomplete

Description

creates an object of class Incomplete, which inherits from class dgCMatrix, a specific instance of class sparseMatrix

Usage

Incomplete(i, j, x)

Arguments

i	row indices
j	column indices
x	a vector of values

Details

The matrix is represented in sparse-matrix format, except the "zeros" represent missing values. Real zeros are represented explicitly as values.

Value

a matrix of class Incomplete which inherits from class dgCMatrix

Author(s)

Trevor Hastie and Rahul Mazumder

See Also

softImpute

Examples

set.seed(101)
n=200
p=100
J=50
np=n*p
missfrac=0.3
x=matrix(rnorm(n*J),n,J)%*%matrix(rnorm(J*p),J,p)+matrix(rnorm(np),n,p)/5
ix=seq(np)
imiss=sample(ix,np*missfrac,replace=FALSE)
xna=x
xna[imiss]=NA
xnaC=as(xna,"Incomplete")
### here we do it a different way to demonstrate Incomplete
### In practise the observed values are stored in this market-matrix format.
i = row(xna)[-imiss]
j = col(xna)[-imiss]
xnaC=Incomplete(i,j,x=x[-imiss])

---

Class "Incomplete"

Description

a sparse matrix inheriting from class dgCMatrix with the NAs represented as zeros

Objects from the Class

Objects can be created by calls of the form new("Incomplete", ...) or by calling the function Incomplete

Slots

i

Object of class "integer"

p

Object of class "integer"

Dim

Object of class "integer"

Dimnames

Object of class "list"

x

Object of class "numeric"

factors

Object of class "list"

Methods

as.matrix

signature(x = "Incomplete"): ...

coerce

signature(from = "matrix", to = "Incomplete"): ...

complete

signature(x = "Incomplete"):...

Author(s)

Trevor Hastie and Rahul Mazumder

See Also

biScale,softImpute,Incomplete,impute,complete

Examples

showClass("Incomplete")
set.seed(101)
n=200
p=100
J=50
np=n*p
missfrac=0.3
x=matrix(rnorm(n*J),n,J)%*%matrix(rnorm(J*p),J,p)+matrix(rnorm(np),n,p)/5
ix=seq(np)
imiss=sample(ix,np*missfrac,replace=FALSE)
xna=x
xna[imiss]=NA
xnaC=as(xna,"Incomplete")

---

compute the smallest value for lambda such that softImpute(x,lambda) returns the zero solution.

Description

this determines the "starting" lambda for a sequence of values for softImpute, and all nonzero solutions would require a smaller value for lambda.

Usage

lambda0(x, lambda = 0, maxit = 100, trace.it = FALSE, thresh = 1e-05)

Arguments

x	An m by n matrix. Large matrices can be in "sparseMatrix" format, as well as "SparseplusLowRank". The latter arise after centering sparse matrices, for example with biScale, as well as in applications such as softImpute.
lambda	As in svd.als, using a value for lambda can speed up iterations. As long as the solution is not zero, the value returned adds back this value.
maxit	maximum number of iterations.
trace.it	with trace.it=TRUE, convergence progress is reported.
thresh	convergence threshold, measured as the relative changed in the Frobenius norm between two successive estimates.

Details

It is the largest singular value for the matrix, with zeros replacing missing values. It uses svd.als with rank=2.

Value

a single number, the largest singular value

Author(s)

Trevor Hastie, Rahul Mazumder
Maintainer: Trevor Hastie [email protected]

References

Rahul Mazumder, Trevor Hastie and Rob Tibshirani (2010) Spectral Regularization Algorithms for Learning Large Incomplete Matrices, https://hastie.su.domains/Papers/mazumder10a.pdf
Journal of Machine Learning Research 11 (2010) 2287-2322

See Also

softImpute,Incomplete, and svd.als.

Examples

set.seed(101)
n=200
p=100
J=50
np=n*p
missfrac=0.3
x=matrix(rnorm(n*J),n,J)%*%matrix(rnorm(J*p),J,p)+matrix(rnorm(np),n,p)/5
ix=seq(np)
imiss=sample(ix,np*missfrac,replace=FALSE)
xna=x
xna[imiss]=NA
lambda0(xna)

---

rdname softImpute-internal

Description

rdname softImpute-internal

Usage

simpute.als(
  x,
  J = 2,
  thresh = 1e-05,
  lambda = 0,
  maxit = 100,
  trace.it = TRUE,
  warm.start = NULL,
  final.svd = TRUE
)

