Tools for tensor analysis and decomposition

Description

This package is centered around the Tensor-class, which defines a S4 class for tensors of arbitrary number of modes. A vignette and/or a possible paper will be included in a future release of this package.

Details

This page will summarize the full functionality of this package. Note that since all the methods associated with S4 class Tensor-class are documented there, we will not duplicate it here.

The remaining functions can be split into two groups: the first is a set of tensor decompositions, and the second is a set of helper functions that are useful in tensor manipulation.

rTensor implements the following tensor decompositions:

cp

Canonical Polyadic (CP) decomposition

tucker

General Tucker decomposition

mpca

Multilinear Principal Component Analysis; note that for 3-Tensors this is also known as Generalized Low Rank Approximation of Matrices(GLRAM)

hosvd

(Truncated-)Higher-order singular value decomposition

t_svd

Tensor singular value decomposition; 3-Tensors only; also note that there is an asociated reconstruction function t_svd_reconstruct

pvd

Population value decomposition of images; 3-Tensors only

rTensor also provides a set functions for tensors multiplication:

ttm

Tensor times matrix, aka m-mode product

ttl

Tensor times list (of matrices)

t_mult

Tensor product based on block circulant unfolding; only implemented for a pair of 3-Tensors

...as well as for matrices:

hadamard_list

Computes the Hadamard (element-wise) product of a list of matrices

kronecker_list

Computes the Kronecker product of a list of matrices

khatri_rao

Computes the Khatri-Rao product of two matrices

khatri_rao_list

Computes the Khatri-Rao product of a list of matrices

fold

General folding of a matrix into a tensor

k_fold

Inverse operation for k_unfold

unmatvec

Inverse operation for matvec

For more information on any of the functions, please consult the individual man pages.

Author(s)

James Li jamesyili@gmail.com, Jacob Bien, and Martin T. Wells

Conformable elementwise operators for Tensor

Description

Overloads elementwise operators for tensors, arrays, and vectors that are conformable (have the same modes).

Usage

## S4 method for signature 'Tensor,Tensor'
Ops(e1, e2)

Arguments

Examples

tnsr <- rand_tensor(c(3,4,5))
tnsr2 <- rand_tensor(c(3,4,5))
tnsrsum <- tnsr + tnsr2
tnsrdiff <- tnsr - tnsr2
tnsrelemprod <- tnsr * tnsr2
tnsrelemquot <- tnsr / tnsr2
for (i in 1:3L){
for (j in 1:4L){
	for (k in 1:5L){
		stopifnot(tnsrsum@data[i,j,k]==tnsr@data[i,j,k]+tnsr2@data[i,j,k])
		stopifnot(tnsrdiff@data[i,j,k]==(tnsr@data[i,j,k]-tnsr2@data[i,j,k]))
		stopifnot(tnsrelemprod@data[i,j,k]==tnsr@data[i,j,k]*tnsr2@data[i,j,k])
		stopifnot(tnsrelemquot@data[i,j,k]==tnsr@data[i,j,k]/tnsr2@data[i,j,k])
}
}
}

S4 Class for a Tensor

Description

An S4 class for a tensor with arbitrary number of modes. The Tensor class extends the base 'array' class to include additional tensor manipulation (folding, unfolding, reshaping, subsetting) as well as a formal class definition that enables more explicit tensor algebra.

Details

This can be seen as a wrapper class to the base array class. While it is possible to create an instance using new, it is also possible to do so by passing the data into as.tensor.

Each slot of a Tensor instance can be obtained using @.

The following methods are overloaded for the Tensor class: dim-methods, head-methods, tail-methods, print-methods, show-methods, element-wise array operations, array subsetting (extract via ‘[’), array subset replacing (replace via ‘[<-’), and tperm-methods, which is a wrapper around the base aperm method.

To sum across any one mode of a tenor, use the function modeSum-methods. To compute the mean across any one mode, use modeMean-methods.

You can always unfold any Tensor into a matrix, and the unfold-methods, k_unfold-methods, and matvec-methods methods are for that purpose. The output can be kept as a Tensor with 2 modes or a matrix object. The vectorization function is also provided as vec. See the attached vignette for a visualization of the different unfoldings.

Conversion from array/matrix to Tensor is facilitated via as.tensor. To convert from a Tensor instance, simply invoke @data.

The Frobenius norm of the Tensor is given by fnorm-methods, while the inner product between two Tensors (of equal modes) is given by innerProd-methods. You can also sum through any one mode to obtain the K-1 Tensor sum using modeSum-methods. modeMean-methods provides similar functionality to obtain the K-1 Tensor mean. These are primarily meant to be used internally but may be useful in doing statistics with Tensors.

