dNTD)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.
You will see that there are four blocks in the data tensor as follows.
Here, we introduce a non-negative tensor decomposition method,
non-negative Tucker decomposition (NTD (Kim 2007;
CICHOCK 2009)). The difference with the NTF is that different
ranks can be specified for factor matrices such as \(A_1\) (\(J1
\times N\)), \(A_2\) (\(J2 \times M\)), and \(A_3\) (\(J3
\times L\)) and that the core tensor can have non-negative values
in the non-diagonal elements. This degree of freedom will show that NTD
fits the second set of data well. For the details, see NTD
function of nnTensor
package.
In BTF by NTD, rank parameters are 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 dNTD, see ?dNTD. After the
calculation, various objects are returned by dNTD. BTF is
achieved by specifying the binary regularization parameter as a large
value like the below:
set.seed(123456)
out_dNTD <- dNTD(X, Bin_A=c(1e+6, 1e+6, 1e+6), thr=1e-20, rank=c(4,4,4))
str(out_dNTD, 2)## List of 6
## $ S :Formal class 'Tensor' [package "rTensor"] with 3 slots
## $ A :List of 3
## ..$ A1: num [1:4, 1:30] 0 0 0.000353 0 0 ...
## ..$ A2: num [1:4, 1:30] 0 1 0 0 0 ...
## ..$ A3: num [1:4, 1:30] 0 1 0 0 0 ...
## $ RecError : Named num [1:101] 1.00e-19 3.10e-14 3.14e-14 3.10e-14 3.07e-14 ...
## ..- attr(*, "names")= chr [1:101] "offset" "1" "2" "3" ...
## $ TrainRecError: Named num [1:101] 1.00e-19 3.10e-14 3.14e-14 3.10e-14 3.07e-14 ...
## ..- attr(*, "names")= chr [1:101] "offset" "1" "2" "3" ...
## $ TestRecError : Named num [1:101] 1e-19 0e+00 0e+00 0e+00 0e+00 ...
## ..- attr(*, "names")= chr [1:101] "offset" "1" "2" "3" ...
## $ RelChange : Named num [1:101] 1.00e-19 8.96e+16 1.42e-02 1.27e-02 8.93e-03 ...
## ..- attr(*, "names")= chr [1:101] "offset" "1" "2" "3" ...After the calculation, various objects are returned by
dNTD.
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_dNTD$RecError[-1]), type="b", main="Reconstruction Error")
plot(log10(out_dNTD$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 dNTD.
The histograms of \(A_{k}\)s show that \(A_{k}\) looks binary.
layout(t(1:3))
hist(out_dNTD$A[[1]], main="A1", breaks=100)
hist(out_dNTD$A[[2]], main="A2", breaks=100)
hist(out_dNTD$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.
In SBTF by NTD, rank parameters are 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_dNTD2 <- dNTD(X2, Bin_A=c(1e-10, 1e-10, 1e+5), rank=c(4,4,4))
str(out_dNTD2, 2)## List of 6
## $ S :Formal class 'Tensor' [package "rTensor"] with 3 slots
## $ A :List of 3
## ..$ A1: num [1:4, 1:30] 1.29e-08 1.22e-08 6.94e-06 1.04e-08 2.54e-08 ...
## ..$ A2: num [1:4, 1:30] 7.89e-02 1.05e+01 1.73e-16 3.85e-02 3.94e-02 ...
## ..$ A3: num [1:4, 1:30] 0.001979 1.019649 0.001033 0.002067 0.000236 ...
## $ RecError : Named num [1:101] 1.00e-09 2.92e+02 2.92e+02 2.92e+02 2.92e+02 ...
## ..- attr(*, "names")= chr [1:101] "offset" "1" "2" "3" ...
## $ TrainRecError: Named num [1:101] 1.00e-09 2.92e+02 2.92e+02 2.92e+02 2.92e+02 ...
## ..- 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 1.12e+07 1.09e-03 9.36e-04 7.81e-04 ...
## ..- 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_dNTD2$RecError[-1]), type="b", main="Reconstruction Error")
plot(log10(out_dNTD2$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 dNTD.
recX <- recTensor(out_dNTD2$S, out_dNTD2$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_dNTD2$A[[1]], main="A1", breaks=100)
hist(out_dNTD2$A[[2]], main="A2", breaks=100)
hist(out_dNTD2$A[[3]], main="A3", breaks=100)
## R version 4.3.1 (2023-06-16)
## Platform: x86_64-pc-linux-gnu (64-bit)
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