In rikenbit/DelayedTensor: R package for sparse and out-of-core arithmetic and decomposition of Tensor

BiocStyle::markdown()

Authors: r packageDescription("DelayedTensor")[["Author"]]
Last modified: r file.info("DelayedTensor_3.Rmd")$mtime
Compiled: r date()

Setting

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)
tmpdir <- tempdir()
setHDF5DumpDir(tmpdir)

Tensor Decomposition

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 r CRANpkg("rTensor") using block processing of r Biocpkg("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 r Biocpkg("DelayedArray") because we are still investigating how to calculate them with out-of-core manner.

Tucker Decomposition

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 r CRANpkg("rTensor").

out_hosvd <- hosvd(darr, ranks=c(2,1,3))
str(out_hosvd)
out_tucker <- tucker(darr, ranks=c(2,3,2))
str(out_tucker)

CANDECOMP/PARAFAC (CP) Decomposition

Suppose 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 r CRANpkg("rTensor").

out_cp <- cp(darr, num_components=2)
str(out_cp)

Multilinear Principal Component Analysis (MPCA)

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 r CRANpkg("rTensor").

out_mpca <- mpca(darr, ranks=c(2,2))
str(out_mpca)

Population Value Decomposition (PVD)

Suppose 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 r CRANpkg("rTensor").

out_pvd <- pvd(darr,
  uranks=rep(2,dim(darr)[3]), wranks=rep(3,dim(darr)[3]), a=2, b=3)
str(out_pvd)

Session information {.unnumbered}

sessionInfo()

rikenbit/DelayedTensor documentation built on Sept. 28, 2024, 5:43 a.m.