Arguments

x	An m by n matrix with NAs. For large matrices can be of class "Incomplete", in which case the missing values are represented as pseudo zeros leading to dramatic storage reduction. x can have been centered and scaled via biScale, and this information is carried along with the solution.
J	Trevor to document this param
thresh	convergence threshold, measured as the relative change in the Frobenius norm between two successive estimates.
lambda	nuclear-norm regularization parameter. If lambda=0, the algorithm reverts to "hardImpute", for which convergence is typically slower, and to local minimum. Ideally lambda should be chosen so that the solution reached has rank slightly less than rank.max. See also lambda0() for computing the smallest lambda with a zero solution.
maxit	maximum number of iterations.
trace.it	with trace.it=TRUE, convergence progress is reported.
warm.start	an svd object can be supplied as a warm start. This is particularly useful when constructing a path of solutions with decreasing values of lambda and increasing rank.max. The previous solution can be provided directly as a warm start for the next.
final.svd	only applicable to type="als". The alternating ridge-regressions do not lead to exact zeros. With the default final.svd=TRUE, at the final iteration, a one step unregularized iteration is performed, followed by soft-thresholding of the singular values, leading to hard zeros.

---

rdname softImpute-internal

Description

rdname softImpute-internal

Usage

simpute.svd(
  x,
  J = 2,
  thresh = 1e-05,
  lambda = 0,
  maxit = 100,
  trace.it = FALSE,
  warm.start = NULL,
  ...
)

Arguments

x	An m by n matrix with NAs. For large matrices can be of class "Incomplete", in which case the missing values are represented as pseudo zeros leading to dramatic storage reduction. x can have been centered and scaled via biScale, and this information is carried along with the solution.
J	Trevor to document this param
thresh	convergence threshold, measured as the relative change in the Frobenius norm between two successive estimates.
lambda	nuclear-norm regularization parameter. If lambda=0, the algorithm reverts to "hardImpute", for which convergence is typically slower, and to local minimum. Ideally lambda should be chosen so that the solution reached has rank slightly less than rank.max. See also lambda0() for computing the smallest lambda with a zero solution.
maxit	maximum number of iterations.
trace.it	with trace.it=TRUE, convergence progress is reported.
warm.start	an svd object can be supplied as a warm start. This is particularly useful when constructing a path of solutions with decreasing values of lambda and increasing rank.max. The previous solution can be provided directly as a warm start for the next.
...	sink argument for unwanted arguments

---

impute missing values for a matrix via nuclear-norm regularization.

Description

fit a low-rank matrix approximation to a matrix with missing values via nuclear-norm regularization. The algorithm works like EM, filling in the missing values with the current guess, and then solving the optimization problem on the complete matrix using a soft-thresholded SVD. Special sparse-matrix classes available for very large matrices.

Usage

softImpute(
  x,
  rank.max = 2,
  lambda = 0,
  type = c("als", "svd"),
  thresh = 1e-05,
  maxit = 100,
  trace.it = FALSE,
  warm.start = NULL,
  final.svd = TRUE
)

Arguments

x	An m by n matrix with NAs. For large matrices can be of class "Incomplete", in which case the missing values are represented as pseudo zeros leading to dramatic storage reduction. x can have been centered and scaled via biScale, and this information is carried along with the solution.
rank.max	This restricts the rank of the solution. If sufficiently large, and with type="svd", the solution solves the nuclear-norm convex matrix-completion problem. In this case the number of nonzero singular values returned will be less than or equal to rank.max. If smaller ranks are used, the solution is not guaranteed to solve the problem, although still results in good local minima. rank.max should be no bigger than min(dim(x)-1.
lambda	nuclear-norm regularization parameter. If lambda=0, the algorithm reverts to "hardImpute", for which convergence is typically slower, and to local minimum. Ideally lambda should be chosen so that the solution reached has rank slightly less than rank.max. See also lambda0() for computing the smallest lambda with a zero solution.
type	two algorithms are implements, type="svd" or the default type="als". The "svd" algorithm repeatedly computes the svd of the completed matrix, and soft thresholds its singular values. Each new soft-thresholded svd is used to re-impute the missing entries. For large matrices of class "Incomplete", the svd is achieved by an efficient form of alternating orthogonal ridge regression. The "als" algorithm uses this same alternating ridge regression, but updates the imputation at each step, leading to quite substantial speedups in some cases. The "als" approach does not currently have the same theoretical convergence guarantees as the "svd" approach.
thresh	convergence threshold, measured as the relative change in the Frobenius norm between two successive estimates.
maxit	maximum number of iterations.
trace.it	with trace.it=TRUE, convergence progress is reported.
warm.start	an svd object can be supplied as a warm start. This is particularly useful when constructing a path of solutions with decreasing values of lambda and increasing rank.max. The previous solution can be provided directly as a warm start for the next.
final.svd	only applicable to type="als". The alternating ridge-regressions do not lead to exact zeros. With the default final.svd=TRUE, at the final iteration, a one step unregularized iteration is performed, followed by soft-thresholding of the singular values, leading to hard zeros.

