dNTF
)In this vignette, we consider approximating a non-negative tensor as a product of binary or non-negative low-rank matrices (a.k.a., factor matrices).
Test data is available from toyModel
.
library("dcTensor")
X <- dcTensor::toyModel("dNTF")
You will see that there are four blocks in the data tensor as follows.
suppressMessages(library("nnTensor"))
plotTensor3D(X)
To decompose a binary tensor (\(\mathcal{X}\)), non-negative CP decomposition (a.k.a. non-negative tensor factorization; NTF (Cichocki 2007; CICHOCK 2009)) can be applied. NTF appoximates \(\mathcal{X}\) (\(N \times M \times L\)) as the mode-product of a core tensor \(S\) (\(J \times J \times J\)) and factor matrices \(A_1\) (\(J \times N\)), \(A_2\) (\(J \times M\)), and \(A_3\) (\(J \times L\)).
\[ \mathcal{X} \approx \mathcal{S} \times_{1} A_1 \times_{2} A_2 \times_{3} A_3\ \mathrm{s.t.}\ \mathcal{S} \geq 0, A_{k} \geq 0\ (k=1 \ldots 3) \]
Note that _{k} is the mode-\(k\) product (CICHOCK 2009) and the core tensor \(S\) has non-negative values only in the diagonal element. For the details, see NTF
function of nnTensor package.
In BTF, a rank parameter \(J\) (\(\leq \min(N, M)\)) is needed to be set in advance. Other settings such as the number of iterations (num.iter
) or factorization algorithm (algorithm
) are also available. For the details of arguments of dNTF, see ?dNTF
. After the calculation, various objects are returned by dNTF
. BTF is achieved by specifying the binary regularization parameter as a large value like the below:
set.seed(123456)
out_dNTF <- dNTF(X, Bin_A=c(1e+2, 1e+2, 1e+2), algorithm="KL", rank=4)
str(out_dNTF, 2)
## List of 6
## $ S : num [1:4] 2.24 2.23 2.24 2.24
## $ A :List of 3
## ..$ : num [1:4, 1:30] 9.99e-01 2.22e-16 2.22e-16 1.00 9.99e-01 ...
## ..$ : num [1:4, 1:30] 1.00 2.22e-16 2.22e-16 2.22e-16 1.00 ...
## ..$ : num [1:4, 1:30] 4.47e-01 9.94e-17 9.93e-17 9.93e-17 4.47e-01 ...
## $ RecError : Named num [1:28] 1.00e-09 2.67e+01 2.45e+01 2.36e+01 2.27e+01 ...
## ..- attr(*, "names")= chr [1:28] "offset" "1" "2" "3" ...
## $ TrainRecError: Named num [1:28] 1.00e-09 2.67e+01 2.45e+01 2.36e+01 2.27e+01 ...
## ..- attr(*, "names")= chr [1:28] "offset" "1" "2" "3" ...
## $ TestRecError : Named num [1:28] 1e-09 0e+00 0e+00 0e+00 0e+00 0e+00 0e+00 0e+00 0e+00 0e+00 ...
## ..- attr(*, "names")= chr [1:28] "offset" "1" "2" "3" ...
## $ RelChange : Named num [1:28] 1.00e-09 2.56e-02 8.80e-02 4.16e-02 3.89e-02 ...
## ..- attr(*, "names")= chr [1:28] "offset" "1" "2" "3" ...
The reconstruction error (RecError
) and relative error (RelChange
, the amount of change from the reconstruction error in the previous step) can be used to diagnose whether the calculation is converged or not.
layout(t(1:2))
plot(log10(out_dNTF$RecError[-1]), type="b", main="Reconstruction Error")
plot(log10(out_dNTF$RelChange[-1]), type="b", main="Relative Change")
The product of core tensor \(S\) and factor matrices \(A_{k}\) shows whether the original data is well-recovered by dNTF
.
recX <- recTensor(out_dNTF$S, out_dNTF$A)
layout(t(1:2))
plotTensor3D(X)
plotTensor3D(recX, thr=0)
The histograms of \(A_{k}\)s show that all the factor matrices \(A_{k}\) looks binary.
layout(t(1:3))
hist(out_dNTF$A[[1]], main="A1", breaks=100)
hist(out_dNTF$A[[2]], main="A2", breaks=100)
hist(out_dNTF$A[[3]], main="A3", breaks=100)
Here, we define this formalization as semi-binary tensor factorization (SBTF). SBTF can capture discrete patterns from non-negative matrices.
To demonstrate SBMF, next we use a non-negative tensor from the nnTensor
package. You will see that there are four blocks in the data tensor as follows.