For Tensors with 3 modes, we also overloaded t (transpose) defined by Kilmer et.al (2013). See t-methods.

To create a Tensor with i.i.d. random normal(0, 1) entries, see rand_tensor.

Slots

num_modes

number of modes (integer)

modes

vector of modes (integer), aka sizes/extents/dimensions

data

actual data of the tensor, which can be 'array' or 'vector'

Methods

[

signature(tnsr = "Tensor"): ...

[<-

signature(tnsr = "Tensor"): ...

matvec

signature(tnsr = "Tensor"): ...

dim

signature(tnsr = "Tensor"): ...

fnorm

signature(tnsr = "Tensor"): ...

head

signature(tnsr = "Tensor"): ...

initialize

signature(.Object = "Tensor"): ...

innerProd

signature(tnsr1 = "Tensor", tnsr2 = "Tensor"): ...

modeMean

signature(tnsr = "Tensor"): ...

modeSum

signature(tnsr = "Tensor"): ...

Ops

signature(e1 = "array", e2 = "Tensor"): ...

Ops

signature(e1 = "numeric", e2 = "Tensor"): ...

Ops

signature(e1 = "Tensor", e2 = "array"): ...

Ops

signature(e1 = "Tensor", e2 = "numeric"): ...

Ops

signature(e1 = "Tensor", e2 = "Tensor"): ...

print

signature(tnsr = "Tensor"): ...

k_unfold

signature(tnsr = "Tensor"): ...

show

signature(tnsr = "Tensor"): ...

t

signature(tnsr = "Tensor"): ...

tail

signature(tnsr = "Tensor"): ...

unfold

signature(tnsr = "Tensor"): ...

tperm

signature(tnsr = "Tensor"): ...

image

signature(tnsr = "Tensor"): ...

Note

All of the decompositions and regression models in this package require a Tensor input.

Author(s)

James Li jamesyili@gmail.com

References

James Li, Jacob Bien, Martin T. Wells (2018). rTensor: An R Package for Multidimensional Array (Tensor) Unfolding, Multiplication, and Decomposition. Journal of Statistical Software, 87(10), 1-31. URL http://www.jstatsoft.org/v087/i10/.

See Also

as.tensor

Examples

tnsr <- rand_tensor()
class(tnsr)
tnsr
print(tnsr)
dim(tnsr)
tnsr@num_modes
tnsr@data

Extract or Replace Subtensors

Description

Extends '[' and '[<-' from the base array class for the Tensor class. Works exactly as it would for the base 'array' class.

Usage

## S4 method for signature 'Tensor'
x[i, j, ..., drop = TRUE]

## S4 replacement method for signature 'Tensor'
x[i, j, ...] <- value

Arguments

Details

x[i,j,...,drop=TRUE]

Value

an object of class Tensor

Examples

tnsr <- rand_tensor()
tnsr[1,2,3]
tnsr[3,1,]
tnsr[,,5]
tnsr[,,5,drop=FALSE]

tnsr[1,2,3] <- 3; tnsr[1,2,3]
tnsr[3,1,] <- rep(0,5); tnsr[3,1,]
tnsr[,2,] <- matrix(0,nrow=3,ncol=5); tnsr[,2,]

Tensor Conversion

Description

Create a Tensor-class object from an array, matrix, or vector.

Usage

as.tensor(x, drop = FALSE)

Arguments

Value

a Tensor-class object

Examples

#From vector
vec <- runif(3); vecT <- as.tensor(vec); vecT
#From matrix
mat <- matrix(runif(2*3),nrow=2,ncol=3)
matT <- as.tensor(mat); matT
#From array
indices <- c(2,3,4)
arr <- array(runif(prod(indices)), dim = indices)
arrT <- as.tensor(arr); arrT

Canonical Polyadic Decomposition

Description

Canonical Polyadic (CP) decomposition of a tensor, aka CANDECOMP/PARAFRAC. Approximate a K-Tensor using a sum of num_components rank-1 K-Tensors. A rank-1 K-Tensor can be written as an outer product of K vectors. There are a total of num_compoents *tnsr@num_modes vectors in the output, stored in tnsr@num_modes matrices, each with num_components columns. This is an iterative algorithm, with two possible stopping conditions: either relative error in Frobenius norm has gotten below tol, or the max_iter number of iterations has been reached. For more details on CP decomposition, consult Kolda and Bader (2009).

Usage

cp(tnsr, num_components = NULL, max_iter = 25, tol = 1e-05)

Arguments

Details

Uses the Alternating Least Squares (ALS) estimation procedure. A progress bar is included to help monitor operations on large tensors.

