# `pls_lda` — PLS-LDA
_Group_: **Classification & GLM** · _Registry tolerance_: `5.0`
## Description
PLS-LDA — LDA on PLS scores (§17 Phase 4)
From the `pls4all.sklearn.PLSLDAClassifier` docstring:
> PLS-LDA on PLS scores (in-sample only).
> **Registry note** — sklearn `PLSRegression(scale=False) -> LinearDiscriminantAnalysis` is the canonical reference. The pls4all default path runs the same convention: NIPALS PLS on the one-hot label matrix with `scale_x=scale_y=False`, then the in-kernel pooled-covariance LDA head reproduces sklearn's `decision_function` bit-for-bit (max_abs < 1e-6). Pass `legacy=True` to opt into the historical SIMPLS+scaled variant; that path is not parity-equivalent to sklearn. R `plsVarSel::lda_from_pls` exists but its return shape differs from pls4all's `decision_scores`.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components` | `int` | `2` | Number of latent components extracted (k). |
| `n_classes` | `int` | `3` | registry benchmark cell value |
## Explanations
### Bibliographic source
Barker, M. & Rayens, W. (2003). *Partial least squares for discrimination*. Journal of Chemometrics 17(3), 166–173.
### Mathematical principle
PLS-LDA is a two-stage classifier: first project $\mathbf{X}$ into the PLS latent space using one-hot encoded class labels as $\mathbf{Y}$, then fit Linear Discriminant Analysis on the resulting scores $\mathbf{T} = \mathbf{X}\mathbf{W}$.
LDA in the latent space is well-conditioned (the score matrix has $k \ll p$ columns by construction), and the PLS projection has already aligned the latent axes with the class separation direction. This is more robust than applying LDA directly to high-dimensional $\mathbf{X}$, where the within-class covariance is singular.
The decision boundary is **linear in the latent space** (and therefore also linear in the original feature space via $\mathbf{W}$). For non-linear class boundaries use PLS-QDA or PLS-logistic.
### Implementation
`n4m_pls_lda_fit`. The reference is composite (sklearn `PLSRegression` + sklearn `LinearDiscriminantAnalysis`); no library exposes a single PLS-LDA call.
MATLAB header (`bindings/matlab/+pls4all/pls_lda.m`):
```text
pls4all.pls_lda Linear Discriminant Analysis on PLS scores.
```
### 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_pls_lda_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 pls_lda_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = pls_lda_fit(ctx, cfg, X, y, n_components=3, y_labels=y_labels)
# 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 PLSLDAClassifier
mdl = PLSLDAClassifier(n_components=2)
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("pls_lda", X, y,
n_components = 3L, params = list(n_classes = 3L))
# 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.pls_lda(X, y, 3);
% see header of bindings/matlab/+pls4all/pls_lda.m for full
% parameter surface:
% res = pls_lda(X, y_labels, n_components, n_classes)
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
_No idiomatic classdef wrapper — invoke `pls4all.fit("pls_lda", X, y, …)` directly from the unified MEX factory._
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`ref.python_scikit_learn`** (python · python) — `scikit-learn` 1.8.0 · qualitative (rmse_rel ≤ 5e+00) — sklearn `PLSRegression -> LinearDiscriminantAnalysis` pipeline. pls4all's PLS-LDA uses a single SIMPLS pass with an internal LDA head; sklearn fits PLS on dummy-encoded targets and feeds the scores into LDA — both are LDA on PLS scores but the latent bases diverge. We compare class boundaries via one-hot decision scores.
:::
### 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**: qualitative — shape/smoke comparison only. The external library and pls4all do not produce numerically equivalent output for this method (see the MethodSpec notes); the `rmse_rel_tol ≤ 5e+00` budget is set wide on purpose. Treat ~ shape as *“we ran both, both finished”*, not as numerical agreement.
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) | 200×40 (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 | ≈ +1e+00 | 2.40 ms🏆 | 1.24 ms🏆 | 14.7 ms | 64.6 ms | 1.59 ms | 3.20 ms | 8.11 ms | 68.6 ms | 314.5 ms🏆 | 31.2 ms🏆 | 322.8 ms | 1.7 s | 138.4 ms | 1.4 s |
pls4all.cpp.blas+omp | ≈ +1e+00 | 3.15 ms | 1.85 ms | 11.5 ms🏆 | 63.9 ms🏆 | 1.58 ms | 2.71 ms🏆 | 5.34 ms🏆 | 66.1 ms | 325.3 ms | 35.0 ms | 309.4 ms🏆 | 1.7 s | 131.6 ms | 1.4 s |
pls4all.cpp.omp | ≈ +1e+00 | 2.58 ms | 1.35 ms | 12.7 ms | 65.3 ms | 1.54 ms🏆 | 2.77 ms | 6.16 ms | 63.1 ms🏆 | 339.0 ms | 31.7 ms | 314.2 ms | 1.7 s | 128.6 ms🏆 | 1.4 s |
pls4all.cpp.ref | ≈ +1e+00 | 2.87 ms | 2.39 ms | 13.6 ms | 65.6 ms | 1.58 ms | 2.87 ms | 7.82 ms | 63.5 ms | 318.3 ms | 32.7 ms | 327.6 ms | 1.7 s🏆 | 131.2 ms | 1.4 s🏆 |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 2.59 ms | — | — | — | 1.67 ms | 2.74 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 1e-14 | 3.28 ms | — | — | — | 2.70 ms | 4.72 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 9e-15 | 13.9 ms | — | — | — | 6.73 ms | 13.4 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 9e-15 | 29.9 ms | — | — | — | 6.44 ms | 13.2 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 9e-15 | 22.8 ms | — | — | — | 6.34 ms | 12.1 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 9e-15 | 23.5 ms | — | — | — | 7.58 ms | 12.7 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +2e+00 | 4.53 ms | — | — | — | 3.09 ms | 4.52 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +2e+00 | 4.65 ms | — | — | — | 3.18 ms | 5.01 ms | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | 4.46 ms | — | — | — | 3.08 ms | 4.59 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×40 (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 | ~ shape 1e+00 | — | — | — | — | 1.60 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 1e+00 | — | — | — | — | 1.51 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 1e+00 | — | — | — | — | 1.47 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 1e+00 | — | — | — | — | 2.36 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ 9e-16 | — | — | — | — | 2.38 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 2e-15 | — | — | — | — | 1.82 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 2e-15 | — | — | — | — | 5.58 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 2e-15 | — | — | — | — | 6.36 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 2e-15 | — | — | — | — | 6.45 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 2e-15 | — | — | — | — | 6.12 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +2e+00 | — | — | — | — | 2.61 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +2e+00 | — | — | — | — | 3.56 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | — | — | — | — | 2.68 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×40 (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 | ~ shape 1e+00 | — | — | — | — | 1.41 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 1e+00 | — | — | — | — | 1.40 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 1e+00 | — | — | — | — | 1.43 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 1e+00 | — | — | — | — | 1.39 ms🏆 | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ 9e-16 | — | — | — | — | 1.51 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 2e-15 | — | — | — | — | 1.57 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 2e-15 | — | — | — | — | 3.96 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 2e-15 | — | — | — | — | 4.73 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 2e-15 | — | — | — | — | 4.59 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 2e-15 | — | — | — | — | 4.75 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +2e+00 | — | — | — | — | 2.40 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +2e+00 | — | — | — | — | 2.85 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | — | — | — | — | 2.30 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)