Tensor-Train Decomposition

Description

Tensor-train is a compact representation for higher-order tensors. Some algorithms for performing tensor-train decomposition are available such as TT-SVD, TT-WOPT, and TT-Cross. For the details of the algorithms, see I. V. Oseledets (2011) <doi:10.1137/090752286>, Yuan Longao, et al (2017) <doi:10.48550/arXiv.1709.02641>, I. V. Oseledets (2010) <doi:10.1016/j.laa.2009.07.024>.

Details

The DESCRIPTION file:

Index of help topics:

as_sptensor             Convert to Simple Sparse Tensor
dtensor                 Dense Tensor Creation
maxvol                  maxvol algorithm
skeleton.decomp         Skeleton Decomposition
TTCross                 Tensor-Train Decomposition by TRCross
TTSVD                   Tensor-Train Decomposition by TTSVD
ttTensor-package        Tensor-Train Decomposition
TTWOPT                  Tensor-Train Decomposition by Tensor-train
                        Weighted OPTimization
unfold                  Unfold a Tensor

Author(s)

Koki Tsuyuzaki [aut, cre], Manabu Ishii [aut], Itoshi Nikaido [aut]

Maintainer: Koki Tsuyuzaki <k.t.the-answer@hotmail.co.jp>

References

I. V. Oseledets, (2011). Tensor-Train Decomposition. SIAM J. SCI. COMPUT.

Yuan, Longhao, et. al., (2017). Completion of high order tensor data with missing entries via tensor-train decomposition. International Conference on Neural Information Processing

I. V. Oseledets, et. al., (2010). TT-cross approximation for multidimensional arrays. Linear Algebra and its Applications

Ali Civril, et. al., (2009). On selecting a maximum volume sub-matrix of a matrix and related problems. Theoretical Computer Science

See Also

TTSVD,TTWOPT,TTCross,skeleton.decomp,maxvol

Examples

ls("package:ttTensor")

Tensor-Train Decomposition by TRCross

Description

TTCross incrementaly decomposes the input tensor by skeleton decomposition algorithm. The algorithm only select the row/column indices and any large temporal matrix are genrated in the process. Therefore, this method is suitable for the sparse tensor.

Usage

TTCross(A, Ranks=NULL, thr=1E-10, num.iter=30)

Arguments

Value

G : Core tensors

Author(s)

Koki Tsuyuzaki

References

I. V. Oseledets, et. al., (2010). TT-cross approximation for multidimensional arrays. Linear Algebra and its Applications

Examples

# TTCross requires sparse tensor input
# Creating a simple example
library("rTensor")
X1 <- array(rnorm(3*4*5), c(3,4,5))
X1 <- as.tensor(X1)
# Convert to sparse format
X2 <- as_sptensor(dtensor(X1@data))
# TT-ranks (should be less than dimensions)
Ranks <- c(p=2, q=2)
# Note: TTCross is designed for sparse tensors
# and may have numerical issues with some inputs
tryCatch({
  out.TTCross <- TTCross(X2, Ranks, num.iter=2)
  print("TTCross completed")
}, error = function(e) {
  print("TTCross encountered an error - this function is experimental")
})

Tensor-Train Decomposition by TTSVD

Description

TTSVD incrementaly decomposes the input tensor by singular value decomposition (SVD).

Usage

TTSVD(A, Ranks=NULL, accuracy=NULL)

Arguments

Value

G : Core tensors

Author(s)

Koki Tsuyuzaki

References

I. V. Oseledets, (2011). Tensor-Train Decomposition. SIAM J. SCI. COMPUT.

Examples

library("rTensor")
# Tensor data
X1 <- array(rnorm(3*5*7*9*11), c(3,5,7,9,11))
dimnames(X1) <- list(
    I=paste0("i", 1:3),
    J=paste0("j", 1:5),
    K=paste0("k", 1:7),
    L=paste0("l", 1:9),
    M=paste0("m", 1:11)
    )
X1 <- as.tensor(X1)
# TT-ranks
Ranks <- c(p=2, q=4, r=6, s=8)
# TTSVD
out.TTSVD <- TTSVD(X1, Ranks)
out.TTSVD <- TTSVD(X1, accuracy=1E-10)

Tensor-Train Decomposition by Tensor-train Weighted OPTimization

Description

TTWOPT incrementaly decomposes the input tensor by gradient desecent. The tensor with missing entries is also specified with weight tensor W.