Value

a list containing the following

lambdas

a vector of normalizing constants, one for each component

U

a list of matrices - one for each mode - each matrix with num_components columns

conv

whether or not resid < tol by the last iteration

norm_percent

the percent of Frobenius norm explained by the approximation

est

estimate of tnsr after compression

fnorm_resid

the Frobenius norm of the error fnorm(est-tnsr)

all_resids

vector containing the Frobenius norm of error for all the iterations

References

T. Kolda, B. Bader, "Tensor decomposition and applications". SIAM Applied Mathematics and Applications 2009.

See Also

tucker

Examples

### How to retrieve faces_tnsr from figshare
# faces_tnsr <- load_orl()
# subject <- faces_tnsr[,,14,]
dummy_faces_tnsr <- rand_tensor(c(92,112,40,10))
subject <- dummy_faces_tnsr[,,14,]
cpD <- cp(subject, num_components=3)
cpD$conv
cpD$norm_percent
plot(cpD$all_resids)

Column Space Folding of Matrix

Description

DEPRECATED. Please see unmatvec

Usage

cs_fold(mat, m = NULL, modes = NULL)

Arguments

Tensor Column Space Unfolding

Description

DEPRECATED. Please see matvec-methods and unfold-methods.

Usage

cs_unfold(tnsr, m)

## S4 method for signature 'Tensor'
cs_unfold(tnsr, m = NULL)

Arguments

Details

cs_unfold(tnsr,m=NULL)

Mode Getter for Tensor

Description

Return the vector of modes from a tensor

Usage

## S4 method for signature 'Tensor'
dim(x)

Arguments

Details

dim(x)

Value

an integer vector of the modes associated with x

Examples

tnsr <- rand_tensor()
dim(tnsr)

Tensor Frobenius Norm

Description

Returns the Frobenius norm of the Tensor instance.

Usage

fnorm(tnsr)

## S4 method for signature 'Tensor'
fnorm(tnsr)

Arguments

Details

fnorm(tnsr)

Value

numeric Frobenius norm of x

Examples

tnsr <- rand_tensor()
fnorm(tnsr)

General Folding of Matrix

Description

General folding of a matrix into a Tensor. This is designed to be the inverse function to unfold-methods, with the same ordering of the indices. This amounts to following: if we were to unfold a Tensor using a set of row_idx and col_idx, then we can fold the resulting matrix back into the original Tensor using the same row_idx and col_idx.

Usage

fold(mat, row_idx = NULL, col_idx = NULL, modes = NULL)

Arguments

Details

This function uses aperm as the primary workhorse.

Value

Tensor object with modes given by modes

References

T. Kolda, B. Bader, "Tensor decomposition and applications". SIAM Applied Mathematics and Applications 2009.

See Also

unfold-methods, k_fold, unmatvec

Examples

tnsr <- new("Tensor",3L,c(3L,4L,5L),data=runif(60))
matT3<-unfold(tnsr,row_idx=2,col_idx=c(3,1))
identical(fold(matT3,row_idx=2,col_idx=c(3,1),modes=c(3,4,5)),tnsr)

List hadamard Product

Description

Returns the hadamard (element-wise) product from a list of matrices or vectors. Commonly used for n-mode products and various Tensor decompositions.

Usage

hadamard_list(L)

Arguments

Value

matrix that is the hadamard product

Note

The modes/dimensions of each element in the list must match.

See Also

kronecker_list, khatri_rao_list

Examples

lizt <- list('mat1' = matrix(runif(40),ncol=4), 
'mat2' = matrix(runif(40),ncol=4),
'mat3' = matrix(runif(40),ncol=4))
dim(hadamard_list(lizt))

Head for Tensor

Description

Extend head for Tensor

Usage

## S4 method for signature 'Tensor'
head(x, ...)

Arguments

Details

head(x,...)

See Also

tail-methods

Examples

tnsr <- rand_tensor()
head(tnsr)

(Truncated-)Higher-order SVD

Description

Higher-order SVD of a K-Tensor. Write the K-Tensor as a (m-mode) product of a core Tensor (possibly smaller modes) and K orthogonal factor matrices. Truncations can be specified via ranks (making them smaller than the original modes of the K-Tensor will result in a truncation). For the mathematical details on HOSVD, consult Lathauwer et. al. (2000).

Usage

hosvd(tnsr, ranks = NULL)

Arguments

Details

A progress bar is included to help monitor operations on large tensors.

Value

a list containing the following:

Z

core tensor with modes speficied by ranks

U

a list of orthogonal matrices, one for each mode

est

estimate of tnsr after compression

fnorm_resid

the Frobenius norm of the error fnorm(est-tnsr) - if there was no truncation, then this is on the order of mach_eps * fnorm.