Details

SoftImpute solves the following problem for a matrix $X$ with missing entries:

$\min||X-M||_o^2 +\lambda||M||_*.$

Here $||\cdot||_o$ is the Frobenius norm, restricted to the entries corresponding to the non-missing entries of $X$, and $||M||_*$ is the nuclear norm of $M$ (sum of singular values). For full details of the "svd" algorithm are described in the reference below. The "als" algorithm will be described in a forthcoming article. Both methods employ special sparse-matrix tricks for large matrices with many missing values. This package creates a new sparse-matrix class "SparseplusLowRank" for matrices of the form

$x+ab',$

where $x$ is sparse and $a$ and $b$ are tall skinny matrices, hence $ab'$ is low rank. Methods for efficient left and right matrix multiplication are provided for this class. For large matrices, the function Incomplete() can be used to build the appropriate sparse input matrix from market-format data.

Value

An svd object is returned, with components "u", "d", and "v". If the solution has zeros in "d", the solution is truncated to rank one more than the number of zeros (so the zero is visible). If the input matrix had been centered and scaled by biScale, the scaling details are assigned as attributes inherited from the input matrix.

Author(s)

Trevor Hastie, Rahul Mazumder
Maintainer: Trevor Hastie [email protected]

References

Rahul Mazumder, Trevor Hastie and Rob Tibshirani (2010) Spectral Regularization Algorithms for Learning Large Incomplete Matrices, https://hastie.su.domains/Papers/mazumder10a.pdf
Journal of Machine Learning Research, 11, 2287-2322
Trevor Hastie, Rahul Mazumder, Jason Lee, Reza Zadeh (2015) Matrix Completion and Low-rank SVD via Fast Alternating Least Squares, https://arxiv.org/abs/1410.2596
Journal of Machine Learning Research, 16, 3367-3402

See Also

biScale, svd.als,Incomplete, lambda0, impute, complete

Examples

set.seed(101)
n=200
p=100
J=50
np=n*p
missfrac=0.3
x=matrix(rnorm(n*J),n,J)%*%matrix(rnorm(J*p),J,p)+matrix(rnorm(np),n,p)/5
ix=seq(np)
imiss=sample(ix,np*missfrac,replace=FALSE)
xna=x
xna[imiss]=NA
###uses regular matrix method for matrices with NAs
fit1=softImpute(xna,rank=50,lambda=30)
###uses sparse matrix method for matrices of class "Incomplete"
xnaC=as(xna,"Incomplete")
fit2=softImpute(xnaC,rank=50,lambda=30)
###uses "svd" algorithm
fit3=softImpute(xnaC,rank=50,lambda=30,type="svd")
ximp=complete(xna,fit1)
### first scale xna
xnas=biScale(xna)
fit4=softImpute(xnas,rank=50,lambda=10)
ximp=complete(xna,fit4)
impute(fit4,i=c(1,3,7),j=c(2,5,10))
impute(fit4,i=c(1,3,7),j=c(2,5,10),unscale=FALSE)#ignore scaling and centering

---

rdname softImpute-internal

Description

rdname softImpute-internal

Usage

softImpute.x.Incomplete(
  x,
  J,
  lambda,
  type,
  thresh,
  maxit,
  trace.it,
  warm.start,
  final.svd
)

