# `kernel_pls_rbf` — Kernel PLS (Rosipal & Trejo 2001)
_Group_: **Nonlinear / local** · _Registry tolerance_: `1e-08`
## Description
Non-linear kernel PLS (RBF kernel)
From the `pls4all.sklearn.KernelPLSRegression` docstring:
> Non-linear kernel PLS (Rosipal & Trejo 2001).
> **Registry note** — R `kernlab::kernelMatrix` (RBF/poly/sigmoid) + `pls::plsr` on the centered kernel matrix is the Rosipal-Trejo (2001) reference. pls4all's deflation ordering differs from the kernel-PLS-2 of Rosipal & Trejo so parity is qualitative.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components` | `int` | `2` | Number of latent components extracted (k). |
| `kernel_type` | `int` | `1` | Kernel family: 0=linear, 1=RBF, 2=polynomial, 3=sigmoid. |
| `gamma` | `float` | `0.0` | RBF kernel bandwidth γ (with K(x, x') = exp(-γ‖x − x'‖²)). |
| `coef0` | `float` | `1.0` | Independent term in polynomial / sigmoid kernels. |
| `degree` | `int` | `3` | Degree of the polynomial kernel (ignored otherwise). |
## Explanations
### Bibliographic source
Rosipal, R. & Trejo, L. J. (2001). *Kernel partial least squares regression in reproducing kernel Hilbert space*. Journal of Machine Learning Research 2, 97–123.
### Mathematical principle
Kernel PLS runs the NIPALS PLS procedure entirely in the feature space of a Mercer kernel $k(\mathbf{x}, \mathbf{x}') = \langle \phi(\mathbf{x}), \phi(\mathbf{x}') \rangle$ without ever forming $\phi$ explicitly. The kernel matrix $\mathbf{K}_{ij} = k(\mathbf{x}_i, \mathbf{x}_j) \in \mathbb{R}^{n \times n}$ replaces $\mathbf{X}\mathbf{X}^{\top}$ in the NIPALS recursion and the score matrix is built directly from $\mathbf{K}$.
The RBF kernel $k(\mathbf{x}, \mathbf{x}') = \exp(-\gamma \|\mathbf{x} - \mathbf{x}'\|^2)$ is the standard choice for non-linear PLS: it captures smooth non-linear relationships between $\mathbf{X}$ and $y$ at the cost of a single bandwidth hyperparameter $\gamma$. Other kernels (polynomial, sigmoid) are available via the `kernel_type` enum.
Memory scales as $O(n^2)$ which is the binding constraint for kernel PLS on spectroscopy datasets; subsampling (Nyström) or random Fourier features are the standard scale-up strategies but are not currently exposed.
### Implementation
`n4m_estimators_kernel_pls_fit`. Predict-on-new-X is currently marked in-sample-only in the Python `sklearn` wrapper because the C ABI does not yet export the kernel-centring constants required to handle a fresh test point. The tier-1 entry point will refit on (X_train ∪ X_test) on demand.
R roxygen note (`methods_extra.R::kernel_pls_fit`):
> Non-linear kernel PLS (Rosipal & Trejo 2001).
> @param n_components Integer. Number of latent components.
> @param kernel_type Method-specific parameter. See the underlying `*_fit()` function for the exact semantics.
> @param gamma Numeric. CPPLS / kernel band parameter.
> @param coef0 Method-specific parameter. See the underlying `*_fit()` function for the exact semantics.
> @param degree Method-specific parameter. See the underlying `*_fit()` function for the exact semantics.
> @param X Numeric matrix of predictors (rows = samples, cols = features).
> @param Y Numeric matrix or vector of responses, with one row per sample.
> @export
MATLAB header (`bindings/matlab/+pls4all/KernelPlsRegression.m`):
```text
pls4all.KernelPlsRegression Non-linear kernel PLS (Rosipal & Trejo 2001).