Note

The length of ranks must match tnsr@num_modes.

References

L. Lathauwer, B.Moor, J. Vanderwalle "A multilinear singular value decomposition". Journal of Matrix Analysis and Applications 2000.

See Also

tucker

Examples

tnsr <- rand_tensor(c(6,7,8))
hosvdD <- hosvd(tnsr)
plot(hosvdD$fnorm_resid)
hosvdD2 <- hosvd(tnsr,ranks=c(3,3,4))
plot(hosvdD2$fnorm_resid)

Initializes a Tensor instance

Description

Not designed to be called by the user. Use as.tensor instead.

Usage

## S4 method for signature 'Tensor'
initialize(.Object, num_modes = NULL, modes = NULL,
  data = NULL)

Arguments

See Also

as.tensor

Tensors Inner Product

Description

Returns the inner product between two Tensors

Usage

innerProd(tnsr1, tnsr2)

## S4 method for signature 'Tensor,Tensor'
innerProd(tnsr1, tnsr2)

Arguments

Details

innerProd(tnsr1,tnsr2)

Value

inner product between x1 and x2

Examples

tnsr1 <- rand_tensor()
tnsr2 <- rand_tensor()
innerProd(tnsr1,tnsr2)

k-mode Folding of Matrix

Description

k-mode folding of a matrix into a Tensor. This is the inverse funtion to k_unfold in the m mode. In particular, k_fold(k_unfold(tnsr, m),m,getModes(tnsr)) will result in the original Tensor.

Usage

k_fold(mat, m = NULL, modes = NULL)

Arguments

Details

This is a wrapper function to fold.

Value

Tensor object with modes given by modes

References

T. Kolda, B. Bader, "Tensor decomposition and applications". SIAM Applied Mathematics and Applications 2009.

See Also

k_unfold-methods, fold, unmatvec

Examples

tnsr <- new("Tensor",3L,c(3L,4L,5L),data=runif(60))
matT2<-k_unfold(tnsr,m=2)
identical(k_fold(matT2,m=2,modes=c(3,4,5)),tnsr)

Tensor k-mode Unfolding

Description

Unfolding of a tensor by mapping the kth mode (specified through parameter m), and all other modes onto the column space. This the most common type of unfolding operation for Tucker decompositions and its variants. Also known as k-mode matricization.

Usage

k_unfold(tnsr, m)

## S4 method for signature 'Tensor'
k_unfold(tnsr, m = NULL)

Arguments

Details

k_unfold(tnsr,m=NULL)

Value

matrix with x@modes[m] rows and prod(x@modes[-m]) columns

References

T. Kolda and B. Bader, "Tensor decomposition and applications". SIAM Applied Mathematics and Applications 2009.

See Also

matvec-methods and unfold-methods

Examples

tnsr <- rand_tensor()
matT2<-rs_unfold(tnsr,m=2)

Khatri-Rao Product

Description

Returns the Khatri-Rao (column-wise Kronecker) product of two matrices. If the inputs are vectors then this is the same as the Kronecker product.

Usage

khatri_rao(x, y)

Arguments

Value

matrix that is the Khatri-Rao product

Note

The number of columns must match in the two inputs.

See Also

kronecker, khatri_rao_list

Examples

dim(khatri_rao(matrix(runif(12),ncol=4),matrix(runif(12),ncol=4)))

List Khatri-Rao Product

Description

Returns the Khatri-Rao product from a list of matrices or vectors. Commonly used for n-mode products and various Tensor decompositions.

Usage

khatri_rao_list(L, reverse = FALSE)

Arguments

Value

matrix that is the Khatri-Rao product

Note

The number of columns must match in every element of the input list.

See Also

khatri_rao

Examples

smalllizt <- list('mat1' = matrix(runif(12),ncol=4), 
'mat2' = matrix(runif(12),ncol=4),
'mat3' = matrix(runif(12),ncol=4))
dim(khatri_rao_list(smalllizt))

List Kronecker Product

Description

Returns the Kronecker product from a list of matrices or vectors. Commonly used for n-mode products and various Tensor decompositions.

Usage

kronecker_list(L)

Arguments

Value

matrix that is the Kronecker product

See Also

hadamard_list, khatri_rao_list, kronecker

Examples

smalllizt <- list('mat1' = matrix(runif(12),ncol=4), 
'mat2' = matrix(runif(12),ncol=4),
'mat3' = matrix(runif(12),ncol=4))
dim(kronecker_list(smalllizt))

ORL Database of Faces

Description

A dataset containing pictures of 40 individuals under 10 different lightings. Each image has 92 x 112 pixels. Structured as a 4-tensor with modes 92 x 112 x 40 x 10. The data is now stored in figshare https://ndownloader.figshare.com/files/28005669

Usage

load_orl()

Format

A Tensor object with modes 92 x 112 x 40 x 10. The first two modes correspond to the image pixels, the third mode corresponds to the individual, and the last mode correpsonds to the lighting.

