# `ridge_pls` — Ridge-augmented PLS _Group_: **Regularised** · _Registry tolerance_: `1e-08` ## 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_estimators_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. R roxygen note (`sklearn_extra.R::ridge_pls`): > Ridge PLS — formula entry point. MATLAB header (`bindings/matlab/+pls4all/RidgePlsRegression.m`): ```text pls4all.RidgePlsRegression L2-augmented PLS regression. ``` ### 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_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} R · pls4all (raw fn) :sync: r-raw :class-label: lang-r ```r library(pls4all) res <- ridge_pls_fit(X, Y, n_components, ridge_lambda = 1.0) yhat <- pls4all_predict(res, X_test) ``` ::: :::{tab-item} R · pls4all (formula+S3) :sync: r-formula :class-label: lang-r ```r library(pls4all) fit <- ridge_pls(y ~ ., data = train, ncomp = 4L) yhat <- predict(fit, newdata = test) summary(fit) ``` ::: :::{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 · strict (rmse_rel ≤ 1e-08) — 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  ·  ⇄ 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
BackendParity200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 6e-161.75 ms
Python · pls4all
pls4all.python✓ bind1.83 ms
pls4all.sklearn✓ bind1.75 ms🏆
R · pls4all
pls4all.R✓ 1e-144.62 ms
pls4all.R.formula✓ 1e-146.20 ms
pls4all.R.mdatools✓ 1e-145.56 ms
pls4all.R.pls✓ 1e-145.59 ms
Python · external
📐ref.python_scikit_learnsource2.19 ms
::: :::{tab-item} 3 threads :sync: threads-3
BackendParity200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 6e-161.80 ms🏆
Python · pls4all
pls4all.python✓ bind1.80 ms
pls4all.sklearn✓ bind1.82 ms
R · pls4all
pls4all.R✓ 1e-144.67 ms
pls4all.R.formula✓ 1e-145.85 ms
pls4all.R.mdatools✓ 1e-145.48 ms
pls4all.R.pls✓ 1e-145.74 ms
Python · external
📐ref.python_scikit_learnsource2.20 ms
::: :::{tab-item} 10 threads :sync: threads-10
BackendParity200×50 (ms)
C++ native · libn4m
pls4all.cpp.blas+omp✓ ref 6e-161.83 ms🏆
Python · pls4all
pls4all.python✓ bind1.87 ms
pls4all.sklearn✓ bind1.85 ms
R · pls4all
pls4all.R✓ 1e-144.60 ms
pls4all.R.formula✓ 1e-145.61 ms
pls4all.R.mdatools✓ 1e-145.92 ms
pls4all.R.pls✓ 1e-146.40 ms
Python · external
📐ref.python_scikit_learnsource2.20 ms
::: :::: --- _See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)