X2 <- nnTensor::toyModel("CP")
plotTensor3D(X2)
In SBTF, a rank parameter \(J\) (\(\leq \min(N, M)\)) is needed to be set in advance. Other settings such as the number of iterations (num.iter
) or factorization algorithm (algorithm
) are also available. For the details of arguments of dNTF, see ?dNTF
. After the calculation, various objects are returned by dNTF
. SBTF is achieved by specifying the binary regularization parameter as a large value like the below:
set.seed(123456)
out_dNTF2 <- dNTF(X2, Bin_A=c(1e+5, 1e+5, 1e-10), algorithm="KL", rank=4)
str(out_dNTF2, 2)
## List of 6
## $ S : num [1:4] 13.1 31.7 112.1 1474.1
## $ A :List of 3
## ..$ : num [1:4, 1:30] 0.00704 0.00175 0.47548 0.00303 0.00653 ...
## ..$ : num [1:4, 1:30] 0.00905 0.00602 0.10119 0.00226 0.0092 ...
## ..$ : num [1:4, 1:30] 0.1385 0.2206 0.0447 0.0048 0.1523 ...
## $ RecError : Named num [1:101] 1.00e-09 2.46e+04 3.90e+03 2.96e+03 6.38e+03 ...
## ..- attr(*, "names")= chr [1:101] "offset" "1" "2" "3" ...
## $ TrainRecError: Named num [1:101] 1.00e-09 2.46e+04 3.90e+03 2.96e+03 6.38e+03 ...
## ..- attr(*, "names")= chr [1:101] "offset" "1" "2" "3" ...
## $ TestRecError : Named num [1:101] 1e-09 0e+00 0e+00 0e+00 0e+00 0e+00 0e+00 0e+00 0e+00 0e+00 ...
## ..- attr(*, "names")= chr [1:101] "offset" "1" "2" "3" ...
## $ RelChange : Named num [1:101] 1.00e-09 8.23e-01 5.30 3.19e-01 5.36e-01 ...
## ..- attr(*, "names")= chr [1:101] "offset" "1" "2" "3" ...
RecError
and RelChange
can be used to diagnose whether the calculation is converged or not.
layout(t(1:2))
plot(log10(out_dNTF2$RecError[-1]), type="b", main="Reconstruction Error")
plot(log10(out_dNTF2$RelChange[-1]), type="b", main="Relative Change")
The product of core tensor \(S\) and factor matrices \(A_{k}\) shows whether the original data is well-recovered by dNTF
.
recX <- recTensor(out_dNTF2$S, out_dNTF2$A)
layout(t(1:2))
plotTensor3D(X2)
plotTensor3D(recX, thr=0)
The histograms of \(A_{k}\)s show that \(A_{k}\) looks binary.
layout(t(1:3))
hist(out_dNTF2$A[[1]], main="A1", breaks=100)
hist(out_dNTF2$A[[2]], main="A2", breaks=100)
hist(out_dNTF2$A[[3]], main="A3", breaks=100)
## R version 4.4.3 (2025-02-28)
## Platform: x86_64-pc-linux-gnu
## Running under: Rocky Linux 9.5 (Blue Onyx)
##
## Matrix products: default
## BLAS: /opt/R/4.4.3/lib64/R/lib/libRblas.so
## LAPACK: /opt/R/4.4.3/lib64/R/lib/libRlapack.so; LAPACK version 3.12.0
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
## [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
##
## time zone: Asia/Tokyo
## tzcode source: system (glibc)
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] nnTensor_1.3.0 fields_16.3.1 viridisLite_0.4.2 spam_2.11-1
## [5] dcTensor_1.3.1
##
## loaded via a namespace (and not attached):
## [1] gtable_0.3.6 jsonlite_1.8.9 dplyr_1.1.4 compiler_4.4.3
## [5] maps_3.4.3 tidyselect_1.2.1 Rcpp_1.1.0 plot3D_1.4.2
## [9] tagcloud_0.7.0 jquerylib_0.1.4 scales_1.3.0 yaml_2.3.10
## [13] fastmap_1.2.0 ggplot2_3.5.1 R6_2.6.1 generics_0.1.3
## [17] tcltk_4.4.3 knitr_1.50 MASS_7.3-65 dotCall64_1.1-1
## [21] misc3d_0.9-1 tibble_3.3.0 munsell_0.5.1 pillar_1.10.1
## [25] bslib_0.9.0 RColorBrewer_1.1-3 rlang_1.1.6 cachem_1.1.0
## [29] xfun_0.53 sass_0.4.10 cli_3.6.5 magrittr_2.0.3
## [33] digest_0.6.37 grid_4.4.3 rTensor_1.4.9 lifecycle_1.0.4
## [37] vctrs_0.6.5 evaluate_1.0.3 glue_1.8.0 colorspace_2.1-1
## [41] rmarkdown_2.29 pkgconfig_2.0.3 tools_4.4.3 htmltools_0.5.8.1
CICHOCK, A. et al. 2009. Nonnegative Matrix and Tensor Factorizations. Wiley.
Cichocki, A. et al. 2007. “Non-Negative Tensor Factorization Using Alpha and Beta Divergence.” ICASSP ’07, III–1393–III–1396.