Source

https://www.kaggle.com/kasikrit/att-database-of-faces

References

AT&T Laboratories Cambridge. https://www.kaggle.com/kasikrit/att-database-of-faces

F. Samaria, A. Harter, "Parameterisation of a Stochastic Model for Human Face Identification". IEEE Workshop on Applications of Computer Vision 1994.

See Also

plot_orl

Tensor Matvec Unfolding

Description

For 3-tensors only. Stacks the slices along the third mode. This is the prevalent unfolding for T-SVD and T-MULT based on block circulant matrices.

Usage

matvec(tnsr)

## S4 method for signature 'Tensor'
matvec(tnsr)

Arguments

Details

matvec(tnsr)

Value

matrix with prod(x@modes[-m]) rows and x@modes[m] columns

References

M. Kilmer, K. Braman, N. Hao, and R. Hoover, "Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging". SIAM Journal on Matrix Analysis and Applications 2013.

See Also

k_unfold-methods and unfold-methods

Examples

tnsr <- rand_tensor(c(2,3,4))
matT1<- matvec(tnsr)

Tensor Mean Across Single Mode

Description

Given a mode for a K-tensor, this returns the K-1 tensor resulting from taking the mean across that particular mode.

Usage

modeMean(tnsr, m, drop)

## S4 method for signature 'Tensor'
modeMean(tnsr, m = NULL, drop = FALSE)

Arguments

Details

modeMean(tnsr,m=NULL,drop=FALSE)

Value

K-1 or K Tensor, where K = x@num_modes

See Also

modeSum

Examples

tnsr <- rand_tensor()
modeMean(tnsr,1,drop=TRUE)

Tensor Sum Across Single Mode

Description

Given a mode for a K-tensor, this returns the K-1 tensor resulting from summing across that particular mode.

Usage

modeSum(tnsr, m, drop)

## S4 method for signature 'Tensor'
modeSum(tnsr, m = NULL, drop = FALSE)

Arguments

Details

modeSum(tnsr,m=NULL,drop=FALSE)

Value

K-1 or K tensor, where K = x@num_modes

See Also

modeMean

Examples

tnsr <- rand_tensor()
modeSum(tnsr,3,drop=TRUE)

Multilinear Principal Components Analysis

Description

This is basically the Tucker decomposition of a K-Tensor, tucker, with one of the modes uncompressed. If K = 3, then this is also known as the Generalized Low Rank Approximation of Matrices (GLRAM). This implementation assumes that the last mode is the measurement mode and hence uncompressed. This is an iterative algorithm, with two possible stopping conditions: either relative error in Frobenius norm has gotten below tol, or the max_iter number of iterations has been reached. For more details on the MPCA of tensors, consult Lu et al. (2008).

Usage

mpca(tnsr, ranks = NULL, max_iter = 25, tol = 1e-05)

Arguments

Details

Uses the Alternating Least Squares (ALS) estimation procedure. A progress bar is included to help monitor operations on large tensors.

Value

a list containing the following:

Z_ext

the extended core tensor, with the first K-1 modes given by ranks

U

a list of K-1 orthgonal factor matrices - one for each compressed mode, with the number of columns of the matrices given by ranks

conv

whether or not resid < tol by the last iteration

est

estimate of tnsr after compression

norm_percent

the percent of Frobenius norm explained by the approximation

fnorm_resid

the Frobenius norm of the error fnorm(est-tnsr)

all_resids

vector containing the Frobenius norm of error for all the iterations

Note

The length of ranks must match tnsr@num_modes-1.

References

H. Lu, K. Plataniotis, A. Venetsanopoulos, "Mpca: Multilinear principal component analysis of tensor objects". IEEE Trans. Neural networks, 2008.

See Also

tucker, hosvd

Examples

### How to retrieve faces_tnsr from figshare
# faces_tnsr <- load_orl()
# subject <- faces_tnsr[,,21,]
dummy_faces_tnsr <- rand_tensor(c(92,112,40,10))
subject <- dummy_faces_tnsr[,,21,]
mpcaD <- mpca(subject, ranks=c(10, 10))
mpcaD$conv
mpcaD$norm_percent
plot(mpcaD$all_resids)

Function to plot the ORL Database of Faces

Description

A wrapper function to image() to allow easy visualization of faces_tnsr, the ORL Face Dataset. The data is now stored in figshare https://ndownloader.figshare.com/files/28005669

Usage

plot_orl(subject = 1, condition = 1)

Arguments

References

AT&T Laboratories Cambridge. https://www.kaggle.com/kasikrit/att-database-of-faces

F. Samaria, A. Harter, "Parameterisation of a Stochastic Model for Human Face Identification". IEEE Workshop on Applications of Computer Vision 1994.