Arguments

x	An m by n matrix with NAs. For large matrices can be of class "Incomplete", in which case the missing values are represented as pseudo zeros leading to dramatic storage reduction. x can have been centered and scaled via biScale, and this information is carried along with the solution.
J	Trevor to document this param
lambda	nuclear-norm regularization parameter. If lambda=0, the algorithm reverts to "hardImpute", for which convergence is typically slower, and to local minimum. Ideally lambda should be chosen so that the solution reached has rank slightly less than rank.max. See also lambda0() for computing the smallest lambda with a zero solution.
type	two algorithms are implements, type="svd" or the default type="als". The "svd" algorithm repeatedly computes the svd of the completed matrix, and soft thresholds its singular values. Each new soft-thresholded svd is used to re-impute the missing entries. For large matrices of class "Incomplete", the svd is achieved by an efficient form of alternating orthogonal ridge regression. The "als" algorithm uses this same alternating ridge regression, but updates the imputation at each step, leading to quite substantial speedups in some cases. The "als" approach does not currently have the same theoretical convergence guarantees as the "svd" approach.
thresh	convergence threshold, measured as the relative change in the Frobenius norm between two successive estimates.
maxit	maximum number of iterations.
trace.it	with trace.it=TRUE, convergence progress is reported.
warm.start	an svd object can be supplied as a warm start. This is particularly useful when constructing a path of solutions with decreasing values of lambda and increasing rank.max. The previous solution can be provided directly as a warm start for the next.
final.svd	only applicable to type="als". The alternating ridge-regressions do not lead to exact zeros. With the default final.svd=TRUE, at the final iteration, a one step unregularized iteration is performed, followed by soft-thresholding of the singular values, leading to hard zeros.

---

Internal softImpute functions

Description

These functions are not intended to be called directly, but they can be useful for understanding the structure of the models used. rdname softImpute-internal

Usage

softImpute.x.matrix(
  x,
  J,
  lambda,
  type,
  thresh,
  maxit,
  trace.it,
  warm.start,
  final.svd
)

Arguments

x	An m by n matrix with NAs. For large matrices can be of class "Incomplete", in which case the missing values are represented as pseudo zeros leading to dramatic storage reduction. x can have been centered and scaled via biScale, and this information is carried along with the solution.
J	Trevor to document this param
lambda	nuclear-norm regularization parameter. If lambda=0, the algorithm reverts to "hardImpute", for which convergence is typically slower, and to local minimum. Ideally lambda should be chosen so that the solution reached has rank slightly less than rank.max. See also lambda0() for computing the smallest lambda with a zero solution.
type	two algorithms are implements, type="svd" or the default type="als". The "svd" algorithm repeatedly computes the svd of the completed matrix, and soft thresholds its singular values. Each new soft-thresholded svd is used to re-impute the missing entries. For large matrices of class "Incomplete", the svd is achieved by an efficient form of alternating orthogonal ridge regression. The "als" algorithm uses this same alternating ridge regression, but updates the imputation at each step, leading to quite substantial speedups in some cases. The "als" approach does not currently have the same theoretical convergence guarantees as the "svd" approach.
thresh	convergence threshold, measured as the relative change in the Frobenius norm between two successive estimates.
maxit	maximum number of iterations.
trace.it	with trace.it=TRUE, convergence progress is reported.
warm.start	an svd object can be supplied as a warm start. This is particularly useful when constructing a path of solutions with decreasing values of lambda and increasing rank.max. The previous solution can be provided directly as a warm start for the next.
final.svd	only applicable to type="als". The alternating ridge-regressions do not lead to exact zeros. With the default final.svd=TRUE, at the final iteration, a one step unregularized iteration is performed, followed by soft-thresholding of the singular values, leading to hard zeros.

---

Class "SparseplusLowRank"

Description

A structured matrix made up of a sparse part plus a low-rank part, all which can be stored and operated on efficiently.

Objects from the Class

Objects can be created by calls of the form new("SparseplusLowRank", ...) or by a call to splr

Slots

x

Object of class "sparseMatrix"

a

Object of class "matrix"

b

Object of class "matrix"

Methods

%*%

signature(x = "ANY", y = "SparseplusLowRank"): ...

%*%

signature(x = "SparseplusLowRank", y = "ANY"): ...

%*%

signature(x = "Matrix", y = "SparseplusLowRank"): ...

%*%

signature(x = "SparseplusLowRank", y = "Matrix"): ...

as.matrix

signature(x = "SparseplusLowRank"): ...

colMeans

signature(x = "SparseplusLowRank"): ...

colSums

signature(x = "SparseplusLowRank"): ...

dim

signature(x = "SparseplusLowRank"): ...

norm

signature(x = "SparseplusLowRank", type = "character"): ...

rowMeans

signature(x = "SparseplusLowRank"): ...

rowSums

signature(x = "SparseplusLowRank"): ...

svd.als

signature(x = "SparseplusLowRank"): ...

