# `ridge_pls` — Ridge-augmented PLS
_Group_: **Regularised** · _Registry tolerance_: `0.1`
## Description
Ridge-augmented PLS
From the `pls4all.sklearn.RidgePLSRegression` docstring:
> L2-augmented PLS regression.
> **Registry note** — sklearn PLSRegression on the (X augmented with sqrt(λ)·I, Y augmented with zeros) is the standard data-augmentation trick for L2-penalized PLS. pls4all now defaults to NIPALS on the augmented matrix to match the reference bit-for-bit; SIMPLS on the same augmented matrix introduces a different FP reduction order and diverges by ~1e-3 on small sizes.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components` | `int` | `2` | Number of latent components extracted (k). |
| `ridge_lambda` | `float` | `1.0` | L2 (ridge) penalty added to the SIMPLS augmented system. |
## Explanations
### Bibliographic source
Hoerl, A. E. & Kennard, R. W. (1970). *Ridge regression: biased estimation for nonorthogonal problems*. Technometrics 12(1), 55–67. — combined with PLS via Tikhonov regularisation of the inner regression.
### Mathematical principle
When the number of components $k$ approaches the rank of $\mathbf{X}$, the inner regression of $\mathbf{Y}$ on the PLS scores becomes ill-conditioned. Ridge-augmented PLS adds an L2 penalty to that inner regression: $\hat{\mathbf{Q}} = (\mathbf{T}^{\top}\mathbf{T} + \lambda \mathbf{I})^{-1}\mathbf{T}^{\top}\mathbf{Y}$, yielding a shrinkage-stabilised coefficient matrix.
Setting $\lambda$ from cross-validation on a logarithmic grid is the standard procedure. The combined method is more forgiving than pure PLS to a slightly over-specified $k$: pure PLS over-fits hard at $k > k_{\mathrm{opt}}$ while ridge-augmented degrades smoothly. Conceptually it is a continuous interpolation between PLS ($\lambda=0$) and a heavily-regularised low-rank ridge regression in latent space.
When $\lambda$ is set per component via the SVD spectrum of $\mathbf{T}$, ridge PLS is closely related to Krylov-subspace PCR with shrinkage.
### Implementation
`n4m_ridge_pls_fit` (in-sample only). No widely installable reference for this exact formulation; the test compares against an sklearn `PLSRegression` + manual Tikhonov inner regression.
MATLAB header (`bindings/matlab/+pls4all/RidgePlsRegression.m`):
```text
pls4all.RidgePlsRegression L2-augmented PLS regression.
```
### Usage
Direct `n4m` Python helper:
```python
import n4m
res = n4m.ridge_pls(
X,
y,
n_components=4,
ridge_lambda=0.5,
)
y_hat = res["predictions"]
coef = res["coefficients"]
```
The `n4m.sklearn.NativeRidgePLSRegressor` wrapper replays predictions from the
returned coefficients plus reconstructed intercept.
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_ridge_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 ridge_pls_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = ridge_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 RidgePLSRegression
mdl = RidgePLSRegression(n_components=2, ridge_lambda=1.0)
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("ridge_pls", X, y,
n_components = 4L, params = list(ridge_lambda = 0.5))
# 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.ridge_pls(X, y, 4);
% see header of bindings/matlab/+pls4all/ridge_pls.m for full
% parameter surface:
% res = ridge_pls(X, Y, n_components, ridge_lambda)
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
```matlab
mdl = pls4all.fit("ridge_pls", X, y, "NumComponents", 4);
yhat = predict(mdl, Xtest);
```
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`ref.python_scikit_learn`** (python · python) — `scikit-learn` 1.4.2 · qualitative (rmse_rel ≤ 1e-01) — Ridge-augmented PLS via sklearn PLSRegression on the (X aug, Y aug) matrices — standard data-augmentation trick to fold an L2 penalty into a least-squares-style algorithm.
:::
### 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 ≤ 1e-01` 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) | 200×50 (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 | ≈ +9e-16 | 3.31 ms | 1.04 ms🏆 | 11.3 ms🏆 | 255.1 ms | 1.92 ms🏆 | 2.73 ms | 5.39 ms | 53.3 ms | 434.2 ms🏆 | 24.2 ms | 241.2 ms | 1.4 s🏆 | 106.7 ms | 1.0 s🏆 |
pls4all.cpp.blas+omp | ≈ +9e-16 | 3.26 ms | 1.70 ms | 13.2 ms | 242.9 ms🏆 | 2.19 ms | 2.60 ms | 5.52 ms | 50.1 ms🏆 | 482.2 ms | 34.3 ms | 237.1 ms🏆 | 1.5 s | 101.1 ms🏆 | 1.1 s |
pls4all.cpp.omp | ≈ +1e-15 | 3.68 ms | 1.11 ms | 13.9 ms | 289.2 ms | 1.96 ms | 2.79 ms | 4.68 ms🏆 | 52.2 ms | 488.9 ms | 24.0 ms🏆 | 261.4 ms | 1.5 s | 106.8 ms | 1.1 s |
pls4all.cpp.ref | ≈ +1e-15 | 3.98 ms | 1.13 ms | 12.2 ms | 295.4 ms | 1.94 ms | 2.59 ms🏆 | 6.09 ms | 52.3 ms | 487.2 ms | 25.5 ms | 250.3 ms | 1.6 s | 109.4 ms | 1.1 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 3.08 ms🏆 | — | — | — | 2.17 ms | 2.64 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ≈ +4e-04 | 3.71 ms | — | — | — | 2.39 ms | 2.82 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ≈ +4e-04 | 12.0 ms | — | — | — | 8.04 ms | 10.3 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ≈ +4e-04 | 19.8 ms | — | — | — | 10.9 ms | 9.34 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ≈ +4e-04 | 22.7 ms | — | — | — | 7.86 ms | 8.75 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ≈ +4e-04 | 21.2 ms | — | — | — | 10.4 ms | 9.78 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | 4.65 ms | — | — | — | 3.62 ms | 4.27 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | 5.32 ms | — | — | — | 3.97 ms | 5.26 ms | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | 4.26 ms | — | — | — | 2.97 ms | 3.03 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×50 (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 6e-16 | — | — | — | — | 2.22 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 6e-16 | — | — | — | — | 2.31 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 1e-15 | — | — | — | — | 1.99 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 1e-15 | — | — | — | — | 1.98 ms🏆 | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 2.13 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ≈ +4e-04 | — | — | — | — | 3.27 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ≈ +4e-04 | — | — | — | — | 6.92 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ≈ +4e-04 | — | — | — | — | 9.11 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ≈ +4e-04 | — | — | — | — | 8.47 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ≈ +4e-04 | — | — | — | — | 8.63 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 3.20 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 4.25 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | — | — | — | — | 2.42 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 200×50 (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 6e-16 | — | — | — | — | 1.79 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ~ shape 6e-16 | — | — | — | — | 1.79 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ~ shape 1e-15 | — | — | — | — | 1.77 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ~ shape 1e-15 | — | — | — | — | 1.78 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ 1e-14 | — | — | — | — | 1.84 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ≈ +4e-04 | — | — | — | — | 1.89 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ≈ +4e-04 | — | — | — | — | 5.12 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ≈ +4e-04 | — | — | — | — | 5.97 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ≈ +4e-04 | — | — | — | — | 6.00 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ≈ +4e-04 | — | — | — | — | 6.04 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 2.85 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 3.24 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | — | — | — | — | 2.19 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)