DelayedTensor 1.0.0
Authors: Koki Tsuyuzaki [aut, cre]
Last modified: 2021-10-26 15:33:21
Compiled: Tue Oct 26 17:15:53 2021
suppressPackageStartupMessages(library("DelayedTensor"))
suppressPackageStartupMessages(library("DelayedArray"))
suppressPackageStartupMessages(library("HDF5Array"))
suppressPackageStartupMessages(library("DelayedRandomArray"))
darr <- RandomUnifArray(c(3,4,5))
setVerbose(FALSE)
setSparse(FALSE)
setAutoBlockSize(1E+8)## automatic block size set to 1e+08 bytes (was 1e+08)tmpdir <- tempdir()
setHDF5DumpDir(tmpdir)Tensor decomposition models decompose multiple factor matrices and core tensor. Each factor matrix means the patterns of each mode and is used for the visualization and the downstream analysis. Core tensor means the intensity of the patterns and is used to decide which patterns are informative.
We reimplemented some of the tensor decomposition functions of rTensor using block processing of DelayedArray.
Only tensor decomposition algorithms and utility functions that require
Fast Fourier Transform (e.g., t_mult, t_svd, and t_svd_reconstruct)
are exceptions and have not yet been implemented in DelayedArray
because we are still investigating how to calculate them with
out-of-core manner.
Suppose a tensor \(\mathcal{X} \in \Re^{I \times J \times K}\). Tucker decomposition models decomposes a tensor \(\mathcal{X}\) into a core tensor \(\mathcal{G} \in \Re^{p \times q \times r}\), and multiple factor matrices \(A \in \Re^{I \times p}\), \(B \in \Re^{J \times q}\), and \(C \in \Re^{K \times r}\) (\(p \leq I, q \leq J, \textrm{and}\ r \leq K\)).
\[ \mathcal{X} = \mathcal{G} \times_{1} A \times_{2} B \times_{3} C \]
For simplicity, here we will use a third-order tensor but the Tucker decomposition can be applied to tensors of larger orders.
There are well-known two algorithms; Higher-Order Singular Value Decomposition (HOSVD) and Higher-order Orthogonal Iteration (HOOI).
hosvd performs the HOSVD Tucker decomposition
and tucker performs HOOI Tucker decomposition.
For the details, check the hosvd and tucker functions of
rTensor.
out_hosvd <- hosvd(darr, ranks=c(2,1,3))##
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## $ fnorm_resid: num 1.84out_tucker <- tucker(darr, ranks=c(2,3,2))##
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## $ conv : logi TRUE
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## $ norm_percent: num 68.6
## $ fnorm_resid : num 1.42
## $ all_resids : num [1:3] 1.43 1.42 1.42Suppose a tensor \(\mathcal{X} \in \Re^{I \times J \times K}\). CP decomposition models decomposes a tensor \(X\) into a core diagonal tensor \(\mathcal{G} \in \Re^{r \times r \times r}\), and multiple factor matrices \(A \in \Re^{I \times r}\), \(B \in \Re^{J \times r}\), and \(C \in \Re^{K \times r}\) (\(r \leq \min{\left(I,J,K\right)}\)).
\[ \mathcal{X} = \mathcal{G} \times_{1} A \times_{2} B \times_{3} C \]
For simplicity, here we will use a third-order tensor but the CP decomposition can be applied to tensors of larger orders.
Alternating least squares (ALS) is a well-known algorithm for CP decomposition
and cp performs the ALS CP decomposition.
For the details, check the cp function of rTensor.
out_cp <- cp(darr, num_components=2)##
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## $ conv : logi FALSE
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## $ norm_percent: num 65.3
## $ fnorm_resid : num 1.57
## $ all_resids : num [1:24] 2.71 1.84 1.81 1.79 1.78 ...MPCA is a kind of Tucker decomposition and when the order of tensor is 3,
this is also known as the
Generalized Low-Rank Approximation of Matrices (GLRAM).
For the details, check the mpca function of rTensor.
out_mpca <- mpca(darr, ranks=c(2,2))##
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## $ conv : logi TRUE
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## $ norm_percent: num 68.2
## $ fnorm_resid : num 1.44
## $ all_resids : num [1:3] 1.44 1.44 1.44Suppose a series of 2D data X_{j}, where \(X_{j} \in \Re^{n_{1} \times n_{2}}\), and \(j = 1, \ldots , n_{3}\). PVD models decomposes a tensor \(X_{j}\) into two common factor matrices across all \(X_{j}\), \(P \in \Re^{n_{1} \times r_{1}}\) and \(D \in \Re^{n_{2} \times r_{2}}\), and a \(X_{j}\) specific factor matrix \(V_{j} \in \Re^{r_{1} \times r_{2}}\) (\(r_{1} \leq n_{1}, \textrm{and}\ r_{2} \leq n_{2}\)).
\[ X_{j} = P \times V_{j} \times D \]
For the details, check the pvd function of rTensor.
out_pvd <- pvd(darr,
uranks=rep(2,dim(darr)[3]), wranks=rep(3,dim(darr)[3]), a=2, b=3)##
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## .. ..@ seed: num [1:3, 1:4, 1:5] 0.761 0.847 0.599 0.962 0.754 ...
## $ norm_percent: num 71.7
## $ fnorm_resid : num 1.28## R version 4.1.1 (2021-08-10)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 20.04.3 LTS
##
## Matrix products: default
## BLAS: /home/biocbuild/bbs-3.14-bioc/R/lib/libRblas.so
## LAPACK: /home/biocbuild/bbs-3.14-bioc/R/lib/libRlapack.so
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_GB LC_COLLATE=C
## [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] stats4 stats graphics grDevices utils datasets methods
## [8] base
##
## other attached packages:
## [1] DelayedRandomArray_1.2.0 HDF5Array_1.22.0 rhdf5_2.38.0
## [4] DelayedArray_0.20.0 IRanges_2.28.0 S4Vectors_0.32.0
## [7] MatrixGenerics_1.6.0 matrixStats_0.61.0 BiocGenerics_0.40.0
## [10] Matrix_1.3-4 DelayedTensor_1.0.0 BiocStyle_2.22.0
##
## loaded via a namespace (and not attached):
## [1] Rcpp_1.0.7 rTensor_1.4.8 bslib_0.3.1
## [4] compiler_4.1.1 BiocManager_1.30.16 jquerylib_0.1.4
## [7] rhdf5filters_1.6.0 tools_4.1.1 digest_0.6.28
## [10] jsonlite_1.7.2 evaluate_0.14 lattice_0.20-45
## [13] rlang_0.4.12 parallel_4.1.1 yaml_2.2.1
## [16] xfun_0.27 fastmap_1.1.0 stringr_1.4.0
## [19] knitr_1.36 sass_0.4.0 grid_4.1.1
## [22] R6_2.5.1 BiocParallel_1.28.0 rmarkdown_2.11
## [25] bookdown_0.24 irlba_2.3.3 Rhdf5lib_1.16.0
## [28] magrittr_2.0.1 BiocSingular_1.10.0 htmltools_0.5.2
## [31] rsvd_1.0.5 beachmat_2.10.0 dqrng_0.3.0
## [34] ScaledMatrix_1.2.0 stringi_1.7.5 einsum_0.1.0