Multi-Way Component Analysis

Description

For single tensor data, any matrix factorization method can be specified the matricised tensor in each dimension by Multi-way Component Analysis (MWCA). An originally extended MWCA is also implemented to specify and decompose multiple matrices and tensors simultaneously (CoupledMWCA). See the reference section of GitHub README.md <https://github.com/rikenbit/mwTensor>, for details of the methods.

Details

The DESCRIPTION file:

Index of help topics:

CoupledMWCA             Coupled Multi-way Component Analysis
                        (CoupledMWCA)
CoupledMWCAParams-class
                        Class "CoupledMWCAParams"
CoupledMWCAResult-class
                        Class "CoupledMWCAResult"
MWCA                    Multi-way Component Analysis (MWCA)
MWCAParams-class        Class "MWCAParams"
MWCAResult-class        Class "MWCAResult"
defaultCoupledMWCAParams
                        Default parameters for CoupledMWCA
defaultMWCAParams       Default parameters for MWCA
mwTensor-package        Multi-Way Component Analysis
myALS_SVD               Alternating Least Square Singular Value
                        Decomposition (ALS-SVD) as an example of
                        user-defined matrix decomposition.
myCX                    CX Decomposition as an example of user-defined
                        matrix decomposition.
myICA                   Independent Component Analysis (ICA) as an
                        example of user-defined matrix decomposition.
myNMF                   Independent Component Analysis (ICA) as an
                        example of user-defined matrix decomposition.
mySVD                   Singular Value Decomposition (SVD) as an
                        example of user-defined matrix decomposition.
plotTensor3Ds           Plot function for visualization of tensor data
                        structure
toyModel                Toy model of coupled tensor data

Author(s)

NA

Maintainer: NA

References

Andrzej Cichocki et al., (2016). Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Andrzej Cichocki et al., (2015). Tensor Decompositions for Signal Processing Applications, IEEE SIGNAL PROCESSING MAGAZINE

Gene H. Golub et al., (2012). Matrix Computation (Johns Hopkins Studies in the Mathematical Sciences), Johns Hopkins University Press

Madeleine Udell et al., (2016). Generalized Low Rank Models, Foundations and Trends in Machine Learning, 9(1).

Andrzej CICHOCK, et. al., (2009). Nonnegative Matrix and Tensor Factorizations.

A. Hyvarinen. (1999). Fast and Robust Fixed-Point Algorithms for Independent Component Analysis. IEEE Transactions on Neural Networks, 10(3), 626-634.

Petros Drineas et al., (2008). Relative-Error CUR Matrix Decompositions, SIAM Journal on Matrix Analysis and Applications, 30(2), 844-881.

See Also

mySVD, myALS_SVD, myNMF, myICA, myCX, MWCA, CoupledMWCA, plotTensor3Ds

Examples

ls("package:mwTensor")

Coupled Multi-way Component Analysis (CoupledMWCA)

Description

The input is assumed to be a CoupledMWCAParams object.

Usage

CoupledMWCA(params)

Arguments

Value

CoupledMWCAResult object.

Author(s)

Koki Tsuyuzaki

See Also

CoupledMWCAParams-class and CoupledMWCAResult-class.

Examples

  if(interactive()){
    # Test data (multiple arrays)
    Xs=list(
        X1=array(runif(7*4), dim=c(7,4)),
        X2=array(runif(4*5*6), dim=c(4,5,6)),
        X3=array(runif(6*8), dim=c(6,8)))
    # Setting of factor matrices
    common_model=list(
        X1=list(I1="A1", I2="A2"),
        X2=list(I2="A2", I3="A3", I4="A4"),
        X3=list(I4="A4", I5="A5"))
    # Default Parameters
    params <- defaultCoupledMWCAParams(Xs=Xs, common_model=common_model)
    # Perform Coupled MWCA
    out <- CoupledMWCA(params)
  }

Class "CoupledMWCAParams"

Description

The parameter object to be specified against CoupledMWCA function.

Objects from the Class

Objects can be created by calls of the form new("CoupledMWCAParams", ...).

Slots

MWCAParams has four settings as follows. For each setting, the list must have the same structure.

1. Data-wise setting Each item must be a list object that is as long as the number of data and is named after the data.

Xs:

A list containing multiple high-dimensional arrays.

mask:

A list containing multiple high-dimensional arrays, in which 0 or 1 values are filled to specify the missing elements.

pseudocount:

The pseudo count to avoid zero division, when the element is zero (Default: Machine Epsilon).

weights:

A list containing multiple high-dimensional arrays, in which some numeric values are specified to weigth each data.

