Automatic Algorithm Selection in bigPLSR

We choose between XtX (SIMPLS), XX^T (wide-kernel), and NIPALS using $(n,p)$ and a RAM budget.

# In pls_fit(), after arg parsing:
if (identical(algo_in, "auto")) {
  algo_in <- .choose_algorithm_auto(backend, X, y, ncomp)
}

.mem_bytes <- function() {
  gb <- getOption("bigPLSR.mem_budget_gb", 8)
  as.numeric(gb) * (1024^3)
}
.dims_of <- function(X) {
  if (inherits(X, "big.matrix")) c(nrow(X), ncol(X)) else c(NROW(X), NCOL(X))
}

.choose_algorithm_auto <- function(backend, X, y, ncomp) {
  is_big_local <- inherits(X, "big.matrix") || inherits(X, "big.matrix.descriptor")
  dims <- .dims_of(X); n <- as.integer(dims[1]); p <- as.integer(dims[2])
  B <- .mem_bytes()
  bytes <- 8
  need_XtX <- bytes * as.double(p) * as.double(p)      # bytes for p x p
  need_XXt <- bytes * as.double(n) * as.double(n)      # bytes for n x n
  can_XtX  <- need_XtX <= M
  can_XXt  <- need_XXt <= M
  shape_XtX <- (p <= 4L * n)
  shape_XXt <- (n <= 4L * p)
  if (can_XtX && shape_XtX) {
    algo_in <- "simpls"
  } else if (can_XXt && shape_XXt) {
    algo_in <- "widekernelpls"
  } else {
    algo_in <- "nipals"
  }
}

Users can override the memory budget:

options(bigPLSR.memory_budget_bytes = 8L * 1024^3)  # 8 GiB

When does each win?

XtX (SIMPLS): moderate $p$ (fits $p^2$ in RAM). Fast BLAS-3; excellent for $n \gg p$ .
XXᵗ (wide-kernel): moderate $n$ (fits $n^2$ ). Great when $p\gg n$ (wide problems).
NIPALS / streaming: both $p^2$ and $n^2$ exceed budget; supports file-backed scores and large-scale chunked BLAS.

Sanity check

set.seed(1)
n <- 1e5; p <- 200
X <- matrix(rnorm(n*p), n, p)
y <- X[,1]*0.5 + rnorm(n)
bmX <- bigmemory::as.big.matrix(X)
bmy <- bigmemory::as.big.matrix(matrix(y, n, 1))

options(bigPLSR.memory_budget_bytes = 2L*1024^3)
fit <- pls_fit(bmX, bmy, ncomp=3, backend="bigmem", scores="r")
fit$algorithm

Overview

bigPLSR::pls_fit() can automatically choose an algorithm based on problem shape and a user-configurable memory budget:

SIMPLS (XtX route) when forming the p × p cross-product fits in memory.
SIMPLS (XXt / kernel route) when XtX does not fit but XXt (n × n) does.
NIPALS (streaming) when neither XtX nor XXt comfortably fit.

This selection only applies when algorithm = "auto" (the default). Any explicit algorithm = overrides the decision.

Why these choices?

SIMPLS works entirely from centered cross-products, which is fast and numerically robust when the target cross-product fits (either p×p or n×n).
Using XtX is efficient when p is moderate; using XXt is efficient for “wide” problems (p ≫ n) but still bounded by n^2 memory.
NIPALS avoids materializing any large cross-product and can stream from big.memory with fixed working memory; it is the safe fallback when memory is tight.

The decision rule

Let the memory budget be B bytes (defaults to 8 GB, configurable via options(bigPLSR.mem_budget_gb = ...)). With doubles (8 bytes), we estimate the size of each symmetric matrix as:

need_XtX = 8 * p^2
need_XXt = 8 * n^2

Then:

  if (can_XtX && shape_XtX) { algo_in <- "simpls"}.    # XtX
  if (can_XXt && shape_XXt) { algo_in <- "widekernelpls"}. XXt (a.k.a. "kernel" route)
  else                      { algorithm <- "nipals"}         # streaming

Configuring the memory budget

# Use 16 GB as selection budget
options(bigPLSR.mem_budget_gb = 16)

This does not change R’s actual memory limit; it only controls the selection.

Reproducibility knobs

For tight numerical parity in tests:

set.seed(1)
if (requireNamespace("RhpcBLASctl", quietly = TRUE)) {
  RhpcBLASctl::blas_set_num_threads(1L)
  RhpcBLASctl::omp_set_num_threads(1L)
}
# otherwise, you can try environment variables:
# Sys.setenv(OPENBLAS_NUM_THREADS = "1", OMP_NUM_THREADS = "1")

Examples

library(bigPLSR)

n <- 2e3; p <- 5e2
X <- matrix(rnorm(n*p), n, p)
y <- X[,1] - 0.5*X[,2] + rnorm(n)

# Auto will likely pick SIMPLS (XtX) here
fit <- pls_fit(X, y, ncomp = 10, algorithm = "auto")
fit$algorithm  # "simpls"

Wide case:

n <- 200; p <- 4000
X <- matrix(rnorm(n*p), n, p)
y <- rnorm(n)

# If budget is small, auto picks kernel (XXt) or NIPALS
options(bigPLSR.mem_budget_gb = 2)  # small budget
fit <- pls_fit(X, y, ncomp = 5, algorithm = "auto")
fit$algorithm  # "kernelpls" or "nipals" depending on n^2 vs budget

Big-matrix streaming:

library(bigmemory)
n <- 1e6; p <- 50
# (example only; allocate according to your RAM)
# bmX <- big.matrix(n, p, type = "double")
# bmy <- big.matrix(n, 1, type = "double")
# fit <- pls_fit(bmX, bmy, ncomp = 10, backend = "bigmem", algorithm = "auto")
# fit$algorithm  # "simpls" or "nipals"

References

de Jong, S. (1993). SIMPLS: An alternative approach to partial least squares regression. Chemometrics and Intelligent Laboratory Systems, 18(3), 251–263.
Dayal, B., & MacGregor, J. F. (1997). Improved PLS algorithms. Journal of Chemometrics, 11(1), 73–85.
Rosipal, R., & Trejo, L. J. (2001). Kernel Partial Least Squares Regression in Reproducing Kernel Hilbert Space. Journal of Machine Learning Research, 2, 97–123.
Wold, H. (1966, 1985). NIPALS algorithm (original PLS formulation).

Appendix: streaming Gram math

For column blocks $J$ , $K \approx \sum_{J} X_{[:,J]} X_{[:,J]}^\top,\quad (Kv) \leftarrow (Kv) + X_{[:,J]} \big(X_{[:,J]}^\top v\big).$

For row blocks $B$ , $K \approx \sum_{B} X_B X^\top,\quad (Kv) \leftarrow (Kv) + X_B \big(X^\top v\big)_B.$

Center on the fly: $H K H v = K v - \tfrac{1}{n}\mathbf{1}\mathbf{1}^\top K v - \tfrac{1}{n}K\mathbf{1}\mathbf{1}^\top v + \tfrac{1}{n^2}\mathbf{1}\mathbf{1}^\top K \mathbf{1}\,\mathbf{1}^\top v$ . Maintain the needed aggregated vectors once per pass.

Frédéric Bertrand

2025-11-18