Print for Tensor

Description

Extend print for Tensor

Usage

## S4 method for signature 'Tensor'
print(x, ...)

Arguments

Details

print(x,...)

See Also

show

Examples

tnsr <- rand_tensor()
print(tnsr)

Population Value Decomposition

Description

The default Population Value Decomposition (PVD) of a series of 2D images. Constructs population-level matrices P, V, and D to account for variances within as well as across the images. Structurally similar to Tucker (tucker) and GLRAM (mpca), but retains crucial differences. Requires 2*n3 + 2 parameters to specified the final ranks of P, V, and D, where n3 is the third mode (how many images are in the set). Consult Crainiceanu et al. (2013) for the construction and rationale behind the PVD model.

Usage

pvd(tnsr, uranks = NULL, wranks = NULL, a = NULL, b = NULL)

Arguments

Details

The PVD is not an iterative method, but instead relies on n3 + 2separate PCA decompositions. The third mode is for how many images are in the set.

Value

a list containing the following:

P

population-level matrix P = U%*%t(U), where U is constructed by stacking the truncated left eigenvectors of slicewise PCA along the third mode

V

a list of image-level core matrices

D

population-leve matrix D = W%*%t(W), where W is constructed by stacking the truncated right eigenvectors of slicewise PCA along the third mode

est

estimate of tnsr after compression

norm_percent

the percent of Frobenius norm explained by the approximation

fnorm_resid

the Frobenius norm of the error fnorm(est-tnsr)

References

C. Crainiceanu, B. Caffo, S. Luo, V. Zipunnikov, N. Punjabi, "Population value decomposition: a framework for the analysis of image populations". Journal of the American Statistical Association, 2013.

Examples

### How to retrieve faces_tnsr from figshare
# faces_tnsr <- load_orl()
# subject <- faces_tnsr[,,8,]
dummy_faces_tnsr <- rand_tensor(c(92,112,40,10))
subject <- dummy_faces_tnsr[,,8,]
pvdD <- pvd(subject, uranks=rep(46,10), wranks=rep(56,10), a=46, b=56)
plot(pvdD$fnorm_resid)

Tensor with Random Entries

Description

Generate a Tensor with specified modes with iid normal(0,1) entries.

Usage

rand_tensor(modes = c(3, 4, 5), drop = FALSE)

Arguments

Value

a Tensor object with modes given by modes

Note

Default rand_tensor() generates a 3-Tensor with modes c(3,4,5).

Examples

rand_tensor()
rand_tensor(c(4,4,4))
rand_tensor(c(10,2,1),TRUE)

Row Space Folding of Matrix

Description

DEPRECATED. Please see k_fold.

Usage

rs_fold(mat, m = NULL, modes = NULL)

Arguments

Tensor Row Space Unfolding

Description

DEPRECATED. Please see k_unfold-methods and unfold-methods.

Usage

rs_unfold(tnsr, m)

## S4 method for signature 'Tensor'
rs_unfold(tnsr, m = NULL)

Arguments

Details

rs_unfold(tnsr,m=NULL)

Show for Tensor

Description

Extend show for Tensor

Usage

## S4 method for signature 'Tensor'
show(object)

Arguments

Details

show(object)

See Also

print

Examples

tnsr <- rand_tensor()
tnsr

Tensor Transpose

Description

Implements the tensor transpose based on block circulant matrices (Kilmer et al. 2013) for 3-tensors.

Usage

## S4 method for signature 'Tensor'
t(x)

Arguments

Details

t(x)

Value

tensor transpose of x

References

M. Kilmer, K. Braman, N. Hao, and R. Hoover, "Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging". SIAM Journal on Matrix Analysis and Applications 2013.

Examples

tnsr <- rand_tensor()
identical(t(tnsr)@data[,,1],t(tnsr@data[,,1]))
identical(t(tnsr)@data[,,2],t(tnsr@data[,,5]))
identical(t(t(tnsr)),tnsr)

Tensor Multiplication (T-MULT)

Description

Implements T-MULT based on block circulant matrices (Kilmer et al. 2013) for 3-tensors.

Usage

t_mult(x, y)

Arguments

Details

Uses the Fast Fourier Transform (FFT) speed up suggested by Kilmer et al. 2013 instead of explicitly constructing the block circulant matrix. For the mathematical details of T-MULT, see Kilmer et al. (2013).