2. Common Model setting Each item must be a nested list object that is as long as the number of data and is named after the data.

common_model:

Each element of the list must be a list corresponding the dimention name of data and common factor matrices name.

3. Common Factor matrix-wise setting Each item must be a list object that is as long as the number of common factor matrices and is named after the factor matrices.

common_initial:

The initial values of common factor matrices. If nothing is specified, random matrices are used.

common_algorithms:

Algorithms used to decompose the matricised tensor in each mode.

common_iteration:

The number of iterations.

common_decomp:

If FALSE is specified, unit matrix is used as the common factor matrix.

common_fix:

If TRUE is specified, the common factor matrix is not updated in the iteration.

common_dims:

The lower dimension of each common factor matrix.

common_transpose:

Whether the common factor matrix is transposed to calculate core tensor.

common_coretype:

If "CP" is specified, all the core tensors become diagonal core tensors. If "Tucker" is specified, all the core tensors become dense core tensors.

4. Specific Model setting Each item must be a nested list object that is as long as the number of data and is named after the data.

specific_model:

Each element of the list must be a list corresponding the dimention name of data and data specific factor matrices name.

5. Specific Factor matrix-wise setting Each item must be a list object that is as long as the number of data specific factor matrices and is named after the factor matrices.

specific_initial:

The initial values of data specific factor matrices. If nothing is specified, random matrices are used.

specific_algorithms:

Algorithms used to decompose the matricised tensor in each mode.

specific_iteration:

The number of iterations.

specific_decomp:

If FALSE is specified, unit matrix is used as the data specific factor matrix.

specific_fix:

If TRUE is specified, the data specific factor matrix is not updated in the iteration.

specific_dims:

The lower dimension of each data specific factor matrix.

specific_transpose:

Whether the data specific factor matrix is transposed to calculate core tensor.

specific_coretype:

If "CP" is specified, all the core tensors become diagonal core tensors. If "Tucker" is specified, all the core tensors become dense core tensors.

6. Other option Each item must to be a vector of length 1.

specific:

Whether data specific factor matrices are also calculated.

thr:

The threshold to stop the iteration. The higher the value, the faster the iteration will stop.

viz:

Whether the output is visualized.

figdir:

When viz=TRUE, whether the plot is output in the directory.

verbose:

Whether the process is monitored by verbose messages.

Methods

CoupledMWCA

Function to peform CoupledMWCA.

See Also

CoupledMWCAResult-class, CoupledMWCA

Class "CoupledMWCAResult"

Description

The result object genarated by CoupledMWCA function.

Slots

weights:

weights of CoupledMWCAParams.

common_model:

common_model of CoupledMWCAParams.

common_initial:

common_initial of CoupledMWCAParams.

common_algorithms:

common_algorithms of CoupledMWCAParams.

common_iteration:

common_iteration of CoupledMWCAParams.

common_decomp:

common_decomp of CoupledMWCAParams.

common_fix:

common_fix of CoupledMWCAParams.

common_dims:

common_dims of CoupledMWCAParams.

common_transpose:

common_transpose of CoupledMWCAParams.

common_coretype:

common_coretype of CoupledMWCAParams.

common_factors:

Common factor matrices of CoupledMWCA.

common_cores:

Common core tensors of CoupledMWCA.

specific_model:

specific_model of CoupledMWCAParams.

specific_initial:

specific_initial of CoupledMWCAParams.

specific_algorithms:

specific_algorithms of CoupledMWCAParams.

specific_iteration:

specific_iteration of CoupledMWCAParams.

specific_decomp:

specific_decomp of CoupledMWCAParams.

specific_fix:

specific_fix of CoupledMWCAParams.

specific_dims:

specific_dims of CoupledMWCAParams.

specific_transpose:

specific_transpose of CoupledMWCAParams.

specific_coretype:

specific_coretype of CoupledMWCAParams.

specific_factors:

Data specific factor matrices of CoupledMWCA.

specific_cores:

Data specific core tensors of CoupledMWCA.

specific:

specific of CoupledMWCAParams.

thr:

thr of CoupledMWCAParams.

viz:

viz of CoupledMWCAParams.

figdir:

figdir of CoupledMWCAParams.

verbose:

verbose of CoupledMWCAParams.

rec_error:

The reconstructed error.

train_error:

Training Error. train_error + test_error = rec_error.

test_error:

Test Error. train_error + test_error = rec_error.

rel_change:

The relative change of each iteration step.