Author(s)

Trevor Hastie and Rahul Mazumder

See Also

softImpute,splr

Examples

showClass("SparseplusLowRank")
x=matrix(sample(c(3,0),15,replace=TRUE),5,3)
x=as(x,"sparseMatrix")
a=matrix(rnorm(10),5,2)
b=matrix(rnorm(6),3,2)
new("SparseplusLowRank",x=x,a=a,b=b)
splr(x,a,b)

---

create a SparseplusLowRank object

Description

create an object of class SparseplusLowRank which can be efficiently stored and for which efficient linear algebra operations are possible.

Usage

splr(x, a = NULL, b = NULL)

Arguments

x	sparse matrix with dimension say m x n
a	matrix with m rows and number of columns r less than min(dim(x))
b	matrix with n rows and number of columns r less than min(dim(x))

Value

an object of S4 class SparseplusLowRank is returned with slots x, a and b

Author(s)

Trevor Hastie

See Also

SparseplusLowRank-class, softImpute

Examples

x=matrix(sample(c(3,0),15,replace=TRUE),5,3)
x=as(x,"sparseMatrix")
a=matrix(rnorm(10),5,2)
b=matrix(rnorm(6),3,2)
new("SparseplusLowRank",x=x,a=a,b=b)
splr(x,a,b)

---

rdname softImpute-internal

Description

rdname softImpute-internal

Usage

Ssimpute.als(
  x,
  J = 2,
  thresh = 1e-05,
  lambda = 0,
  maxit = 100,
  trace.it = FALSE,
  warm.start = NULL,
  final.svd = TRUE
)

Arguments

x	An m by n matrix with NAs. For large matrices can be of class "Incomplete", in which case the missing values are represented as pseudo zeros leading to dramatic storage reduction. x can have been centered and scaled via biScale, and this information is carried along with the solution.
J	Trevor to document this param
thresh	convergence threshold, measured as the relative change in the Frobenius norm between two successive estimates.
lambda	nuclear-norm regularization parameter. If lambda=0, the algorithm reverts to "hardImpute", for which convergence is typically slower, and to local minimum. Ideally lambda should be chosen so that the solution reached has rank slightly less than rank.max. See also lambda0() for computing the smallest lambda with a zero solution.
maxit	maximum number of iterations.
trace.it	with trace.it=TRUE, convergence progress is reported.
warm.start	an svd object can be supplied as a warm start. This is particularly useful when constructing a path of solutions with decreasing values of lambda and increasing rank.max. The previous solution can be provided directly as a warm start for the next.
final.svd	only applicable to type="als". The alternating ridge-regressions do not lead to exact zeros. With the default final.svd=TRUE, at the final iteration, a one step unregularized iteration is performed, followed by soft-thresholding of the singular values, leading to hard zeros.

---

rdname softImpute-internal

Description

rdname softImpute-internal

Usage

Ssimpute.svd(
  x,
  J = 2,
  thresh = 1e-05,
  lambda = 0,
  maxit = 100,
  trace.it = FALSE,
  warm.start = NULL,
  ...
)

Arguments

x	An m by n matrix with NAs. For large matrices can be of class "Incomplete", in which case the missing values are represented as pseudo zeros leading to dramatic storage reduction. x can have been centered and scaled via biScale, and this information is carried along with the solution.
J	Trevor to document this param
thresh	convergence threshold, measured as the relative change in the Frobenius norm between two successive estimates.
lambda	nuclear-norm regularization parameter. If lambda=0, the algorithm reverts to "hardImpute", for which convergence is typically slower, and to local minimum. Ideally lambda should be chosen so that the solution reached has rank slightly less than rank.max. See also lambda0() for computing the smallest lambda with a zero solution.
maxit	maximum number of iterations.
trace.it	with trace.it=TRUE, convergence progress is reported.
warm.start	an svd object can be supplied as a warm start. This is particularly useful when constructing a path of solutions with decreasing values of lambda and increasing rank.max. The previous solution can be provided directly as a warm start for the next.
...	sink argument for unwanted arguments

---

rdname softImpute-internal

Description

rdname softImpute-internal

Usage

Ssvd.als(
  x,
  J = 2,
  thresh = 1e-05,
  lambda = 0,
  maxit = 100,
  trace.it = FALSE,
  warm.start = NULL,
  final.svd = TRUE
)