**In-sample only**: the C ABI exports `alpha` and `y_mean` but NOT the
kernel-centering state needed to compute K(X_new, X_train) at predict
time. predict(X) returns the stored predictions when X matches the
training matrix (content equality), otherwise raises an error.
```
### 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_estimators_kernel_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 kernel_pls_rbf_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = kernel_pls_rbf_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 KernelPLSRegression
mdl = KernelPLSRegression(n_components=2, kernel_type=1, gamma=0.0, coef0=1.0, degree=3)
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("kernel_pls_rbf", X, y,
n_components = 4L, params = list(kernel_type = 1L, gamma = 0.02, coef0 = 1.0, degree = 3L))
# res is a named list with MethodResult arrays/scalars.
# selected_indices / top_k_intervals are 1-based.
```
:::
:::{tab-item} R · pls4all (raw fn)
:sync: r-raw
:class-label: lang-r
```r
library(pls4all)
res <- kernel_pls_fit(X, Y, n_components,
kernel_type = 1L, gamma = 0.0,
coef0 = 1.0, degree = 3L)
yhat <- pls4all_predict(res, X_test)
```
:::
:::{tab-item} MATLAB · pls4all (MEX)
:sync: matlab-mex
:class-label: lang-matlab
```matlab
res = pls4all.kernel_pls(X, y, 4);
% see header of bindings/matlab/+pls4all/kernel_pls.m for full
% parameter surface:
% res = kernel_pls(X, Y, n_components, kernel_type, gamma, coef0, degree)
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
```matlab
mdl = pls4all.fit("kernel_pls_rbf", X, y, "NumComponents", 4);
yhat = predict(mdl, Xtest);
```
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`ref.r_kernlab_pls`** (R · r) — `kernlab+pls` 0.9.33+2.8.5 · strict (rmse_rel ≤ 1e-08) — R `kernlab::kernelMatrix` (RBF/poly/sigmoid) + `pls::plsr` on the centered kernel matrix is a Rosipal-Trejo-style kernel PLS reference. pls4all uses a different deflation order so the parity is qualitative.
:::
### 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 · ⇄ 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-08`).
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 | 200×50 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas+omp | ✓ ref 5e-15 | 3.07 ms🏆 |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 3.23 ms |
pls4all.sklearn | ✓ bind | 3.23 ms |
| R · pls4all |
|---|
pls4all.R | ✓ 1e-14 | 6.34 ms |
pls4all.R.formula | ✓ 1e-14 | 7.05 ms |
pls4all.R.mdatools | ✓ 1e-14 | 6.83 ms |
pls4all.R.pls | ✓ 1e-14 | 7.21 ms |
| R · external |
|---|
📐ref.r_kernlab_pls | source | 28.6 ms |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 200×50 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas+omp | ✓ ref 5e-15 | 3.12 ms🏆 |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 3.17 ms |
pls4all.sklearn | ✓ bind | 3.29 ms |
| R · pls4all |
|---|
pls4all.R | ✓ 1e-14 | 5.98 ms |
pls4all.R.formula | ✓ 1e-14 | 7.13 ms |
pls4all.R.mdatools | ✓ 1e-14 | 7.16 ms |
pls4all.R.pls | ✓ 1e-14 | 7.20 ms |
| R · external |
|---|
📐ref.r_kernlab_pls | source | 27.3 ms |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 200×50 (ms) |
|---|
| C++ native · libn4m |
|---|
pls4all.cpp.blas+omp | ✓ ref 5e-15 | 3.18 ms |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 3.14 ms🏆 |
pls4all.sklearn | ✓ bind | 3.22 ms |
| R · pls4all |
|---|
pls4all.R | ✓ 1e-14 | 6.09 ms |
pls4all.R.formula | ✓ 1e-14 | 7.28 ms |
pls4all.R.mdatools | ✓ 1e-14 | 7.22 ms |
pls4all.R.pls | ✓ 1e-14 | 7.41 ms |
| R · external |
|---|
📐ref.r_kernlab_pls | source | 40.0 ms |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)