# `gpr_pls` — Gaussian Process on PLS scores
_Group_: **Nonlinear / local** · _Registry tolerance_: `1e-08`
## Description
GPR-on-PLS — RBF Gaussian Process on PLS scores (§47)
From the `pls4all.sklearn.GPRPLSRegression` docstring:
> Gaussian-process head on SIMPLS training scores.
> **Registry note** — GP head parity (sklearn `GaussianProcessRegressor` with RBF+WhiteKernel, optimizer=None) on the same PLS scores. Architecturally separated to allow GPR-on-AOMPLS reuse of `fit_gp_on_scores`.
### Parameters
| Name | Type | Default | Notes |
|------|------|---------|-------|
| `n_components` | `int` | `2` | Number of latent components extracted (k). |
| `length_scale` | `float` | `1.0` | |
| `noise_level` | `float` | `0.001` | |
| `seed` | `int` | `0` | Random seed for reproducible sampling/initialization. |
## Explanations
### Bibliographic source
Bishop, C. M. (2006). *Pattern Recognition and Machine Learning*, §6.4 (Gaussian Processes). — combined with a preliminary PLS dimensionality reduction for spectroscopy.
### Mathematical principle
Spectroscopic data are too high-dimensional for a direct Gaussian Process: GP inference is $O(n^3)$ in samples but the *kernel quality* degrades rapidly when $p$ exceeds a few hundred — most pairwise distances become near-identical, the kernel matrix loses contrast and the GP under-fits.
GPR-PLS solves this by first projecting $\mathbf{X} \to \mathbf{T} = \mathbf{X}\mathbf{W}$ into a low-dimensional PLS latent space and **then** training a GP on $\mathbf{T}$. The latent space preserves the variance most relevant to $y$, the GP captures smooth non-linear residual structure, and the kernel matrix is well-conditioned because pairwise distances in $\mathbb{R}^k$ remain informative.
Default kernel: RBF with length scale $\ell$ and amplitude $\sigma_f^2$, plus an isotropic noise variance $\sigma_n^2$. Marginal-likelihood maximisation selects the three hyperparameters; pls4all uses a fixed-iteration L-BFGS pass to keep the cost bounded per cell.
### Implementation
`n4m_gpr_pls_fit`. Reference: sklearn `GaussianProcessRegressor` with an RBF kernel applied to the score matrix from a separate sklearn `PLSRegression`.
MATLAB header (`bindings/matlab/+pls4all/gpr_pls.m`):
```text
pls4all.gpr_pls GPR on PLS scores (RBF kernel, single-target Y).
```
### 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_gpr_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 gpr_pls_fit
with pls4all.Context() as ctx, pls4all.Config() as cfg:
res = gpr_pls_fit(ctx, cfg, X, y, n_components=3)
# 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 GPRPLSRegression
mdl = GPRPLSRegression(n_components=2, length_scale=1.0, noise_level=0.001, seed=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("gpr_pls", X, y,
n_components = 3L, params = list(length_scale = 1.0, noise_level = 0.001, seed = 0L))
# 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.gpr_pls(X, y, 3);
% see header of bindings/matlab/+pls4all/gpr_pls.m for full
% parameter surface:
% res = gpr_pls(X, Y, n_components, length_scale, noise_level, seed)
yhat = predict(res, Xtest);
```
:::
:::{tab-item} MATLAB · pls4all (classdef)
:sync: matlab-classdef
:class-label: lang-matlab
_No idiomatic classdef wrapper — invoke `pls4all.fit("gpr_pls", X, y, …)` directly from the unified MEX factory._
:::
::::
**Registry parity references** 📐
:::{card}
:class-card: external-refs
- 📐 **`ref.python_scikit_learn`** (python · python) — `scikit-learn` 1.4.2 · strict (rmse_rel ≤ 1e-08) — sklearn GP head on the same PLS training scores pls4all produces. PLS rotation conventions diverge per-component; comparing the GP head on shared T isolates the novel stage.