Value

tensor product between x and y

Note

This only works (so far) between 3-Tensors.

References

M. Kilmer, K. Braman, N. Hao, and R. Hoover, "Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging". SIAM Journal on Matrix Analysis and Applications 2013.

Examples

tnsr <- new("Tensor",3L,c(3L,4L,5L),data=runif(60))
tnsr2 <- new("Tensor",3L,c(4L,3L,5L),data=runif(60))
t_mult(tnsr, tnsr2)

Tensor Singular Value Decomposition

Description

TSVD for a 3-Tensor. Constructs 3-Tensors U, S, V such that tnsr = t_mult(t_mult(U,S),t(V)). U and V are orthgonal 3-Tensors with orthogonality defined in Kilmer et al. (2013), and S is a 3-Tensor consists of facewise diagonal matrices. For more details on the TSVD, consult Kilmer et al. (2013).

Usage

t_svd(tnsr)

Arguments

Value

a list containing the following:

U

the left orthgonal 3-Tensor

V

the right orthgonal 3-Tensor

S

the middle 3-Tensor consisting of face-wise diagonal matrices

Note

Computation involves complex values, but if the inputs are real, then the outputs are also real. Some loss of precision occurs in the truncation of the imaginary components during the FFT and inverse FFT.

References

M. Kilmer, K. Braman, N. Hao, and R. Hoover, "Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging". SIAM Journal on Matrix Analysis and Applications 2013.

See Also

t_mult, t_svd_reconstruct

Examples

tnsr <- rand_tensor()
tsvdD <- t_svd(tnsr)

Reconstruct Tensor From TSVD

Description

Reconstruct the original 3-Tensor after it has been decomposed into U, S, V via t_svd.

Usage

t_svd_reconstruct(L)

Arguments

Value

a 3-Tensor

See Also

t_svd

Examples

tnsr <- rand_tensor(c(10,10,10))
tsvdD <- t_svd(tnsr)
1 - fnorm(t_svd_reconstruct(tsvdD)-tnsr)/fnorm(tnsr)

Tail for Tensor

Description

Extend tail for Tensor

Usage

## S4 method for signature 'Tensor'
tail(x, ...)

Arguments

Details

tail(x,...)

See Also

head-methods

Examples

tnsr <- rand_tensor()
tail(tnsr)

Mode Permutation for Tensor

Description

Overloads aperm for Tensor class for convenience.

Usage

tperm(tnsr, perm, ...)

## S4 method for signature 'Tensor'
tperm(tnsr, perm, ...)

Arguments

Details

tperm(tnsr,perm=NULL,...)

Examples

tnsr <- rand_tensor(c(3,4,5))
dim(tperm(tnsr,perm=c(2,1,3)))
dim(tperm(tnsr,perm=c(1,3,2)))

Tensor Times List

Description

Contracted (m-Mode) product between a Tensor of arbitrary number of modes and a list of matrices. The result is folded back into Tensor.

Usage

ttl(tnsr, list_mat, ms = NULL)

Arguments

Details

Performs ttm repeated for a single Tensor and a list of matrices on multiple modes. For instance, suppose we want to do multiply a Tensor object tnsr with three matrices mat1, mat2, mat3 on modes 1, 2, and 3. We could do ttm(ttm(ttm(tnsr,mat1,1),mat2,2),3), or we could do ttl(tnsr,list(mat1,mat2,mat3),c(1,2,3)). The order of the matrices in the list should obviously match the order of the modes. This is a common operation for various Tensor decompositions such as CP and Tucker. For the math on the m-Mode Product, see Kolda and Bader (2009).

Value

Tensor object with K modes

Note

The returned Tensor does not drop any modes equal to 1.

References

T. Kolda, B. Bader, "Tensor decomposition and applications". SIAM Applied Mathematics and Applications 2009.

See Also

ttm

Examples

tnsr <- new("Tensor",3L,c(3L,4L,5L),data=runif(60))
lizt <- list('mat1' = matrix(runif(30),ncol=3), 
'mat2' = matrix(runif(40),ncol=4),
'mat3' = matrix(runif(50),ncol=5))
ttl(tnsr,lizt,ms=c(1,2,3))

Tensor Times Matrix (m-Mode Product)

Description

Contracted (m-Mode) product between a Tensor of arbitrary number of modes and a matrix. The result is folded back into Tensor.

Usage

ttm(tnsr, mat, m = NULL)

Arguments

Details

By definition, rs_unfold(ttm(tnsr,mat),m) = mat%*%rs_unfold(tnsr,m), so the number of columns in mat must match the mth mode of tnsr. For the math on the m-Mode Product, see Kolda and Bader (2009).

