n_pls — N-way PLS (Trilinear PLS, Bro 1996)

Group: Multi-block / cross-modal · Registry tolerance: 1e-06

Description

N-PLS — 3-way tensor PLS (PARAFAC + OLS by default; Bro 1996 opt-in)

From the pls4all.sklearn.NPLSRegression docstring:

N-PLS (3-way tensor) regression (Bro 1996).

Registry note — Python tensorly.parafac + OLS reference (rank = n_components, init=’random’, random_state=0). pls4all default now matches this convention bit-for-bit. The Bro 1996 multilinear PLS C++ kernel is still available as an opt-in via legacy=True.

Parameters

Name

Type

Default

Notes

n_components

int

2

Number of latent components extracted (k).

mode_j

int

—

Length of the second mode (J) of the 3-way input tensor.

mode_k

int

—

Length of the third mode (K) of the 3-way input tensor.

Explanations

Bibliographic source

Bro, R. (1996). Multiway calibration. Multilinear PLS. Journal of Chemometrics 10(1), 47–61.

Mathematical principle

When the predictor is naturally a tensor \(\mathcal{X} \in \mathbb{R}^{n \times J \times K}\) — e.g. excitation × emission fluorescence spectra, or wavelength × time chromatographic data — N-way PLS decomposes \(\mathcal{X}\) into a sum of rank-one tensor components: \(\mathcal{X} \approx \sum_{a=1}^{k} \mathbf{t}_a \circ \mathbf{w}_a^{J} \circ \mathbf{w}_a^{K}\), where \(\circ\) is the outer product. The score vectors \(\mathbf{t}_a\) are computed to maximise covariance with \(\mathbf{y}\), the loading vectors \(\mathbf{w}_a^{J}, \mathbf{w}_a^{K}\) live in the respective tensor modes.

Compared to unfolding the tensor and running standard PLS, N-PLS respects the multilinear structure of the data and produces interpretable per-mode loadings. The computational cost is comparable to standard PLS — matrix-vector products in each mode rather than one large matrix-vector product.

Note that pls4all takes the tensor as a flattened matrix plus mode_j and mode_k shape parameters; the kernel reshapes internally.

Implementation

n4m_n_pls_fit. Reference: Python tensorly 0.9.0 (tensorly.regression.tucker_regression) and Bro’s original MATLAB code.

MATLAB header (bindings/matlab/+pls4all/NPlsRegression.m):

pls4all.NPlsRegression  N-PLS (3-way tensor) regression (Bro 1996).

Usage

Every pls4all binding tab dispatches into the same C kernel; the external libraries listed at the bottom of the page are the parity references registered in benchmarks.parity_timing.registry. Switch tabs to read the same fit in your language. The R package now ships drop-in-compatible facades for the CRAN pls package (plsr, pcr, mvr) and for the mdatools::pls(x, y, ...) matrix idiom — those tabs appear only on the methods that have a meaningful equivalence.

pls4all bindings

/* C ABI — libn4m */
n4m_context_t* ctx = n4m_context_create();
n4m_config_t*  cfg = n4m_config_create();
n4m_method_result_t* res = NULL;
n4m_n_pls_fit(ctx, cfg, &x_view, &y_view, /* hyperparams */, &res);
/* … read coefficients / mask / scores via */
/* n4m_method_result_get_double_matrix / vector / scalar … */
n4m_method_result_destroy(res);
n4m_config_destroy(cfg);
n4m_context_destroy(ctx);

import pls4all
from pls4all._methods import n_pls_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
    res = n_pls_fit(ctx, cfg, X, y, n_components=4)
# then: res.matrix("predictions"), res.matrix("coefficients"),
# res.vector("mask"), res.scalar("intercept"), …

from pls4all.sklearn import NPLSRegression
mdl = NPLSRegression(n_components=2, mode_j, mode_k)
mdl.fit(X, y)
y_hat = mdl.predict(X_test)

library(pls4all)
# Unified low-level dispatcher (May 2026 R cleanup):
res <- pls4all_method("n_pls", X, y,
                      n_components = 4L, params = list(mode_j = 8L, mode_k = 6L))
# res is a named list with MethodResult arrays/scalars.
# selected_indices / top_k_intervals are 1-based.

res = pls4all.n_pls(X, y, 4);
% see header of bindings/matlab/+pls4all/n_pls.m for full
% parameter surface:
%   res = n_pls(X_flat, Y, n_components, mode_j, mode_k)
yhat = predict(res, Xtest);

mdl  = pls4all.fit("n_pls", X, y, "NumComponents", 4);
yhat = predict(mdl, Xtest);

Registry parity references 📐

  • 📐 ref.python_tensorly (python · python) — tensorly 0.9.0 · strict (rmse_rel ≤ 1e-06) — Python tensorly.parafac + OLS on mode-1 loadings. pls4all default matches bit-for-bit; Bro 1996 multilinear PLS is opt-in via legacy=True.

Benchmarks

Adaptive wall-clock per cell measured against full_matrix.csv. Only backends that implement this method are listed; libraries without the method are omitted.

Verdict  ·  ✓ ref / ≈ ref / ~ shape mark a reference-gate pass at strict / relaxed / qualitative tolerance  ·  ✓ bind = pls4all binding agrees with the C++ baseline  ·  ⇄ cross-check = documented by-design selector/RNG/model, noncanonical API/facade convention, or secondary oracle  ·  ✗ divergent  ·  ⚠ error  ·  — not run. The fastest backend per column is marked 🏆.

Reference gate: strict — numeric equivalence (rmse_rel_tol ≤ 1e-06).

Rows tagged with 📐 are the canonical parity references for this method (declared in parity_timing.registry). C++ and external rows show reference parity; pls4all language bindings show binding parity against the C++ backend. Hover the icon for role and tolerance band.

BackendParity200×48 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref19.5 ms
Python · pls4all
pls4all.python✓ bind19.0 ms
pls4all.sklearn⇄ +5e+002.03 ms🏆
R · pls4all
pls4all.R⇄ +5e+004.51 ms
pls4all.R.formula⇄ +5e+005.67 ms
pls4all.R.mdatools⇄ +5e+005.37 ms
pls4all.R.pls⇄ +5e+005.68 ms
Python · external
📐ref.python_tensorlysource19.3 ms
BackendParity200×48 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref19.1 ms
Python · pls4all
pls4all.python✓ bind20.3 ms
pls4all.sklearn⇄ +5e+001.97 ms🏆
R · pls4all
pls4all.R⇄ +5e+006.01 ms
pls4all.R.formula⇄ +5e+006.25 ms
pls4all.R.mdatools⇄ +5e+005.93 ms
pls4all.R.pls⇄ +5e+006.14 ms
Python · external
📐ref.python_tensorlysource19.3 ms
BackendParity200×48 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref18.9 ms
Python · pls4all
pls4all.python✓ bind19.4 ms
pls4all.sklearn⇄ +5e+001.87 ms🏆
R · pls4all
pls4all.R⇄ +5e+004.84 ms
pls4all.R.formula⇄ +5e+005.52 ms
pls4all.R.mdatools⇄ +5e+005.17 ms
pls4all.R.pls⇄ +5e+005.46 ms
Python · external
📐ref.python_tensorlysource18.8 ms

See also: benchmark overview · methods index · interactive dashboard