# 5. Discretized Partial Least Squares (`dPLS`)

#### Koki Tsuyuzaki

#### 2024-05-11

# Introduction

In this vignette, we consider approximating multiple matrices as a product of ternary (or non-negative) low-rank matrices (a.k.a., factor matrices).

Test data is available from `toyModel`.

    library("dcTensor")
    X <- dcTensor::toyModel("dPLS_Easy")

You will see that there are five blocks in the data matrix as follows.

    suppressMessages(library("fields"))
    layout(t(1:3))
    image.plot(X[[1]], main="X1", legend.mar=8)
    image.plot(X[[2]], main="X2", legend.mar=8)
    image.plot(X[[3]], main="X3", legend.mar=8)

# Semi-Ternary Simultaneous Matrix Factorization (STSMF)

Here, we introduce the ternary regularization to take {-1,0,1} values in \(V_{k}\) as below:

\[ \max{\mathrm{tr} \left( V_{j}'X_{j}'X_{k}V_{k} \right)}\ \mathrm{s.t.}\ j ≠k, V \in \{-1,0,1\}, \] where \(j\) and \(k\) range from \(1\) to \(K\), \(K\) is the number of matrices, \(X_{k}\) (\(N \times M_{k}\)) is a \(k\)-th data matrix and \(V_{k}\) (\(M_{k} \times J\)) is a \(k\)-th ternary loading matrix. In `dcTensor` package, the object function is optimized by combining gradient-descent algorithm (Tsuyuzaki 2020) and ternary regularization.

## Basic Usage

In STSMF, 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`) are also available. For the details of arguments of dPLS, see `?dPLS`. After the calculation, various objects are returned by `dPLS`. STSMF is achieved by specifying the ternary regularization parameter as a large value like the below:

    set.seed(123456)
    out_dPLS <- dPLS(X, Ter_V=1E+5, J=3)
    str(out_dPLS, 2)

    ## List of 6
    ##  $ U            :List of 3
    ##   ..$ : num [1:100, 1:3] 8722 8926 8821 8626 8589 ...
    ##   ..$ : num [1:100, 1:3] 5888 5898 6044 5910 5695 ...
    ##   ..$ : num [1:100, 1:3] 3879 3904 3961 3806 3909 ...
    ##  $ V            :List of 3
    ##   ..$ : num [1:300, 1:3] 0.96 0.966 0.973 0.95 0.925 ...
    ##   ..$ : num [1:200, 1:3] 0.887 0.892 0.881 0.913 0.913 ...
    ##   ..$ : num [1:150, 1:3] 0.0252 0.0332 0.0286 0.0346 0.0244 ...
    ##  $ RecError     : Named num [1:101] 1.00e-09 1.88e+06 1.87e+06 1.83e+06 1.79e+06 ...
    ##   ..- attr(*, "names")= chr [1:101] "offset" "1" "2" "3" ...
    ##  $ TrainRecError: Named num [1:101] 1.00e-09 1.88e+06 1.87e+06 1.83e+06 1.79e+06 ...
    ##   ..- 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 9.92e-01 5.11e-03 2.20e-02 2.12e-02 ...
    ##   ..- attr(*, "names")= chr [1:101] "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_dPLS$RecError[-1]), type="b", main="Reconstruction Error")
    plot(log10(out_dPLS$RelChange[-1]), type="b", main="Relative Change")

The products of \(U_{k}\) and \(V_{k}\) (\(k = 1 \ldots K\)) show whether the original data matrices are well-recovered by `dPLS`.

    recX <- lapply(seq_along(X), function(x){
      out_dPLS$U[[x]] %*% t(out_dPLS$V[[x]])
    })
    layout(rbind(1:3, 4:6))
    image.plot(t(X[[1]]))
    image.plot(t(X[[2]]))
    image.plot(t(X[[3]]))
    image.plot(t(recX[[1]]))
    image.plot(t(recX[[2]]))
    image.plot(t(recX[[3]]))

The histograms of \(V_{k}\)s show that all the factor matrices \(V_{k}\) looks ternary.

    layout(rbind(1:3, 4:6))
    hist(out_dPLS$U[[1]], breaks=100)
    hist(out_dPLS$U[[2]], breaks=100)
    hist(out_dPLS$U[[3]], breaks=100)
    hist(out_dPLS$V[[1]], breaks=100)
    hist(out_dPLS$V[[2]], breaks=100)
    hist(out_dPLS$V[[3]], breaks=100)

# Session Information

    ## R version 4.3.1 (2023-06-16)
    ## Platform: x86_64-pc-linux-gnu (64-bit)
    ## Running under: Ubuntu 22.04.3 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
    ## LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.20.so;  LAPACK version 3.10.0
    ##
    ## locale:
    ##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    ##  [3] LC_TIME=en_US.UTF-8        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
    ##
    ## time zone: Etc/UTC
    ## tzcode source: system (glibc)
    ##
    ## attached base packages:
    ## [1] stats     graphics  grDevices utils     datasets  methods   base
    ##
    ## other attached packages:
    ## [1] nnTensor_1.2.0    fields_15.2       viridisLite_0.4.2 spam_2.9-1
    ## [5] dcTensor_1.3.0
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] gtable_0.3.4       jsonlite_1.8.7     highr_0.10         compiler_4.3.1
    ##  [5] maps_3.4.1         Rcpp_1.0.11        plot3D_1.4         tagcloud_0.6
    ##  [9] jquerylib_0.1.4    scales_1.2.1       yaml_2.3.7         fastmap_1.1.1
    ## [13] ggplot2_3.4.3      R6_2.5.1           tcltk_4.3.1        knitr_1.43
    ## [17] MASS_7.3-60        dotCall64_1.0-2    misc3d_0.9-1       tibble_3.2.1
    ## [21] munsell_0.5.0      pillar_1.9.0       bslib_0.5.1        RColorBrewer_1.1-3
    ## [25] rlang_1.1.1        utf8_1.2.3         cachem_1.0.8       xfun_0.40
    ## [29] sass_0.4.7         cli_3.6.1          magrittr_2.0.3     digest_0.6.33
    ## [33] grid_4.3.1         rTensor_1.4.8      lifecycle_1.0.3    vctrs_0.6.3
    ## [37] evaluate_0.21      glue_1.6.2         fansi_1.0.4        colorspace_2.1-0
    ## [41] rmarkdown_2.24     pkgconfig_2.0.3    tools_4.3.1        htmltools_0.5.6

# References

Tsuyuzaki, K. et al. 2020. “Benchmarking Principal Component Analysis for Large-Scale Single-Cell RNA-Sequencing.” BMC Genome Biology 21(1): 9.