See Also

CoupledMWCAParams-class, CoupledMWCA

Multi-way Component Analysis (MWCA)

Description

The input is assumed to be a MWCAParams object.

Usage

MWCA(params)

Arguments

Value

MWCAResult object.

Author(s)

Koki Tsuyuzaki

References

Andrzej Cichocki et al., (2016). Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Andrzej Cichocki et al., (2015). Tensor Decompositions for Signal Processing Applications, IEEE SIGNAL PROCESSING MAGAZINE

See Also

MWCAParams-class and MWCAResult-class.

Examples

  if(interactive()){
  	# Test data (single array)
    X <- nnTensor::toyModel("Tucker")@data
    # Default Parameters
    params <- defaultMWCAParams(X)
    # Perform MWCA
    out <- MWCA(params)
  }

Class "MWCAParams"

Description

The parameter object to be specified against MWCA function.

Objects from the Class

Objects can be created by calls of the form new("MWCAParams", ...).

Slots

X:

A high-dimensional array.

mask:

A mask array having the same dimension of X.

pseudocount:

The pseudo count to avoid zero division, when the element is zero (Default: Machine Epsilon).

algorithms:

Algorithms used to decompose the matricised tensor in each mode.

dims:

The lower dimension of each factor matrix.

transpose:

Whether the factor matrix is transposed to calculate core tensor.

viz:

Whether the output is visualized.

figdir:

When viz=TRUE, whether the plot is output in the directory.

Methods

MWCA

Function to peform MWCA.

See Also

MWCAResult-class, MWCA

Class "MWCAResult"

Description

The result object genarated by MWCA function.

Slots

algorithms:

algorithm of MWCAParams.

dims:

dims of MWCAParams.

transpose:

transpose of MWCAParams.

viz:

viz of MWCAParams.

figdir:

figdir of MWCAParams.

factors:

The factor matrices of MWCA.

core:

The core tensor of MWCA.

rec_error:

The reconstructed error.

train_error:

Training Error. train_error + test_error = rec_error.

test_error:

Test Error. train_error + test_error = rec_error.

See Also

MWCAParams-class, MWCA

Default parameters for CoupledMWCA

Description

The input list is assumed to contain multiple arrays.

Usage

defaultCoupledMWCAParams(Xs, common_model)

Arguments

Value

CoupledMWCAParams object.

Author(s)

Koki Tsuyuzaki

References

Andrzej Cichocki et al., (2016). Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Andrzej Cichocki et al., (2015). Tensor Decompositions for Signal Processing Applications, IEEE SIGNAL PROCESSING MAGAZINE

See Also

CoupledMWCAParams-class and MWCAResult-class.

Examples

  if(interactive()){
    # Test data (multiple arrays)
    Xs=list(
        X1=array(runif(7*4), dim=c(7,4)),
        X2=array(runif(4*5*6), dim=c(4,5,6)),
        X3=array(runif(6*8), dim=c(6,8)))
    # Setting of factor matrices
    common_model=list(
        X1=list(I1="A1", I2="A2"),
        X2=list(I2="A2", I3="A3", I4="A4"),
        X3=list(I4="A4", I5="A5"))
    # Default Parameters
    params <- defaultCoupledMWCAParams(Xs=Xs, common_model=common_model)
    # Perform Coupled MWCA
    out <- CoupledMWCA(params)
  }

Default parameters for MWCA

Description

The input is assumed to be an array object.

Usage

defaultMWCAParams(X)

Arguments

Value

MWCAParams object.

Author(s)

Koki Tsuyuzaki

References

Andrzej Cichocki et al., (2016). Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Andrzej Cichocki et al., (2015). Tensor Decompositions for Signal Processing Applications, IEEE SIGNAL PROCESSING MAGAZINE

See Also

MWCAParams-class and MWCAResult-class.

Examples

  if(interactive()){
    # Test data (single array)
    X <- nnTensor::toyModel("Tucker")@data
    # Default Parameters
    params <- defaultMWCAParams(X)
    # Perform MWCA
    out <- MWCA(params)
  }

Alternating Least Square Singular Value Decomposition (ALS-SVD) as an example of user-defined matrix decomposition.

Description

The input data is assumed to be a matrix. When algorithms of MWCAParams and CoupledMWCAParams are specified as "myALS_SVD", This function is called in MWCA and CoupledMWCA.

Usage

myALS_SVD(Xn, k, L2=1e-10, iter=30)

Arguments

Value

The output matrix which has N-rows and k-columns.

