# `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`):
```text
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**
::::{tab-set}
:class: pls4all-bindings
:::{tab-item} C ABI · libn4m
:sync: c
:class-label: lang-c
```c
/* 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);
```
:::
:::{tab-item} Python · pls4all (raw)
:sync: python-raw
:class-label: lang-python
```python
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"), …
```
:::
:::{tab-item} Python · pls4all.sklearn
:sync: python-sklearn
:class-label: lang-python
```python
from pls4all.sklearn import NPLSRegression
mdl = NPLSRegression(n_components=2, mode_j, mode_k)
mdl.fit(X, y)
y_hat = mdl.predict(X_test)
```
:::
:::{tab-item} R · pls4all_method()
:sync: r-dispatcher
:class-label: lang-r
```r
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.
```
:::
:::{tab-item} MATLAB · pls4all (MEX)
:sync: matlab-mex
:class-label: lang-matlab
```matlab
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);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
```matlab
mdl = pls4all.fit("n_pls", X, y, "NumComponents", 4);
yhat = predict(mdl, Xtest);
```
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`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`](../benchmarks/overview.md). 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 · ✗ 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`](../benchmarks/methodology.md)). 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.
::::{tab-set}
:class: parity-tabs
:::{tab-item} 1 thread
:sync: threads-1
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×48 (ms) | 250×50 (ms) | 500×50 (ms) | 500×500 (ms) | 500×2500 (ms) | 2500×50 (ms) | 2500×500 (ms) | 2500×2500 (ms) | 10000×50 (ms) | 10000×500 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas | ≈ | 21.8 ms | 30.1 ms🏆 | 69.3 ms🏆 | 281.8 ms | 22.3 ms | 24.8 ms | 65.7 ms | 271.8 ms | 1.3 s | 408.6 ms | 1.6 s | 9.2 s🏆 | 1.6 s | 8.5 s |
pls4all.cpp.blas+omp | ≈ | 22.4 ms | 36.4 ms | 81.9 ms | 294.8 ms | 20.9 ms | 28.9 ms | 60.2 ms | 295.6 ms | 1.3 s | 389.6 ms🏆 | 1.6 s | 9.6 s | 1.6 s🏆 | 8.4 s🏆 |
pls4all.cpp.omp | ≈ | 21.4 ms🏆 | 36.6 ms | 70.1 ms | 274.7 ms🏆 | 18.5 ms🏆 | 24.3 ms🏆 | 63.2 ms | 262.2 ms🏆 | 1.3 s🏆 | 406.9 ms | 1.6 s🏆 | 9.4 s | 1.6 s | 9.0 s |
pls4all.cpp.ref | ≈ | 22.6 ms | 35.7 ms | 72.9 ms | 281.5 ms | 21.7 ms | 26.1 ms | 59.6 ms🏆 | 268.9 ms | 1.3 s | 418.4 ms | 1.6 s | 9.5 s | 1.6 s | 8.9 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 21.9 ms | — | — | — | 26.0 ms | 24.7 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +7e+00 | 2.77 ms | — | — | — | 2.45 ms | 2.77 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +7e+00 | 11.2 ms | — | — | — | 7.43 ms | 9.51 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +7e+00 | 18.4 ms | — | — | — | 9.52 ms | 10.2 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +7e+00 | 18.8 ms | — | — | — | 8.21 ms | 9.90 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +7e+00 | 18.4 ms | — | — | — | 9.59 ms | 10.0 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +7e+00 | 4.76 ms | — | — | — | 3.27 ms | 4.26 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +7e+00 | 5.51 ms | — | — | — | 3.82 ms | 5.36 ms | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_tensorly | source | 23.2 ms | — | — | — | 25.2 ms | 24.7 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×48 (ms) | 250×50 (ms) | 500×50 (ms) | 500×500 (ms) | 500×2500 (ms) | 2500×50 (ms) | 2500×500 (ms) | 2500×2500 (ms) | 10000×50 (ms) | 10000×500 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas | ✓ ref | — | — | — | — | 19.0 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref | — | — | — | — | 20.4 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref | — | — | — | — | 19.5 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref | — | — | — | — | 19.1 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 20.6 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +5e+00 | — | — | — | — | 2.14 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +5e+00 | — | — | — | — | 6.61 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +5e+00 | — | — | — | — | 7.82 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +5e+00 | — | — | — | — | 7.71 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +5e+00 | — | — | — | — | 9.68 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +7e+00 | — | — | — | — | 4.60 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +7e+00 | — | — | — | — | 3.96 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_tensorly | source | — | — | — | — | 24.1 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×48 (ms) | 250×50 (ms) | 500×50 (ms) | 500×500 (ms) | 500×2500 (ms) | 2500×50 (ms) | 2500×500 (ms) | 2500×2500 (ms) | 10000×50 (ms) | 10000×500 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas | ✓ ref | — | — | — | — | 18.9 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref | — | — | — | — | 18.8 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref | — | — | — | — | 19.4 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref | — | — | — | — | 18.6 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 18.4 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✗ +5e+00 | — | — | — | — | 2.00 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✗ +5e+00 | — | — | — | — | 4.94 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✗ +5e+00 | — | — | — | — | 6.61 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✗ +5e+00 | — | — | — | — | 6.08 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✗ +5e+00 | — | — | — | — | 5.87 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +7e+00 | — | — | — | — | 2.76 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +7e+00 | — | — | — | — | 3.29 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_tensorly | source | — | — | — | — | 19.1 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)