Usage

TTWOPT(X, Ranks, W=NULL, eta=1E-7, thr=1E-10, num.iter=100)

Arguments

Value

G : Core tensors RelChange : The relative change of the error f : The values of the object function RecError : The reconstruction error between data tensor and reconstructed tensor from C, U, and R

Author(s)

Koki Tsuyuzaki

References

Yuan, Longhao, et. al., (2017). Completion of high order tensor data with missing entries via tensor-train decomposition. International Conference on Neural Information Processing

Examples

library("rTensor")
# Tensor data
X1 <- array(rnorm(3*5*7*9*11), c(3,5,7,9,11))
dimnames(X1) <- list(
    I=paste0("i", 1:3),
    J=paste0("j", 1:5),
    K=paste0("k", 1:7),
    L=paste0("l", 1:9),
    M=paste0("m", 1:11)
    )
X1 <- as.tensor(X1)
# TT-ranks
Ranks <- c(p=2, q=4, r=6, s=8)
# TTWOPT
out.TTWOPT <- TTWOPT(X1, Ranks, eta=1E-7)

Convert to Simple Sparse Tensor

Description

Converts an array or matrix to a simple sparse tensor format. This is a minimal implementation to replace the tensorr dependency.

Usage

as_sptensor(x)

Arguments

Details

This function provides a minimal sparse tensor implementation to support the TTCross function without requiring the archived tensorr package. For production use with actual sparse data, consider using specialized sparse tensor packages.

Value

A simple_sparse_tensor object

Examples

# Create a 3D array
x <- array(rnorm(24), dim = c(2, 3, 4))

# Convert to sparse tensor format
sparse_x <- as_sptensor(x)

Dense Tensor Creation

Description

Creates a dense tensor representation. This is a compatibility function that simply returns the input as-is.

Usage

dtensor(x)

Arguments

Details

This function is provided for compatibility with code that previously used the tensorr package. It simply returns the input without modification.

Value

The input array or matrix unchanged

Examples

# Create a 3D array
x <- array(rnorm(24), dim = c(2, 3, 4))

# Create dense tensor (returns x unchanged)
dense_x <- dtensor(x)

maxvol algorithm

Description

maxvol finds the r*r submatrix of maximal volume in C (n*r) by greedily searching the vector of max norm, and subtractction of its projection from the rest of rows. See also http://tensorly.org/stable/_modules/tensorly/contrib/decomposition/mps_decomposition_cross.html#matrix_product_state_cross

Usage

maxvol(C)

Arguments

Value

row_idx : The indices of rows, which make the determinant as large

Author(s)

Koki Tsuyuzaki

References

Ali Civril, et. al., (2009). On selecting a maximum volume sub-matrix of a matrix and related problems. Theoretical Computer Science

See Also

skeleton.decomp

Examples

library("Matrix")
# Matrix data
X3 <- matrix(runif(10*20), nrow=10)
X3 <- as(X3, "sparseMatrix")
# Skeleton Decomposition
out.SKD <- skeleton.decomp(X3, r=3, num.iter=2, thr=1E-5)

Skeleton Decomposition

Description

skeleton.decomp decomposes the input sparse matrix (n*m) and return the three matrices C (n*r), U (r*r), and R (r*m). Only sparse matrix defined by the Matrix package is acceptable as the input.

Usage

skeleton.decomp(A, r, thr=1E-10, num.iter=30)

Arguments

Value

C : A[I, :] U : inverse(A[I, J]) R : A[:, J] rowidx :The indices of rows colidx : The indices of columns RecError : The reconstruction error between data matrix and reconstructed matrix from C, U, and R RelChange : The relative change of the error

Author(s)

Koki Tsuyuzaki

References

I. V. Oseledets, et. al., (2010). TT-cross approximation for multidimensional arrays. Linear Algebra and its Applications

See Also

maxvol

Examples

library("Matrix")
# Matrix data
X3 <- matrix(runif(10*20), nrow=10)
X3 <- as(X3, "sparseMatrix")
# Skeleton Decomposition
out.SKD <- skeleton.decomp(X3, r=3, num.iter=2, thr=1E-5)

Unfold a Tensor

Description

Unfolds a tensor along a specified mode into a matrix representation.

Usage

unfold(x, mode)

Arguments

Details

This function unfolds a tensor along the specified mode into a matrix. It supports simple_sparse_tensor objects, rTensor Tensor objects, and regular arrays/matrices. The function uses rTensor's rs_unfold internally.

Value

A matrix representation of the unfolded tensor

Examples

library(rTensor)

# Create a 3D tensor
x <- array(rnorm(24), dim = c(2, 3, 4))
tensor_x <- as.tensor(x)

# Unfold along mode 1 (using ttTensor's unfold function)
unfolded <- ttTensor::unfold(tensor_x, mode = 1)