Arguments

x	An m by n matrix with NAs. For large matrices can be of class "Incomplete", in which case the missing values are represented as pseudo zeros leading to dramatic storage reduction. x can have been centered and scaled via biScale, and this information is carried along with the solution.
J	Trevor to document this param
thresh	convergence threshold, measured as the relative change in the Frobenius norm between two successive estimates.
lambda	nuclear-norm regularization parameter. If lambda=0, the algorithm reverts to "hardImpute", for which convergence is typically slower, and to local minimum. Ideally lambda should be chosen so that the solution reached has rank slightly less than rank.max. See also lambda0() for computing the smallest lambda with a zero solution.
maxit	maximum number of iterations.
trace.it	with trace.it=TRUE, convergence progress is reported.
warm.start	an svd object can be supplied as a warm start. This is particularly useful when constructing a path of solutions with decreasing values of lambda and increasing rank.max. The previous solution can be provided directly as a warm start for the next.
final.svd	only applicable to type="als". The alternating ridge-regressions do not lead to exact zeros. With the default final.svd=TRUE, at the final iteration, a one step unregularized iteration is performed, followed by soft-thresholding of the singular values, leading to hard zeros.

---

compute a low rank soft-thresholded svd by alternating orthogonal ridge regression

Description

fit a low-rank svd to a complete matrix by alternating orthogonal ridge regression. Special sparse-matrix classes available for very large matrices, including "SparseplusLowRank" versions for row and column centered sparse matrices.

Usage

svd.als(
  x,
  rank.max = 2,
  lambda = 0,
  thresh = 1e-05,
  maxit = 100,
  trace.it = FALSE,
  warm.start = NULL,
  final.svd = TRUE
)

Arguments

x	An m by n matrix. Large matrices can be in "sparseMatrix" format, as well as "SparseplusLowRank". The latter arise after centering sparse matrices, for example with biScale, as well as in applications such as softImpute.
rank.max	The maximum rank for the solution. This is also the dimension of the left and right matrices of orthogonal singular vectors. 'rank.max' should be no bigger than 'min(dim(x)'.
lambda	The regularization parameter. lambda=0 corresponds to an accelerated version of the orthogonal QR-algorithm. With lambda>0 the algorithm amounts to alternating orthogonal ridge regression.
thresh	convergence threshold, measured as the relative changed in the Frobenius norm between two successive estimates.
maxit	maximum number of iterations.
trace.it	with trace.it=TRUE, convergence progress is reported.
warm.start	an svd object can be supplied as a warm start. If the solution requested has higher rank than the warm start, the additional subspace is initialized with random Gaussians (and then orthogonalized wrt the rest).
final.svd	Although in theory, this algorithm converges to the solution to a nuclear-norm regularized low-rank matrix approximation problem, with potentially some singular values equal to zero, in practice only near-zeros are achieved. This final step does one more iteration with lambda=0, followed by soft-thresholding.

Details

This algorithm solves the problem

$\min ||X-M||_F^2 +\lambda ||M||_*$

subject to $rank(M)\leq r$, where $||M||_*$ is the nuclear norm of $M$ (sum of singular values). It achieves this by solving the related problem

$\min ||X-AB'||_F^2 +\lambda/2 (||A||_F^2+||B||_F^2)$

subject to $rank(A)=rank(B)\leq r$. The solution is a rank-restricted, soft-thresholded SVD of $X$.

Value

An svd object is returned, with components "u", "d", and "v".

u	an m by rank.max matrix with the left orthogonal singular vectors
d	a vector of length rank.max of soft-thresholded singular values
v	an n by rank.max matrix with the right orthogonal singular vectors

Author(s)

Trevor Hastie, Rahul Mazumder
Maintainer: Trevor Hastie [email protected]

References

Rahul Mazumder, Trevor Hastie and Rob Tibshirani (2010) Spectral Regularization Algorithms for Learning Large Incomplete Matrices, https://hastie.su.domains/Papers/mazumder10a.pdf
Journal of Machine Learning Research 11 (2010) 2287-2322

See Also

biScale, softImpute, Incomplete, lambda0, impute, complete

Examples

#create a matrix and run the algorithm
set.seed(101)
n=100
p=50
J=25
np=n*p
x=matrix(rnorm(n*J),n,J)%*%matrix(rnorm(J*p),J,p)+matrix(rnorm(np),n,p)/5
fit=svd.als(x,rank=25,lambda=50)
fit$d
pmax(svd(x)$d-50,0)
# now create a sparse matrix and do the same
nnz=trunc(np*.3)
inz=sample(seq(np),nnz,replace=FALSE)
i=row(x)[inz]
j=col(x)[inz]
x=rnorm(nnz)
xS=sparseMatrix(x=x,i=i,j=j)
fit2=svd.als(xS,rank=20,lambda=7)
fit2$d
pmax(svd(as.matrix(xS))$d-7,0)

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