:::
### 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-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 | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 120×25 (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 | ≈ +3e-10 | 2.93 ms | 3.44 ms | 10.5 ms🏆 | 59.3 ms | 1.16 ms🏆 | 7.15 ms | 49.6 ms | 103.6 ms | 333.6 ms🏆 | 21.8 s🏆 | 18.9 s | 18.6 s🏆 | — | 796.9 s |
pls4all.cpp.blas+omp | ≈ +3e-10 | 2.94 ms | 2.94 ms🏆 | 11.4 ms | 58.2 ms🏆 | 1.24 ms | 10.3 ms | 49.2 ms | 97.5 ms🏆 | 342.0 ms | 22.9 s | 19.5 s | 21.3 s | — | 691.6 s🏆 |
pls4all.cpp.omp | ≈ +3e-10 | 3.08 ms | 3.39 ms | 11.5 ms | 58.6 ms | 1.25 ms | 7.30 ms | 48.6 ms🏆 | 102.1 ms | 353.7 ms | 23.0 s | 19.7 s | 18.9 s | — | 718.8 s |
pls4all.cpp.ref | ≈ +3e-10 | 2.81 ms🏆 | 3.34 ms | 11.0 ms | 59.4 ms | 1.19 ms | 6.57 ms🏆 | 50.8 ms | 98.0 ms | 345.8 ms | 22.9 s | 17.7 s🏆 | 21.3 s | — | 822.2 s |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | 3.07 ms | — | — | — | 1.16 ms | 6.73 ms | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | 3.13 ms | — | — | — | 1.60 ms | 7.06 ms | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 2e-13 | 12.1 ms | — | — | — | 2.86 ms | 18.7 ms | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 2e-13 | 23.1 ms | — | — | — | 4.96 ms | 20.4 ms | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 2e-13 | 22.0 ms | — | — | — | 3.70 ms | 14.9 ms | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 2e-13 | 27.6 ms | — | — | — | 4.24 ms | 16.3 ms | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | 4.76 ms | — | — | — | 1.88 ms | 8.13 ms | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | 4.95 ms | — | — | — | 3.18 ms | 9.12 ms | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | 4.24 ms | — | — | — | 2.29 ms | 9.84 ms | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 3 threads
:sync: threads-3
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 120×25 (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 2e-10 | — | — | — | — | 1.23 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref 2e-10 | — | — | — | — | 1.72 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref 2e-10 | — | — | — | — | 1.18 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref 2e-10 | — | — | — | — | 1.20 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ 2e-13 | — | — | — | — | 1.68 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ 2e-13 | — | — | — | — | 1.46 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ bind | — | — | — | — | 2.67 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ bind | — | — | — | — | 4.26 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ bind | — | — | — | — | 3.83 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ bind | — | — | — | — | 4.47 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 1.72 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 2.19 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | — | — | — | — | 2.39 ms | — | — | — | — | — | — | — | — | — |
:::
:::{tab-item} 10 threads
:sync: threads-10
| Backend | Parity | 50×250 (ms) | 100×50 (ms) | 100×500 (ms) | 100×2500 (ms) | 120×25 (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 2e-10 | — | — | — | — | 1.10 ms🏆 | — | — | — | — | — | — | — | — | — |
pls4all.cpp.blas+omp | ✓ ref 2e-10 | — | — | — | — | 1.12 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.omp | ✓ ref 2e-10 | — | — | — | — | 1.12 ms | — | — | — | — | — | — | — | — | — |
pls4all.cpp.ref | ✓ ref 2e-10 | — | — | — | — | 1.10 ms | — | — | — | — | — | — | — | — | — |
| Python · pls4all |
|---|
pls4all.python | ✓ bind | — | — | — | — | 1.97 ms | — | — | — | — | — | — | — | — | — |
pls4all.sklearn | ✓ bind | — | — | — | — | 1.28 ms | — | — | — | — | — | — | — | — | — |
| R · pls4all |
|---|
pls4all.R | ✓ 2e-13 | — | — | — | — | 2.20 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.formula | ✓ 2e-13 | — | — | — | — | 2.82 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.mdatools | ✓ 2e-13 | — | — | — | — | 2.73 ms | — | — | — | — | — | — | — | — | — |
pls4all.R.pls | ✓ 2e-13 | — | — | — | — | 2.83 ms | — | — | — | — | — | — | — | — | — |
| MATLAB · pls4all |
|---|
pls4all.matlab | ✗ +9e+00 | — | — | — | — | 1.61 ms | — | — | — | — | — | — | — | — | — |
pls4all.matlab.classdef | ✗ +9e+00 | — | — | — | — | 1.95 ms | — | — | — | — | — | — | — | — | — |
| Python · external |
|---|
📐ref.python_scikit_learn | source | — | — | — | — | 1.99 ms | — | — | — | — | — | — | — | — | — |
:::
::::
---
_See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)