Author(s)

Koki Tsuyuzaki

References

Madeleine Udell et al., (2016). Generalized Low Rank Models, Foundations and Trends in Machine Learning, 9(1).

Examples

  if(interactive()){
    # Test data
    matdata <- matrix(runif(10*20), nrow=10, ncol=20)
    # Perform ALS-SVD
    myALS_SVD(matdata, k=3, L2=0.1, iter=10)
  }

CX Decomposition as an example of user-defined matrix decomposition.

Description

The input data is assumed to be a matrix. When algorithms of MWCAParams and CoupledMWCAParams are specified as "myCX", This function is called in MWCA and CoupledMWCA.

Usage

myCX(Xn, k)

Arguments

Value

The output matrix which has N-rows and k-columns.

Author(s)

Koki Tsuyuzaki

References

Petros Drineas et al., (2008). Relative-Error CUR Matrix Decompositions, SIAM Journal on Matrix Analysis and Applications, 30(2), 844-881.

Examples

  if(interactive()){
    # Test data
    matdata <- matrix(runif(10*20), nrow=10, ncol=20)
    # Perform CX
    myCX(matdata, k=3)
  }

Independent Component Analysis (ICA) as an example of user-defined matrix decomposition.

Description

The input data is assumed to be a matrix. When algorithms of MWCAParams and CoupledMWCAParams are specified as "myICA", This function is called in MWCA and CoupledMWCA.

Usage

myICA(Xn, k)

Arguments

Value

The output matrix which has N-rows and k-columns.

Author(s)

Koki Tsuyuzaki

References

A. Hyvarinen. (1999). Fast and Robust Fixed-Point Algorithms for Independent Component Analysis. IEEE Transactions on Neural Networks, 10(3), 626-634.

Examples

  if(interactive()){
    # Test data
    matdata <- matrix(runif(10*20), nrow=10, ncol=20)
    # Perform ICA
    myICA(matdata, k=3)
  }

Independent Component Analysis (ICA) as an example of user-defined matrix decomposition.

Description

The input data is assumed to be a matrix. When algorithms of MWCAParams and CoupledMWCAParams are specified as "myNMF", This function is called in MWCA and CoupledMWCA.

Usage

myNMF(Xn, k, L1=1e-10, L2=1e-10)

Arguments

Value

The output matrix which has N-rows and k-columns.

Author(s)

Koki Tsuyuzaki

References

Andrzej CICHOCK, et. al., (2009). Nonnegative Matrix and Tensor Factorizations.

Examples

  if(interactive()){
    # Test data
    matdata <- matrix(runif(10*20), nrow=10, ncol=20)
    # Perform NMF
    myNMF(matdata, k=3, L1=1e-1, L2=1e-2)
  }

Singular Value Decomposition (SVD) as an example of user-defined matrix decomposition.

Description

The input data is assumed to be a matrix. When algorithms of MWCAParams and CoupledMWCAParams are specified as "mySVD", This function is called in MWCA and CoupledMWCA.

Usage

mySVD(Xn, k)

Arguments

Value

The output matrix which has N-rows and k-columns.

Author(s)

Koki Tsuyuzaki

Examples

  if(interactive()){
    # Test data
    matdata <- matrix(runif(10*20), nrow=10, ncol=20)
    # Perform SVD
    mySVD(matdata, k=3)
  }

Plot function for visualization of tensor data structure

Description

Multiple multi-dimensional arrays and matrices are visualized simultaneously.

Usage

plotTensor3Ds(Xs)

Arguments

Author(s)

Koki Tsuyuzaki

See Also

plotTensor3D and plotTensor2D.

Examples

Xs <- toyModel(model = "coupled_CP_Easy")

tmp <- tempdir()

png(filename=paste0(tmp, "/couled_CP.png"))
plotTensor3Ds(Xs)
dev.off()

Toy model of coupled tensor data

Description

A list object containing multiple arrays are generated.

Usage

toyModel(model = "coupled_CP_Easy", seeds=123)

Arguments

Author(s)

Koki Tsuyuzaki

Examples

Xs1 <- toyModel(model = "coupled_CP_Easy", seeds=123)
Xs2 <- toyModel(model = "coupled_CP_Hard", seeds=123)
Xs3 <- toyModel(model = "coupled_Tucker_Easy", seeds=123)
Xs4 <- toyModel(model = "coupled_Tucker_Hard", seeds=123)
Xs5 <- toyModel(model = "coupled_Complex_Easy", seeds=123)
Xs6 <- toyModel(model = "coupled_Complex_Hard", seeds=123)