Value

a Tensor object with K modes

Note

The mth mode of tnsr must match the number of columns in mat. By default, the returned Tensor does not drop any modes equal to 1.

References

T. Kolda, B. Bader, "Tensor decomposition and applications". SIAM Applied Mathematics and Applications 2009.

See Also

ttl, rs_unfold-methods

Examples

tnsr <- new("Tensor",3L,c(3L,4L,5L),data=runif(60))
mat <- matrix(runif(50),ncol=5)
ttm(tnsr,mat,m=3)

Tucker Decomposition

Description

The Tucker decomposition of a tensor. Approximates a K-Tensor using a n-mode product of a core tensor (with modes specified by ranks) with orthogonal factor matrices. If there is no truncation in one of the modes, then this is the same as the MPCA, mpca. If there is no truncation in all the modes (i.e. ranks = tnsr@modes), then this is the same as the HOSVD, hosvd. This is an iterative algorithm, with two possible stopping conditions: either relative error in Frobenius norm has gotten below tol, or the max_iter number of iterations has been reached. For more details on the Tucker decomposition, consult Kolda and Bader (2009).

Usage

tucker(tnsr, ranks = NULL, max_iter = 25, tol = 1e-05)

Arguments

Details

Uses the Alternating Least Squares (ALS) estimation procedure also known as Higher-Order Orthogonal Iteration (HOOI). Intialized using a (Truncated-)HOSVD. A progress bar is included to help monitor operations on large tensors.

Value

a list containing the following:

Z

the core tensor, with modes specified by ranks

U

a list of orthgonal factor matrices - one for each mode, with the number of columns of the matrices given by ranks

conv

whether or not resid < tol by the last iteration

est

estimate of tnsr after compression

norm_percent

the percent of Frobenius norm explained by the approximation

fnorm_resid

the Frobenius norm of the error fnorm(est-tnsr)

all_resids

vector containing the Frobenius norm of error for all the iterations

Note

The length of ranks must match tnsr@num_modes.

References

T. Kolda, B. Bader, "Tensor decomposition and applications". SIAM Applied Mathematics and Applications 2009.

See Also

hosvd, mpca

Examples

tnsr <- rand_tensor(c(4,4,4,4))
tuckerD <- tucker(tnsr,ranks=c(2,2,2,2))
tuckerD$conv 
tuckerD$norm_percent
plot(tuckerD$all_resids)

Tensor Unfolding

Description

Unfolds the tensor into a matrix, with the modes in rs onto the rows and modes in cs onto the columns. Note that c(rs,cs) must have the same elements (order doesn't matter) as x@modes. Within the rows and columns, the order of the unfolding is determined by the order of the modes. This convention is consistent with Kolda and Bader (2009).

Usage

unfold(tnsr, row_idx, col_idx)

## S4 method for signature 'Tensor'
unfold(tnsr, row_idx = NULL, col_idx = NULL)

Arguments

Details

For Row Space Unfolding or m-mode Unfolding, see rs_unfold-methods. For Column Space Unfolding or matvec, see cs_unfold-methods.

vec-methods returns the vectorization of the tensor.

unfold(tnsr,row_idx=NULL,col_idx=NULL)

Value

matrix with prod(row_idx) rows and prod(col_idx) columns

References

T. Kolda, B. Bader, "Tensor decomposition and applications". SIAM Applied Mathematics and Applications 2009.

See Also

k_unfold-methods and matvec-methods

Examples

tnsr <- rand_tensor()
matT3<-unfold(tnsr,row_idx=2,col_idx=c(3,1))

Unmatvec Folding of Matrix

Description

The inverse operation to matvec-methods, turning a matrix into a Tensor. For a full account of matrix folding/unfolding operations, consult Kolda and Bader (2009).

Usage

unmatvec(mat, modes = NULL)

Arguments

Value

Tensor object with modes given by modes

References

T. Kolda, B. Bader, "Tensor decomposition and applications". SIAM Applied Mathematics and Applications 2009.

See Also

matvec-methods, fold, k_fold

Examples

tnsr <- new("Tensor",3L,c(3L,4L,5L),data=runif(60))
matT1<-matvec(tnsr)
identical(unmatvec(matT1,modes=c(3,4,5)),tnsr)

Tensor Vec

Description

Turns the tensor into a single vector, following the convention that earlier indices vary slower than later indices.

Usage

vec(tnsr)

## S4 method for signature 'Tensor'
vec(tnsr)

Arguments

Details

vec(tnsr)

Value

vector with length prod(x@modes)

References

T. Kolda, B. Bader, "Tensor decomposition and applications". SIAM Applied Mathematics and Applications 2009.

Examples

tnsr <- rand_tensor(c(4,5,6,7))
vec(tnsr)