# `fused_sparse_pls` — Fused-sparse PLS _Group_: **Sparse** · _Registry tolerance_: `0.05` · _Parity reference_: **paper-only** (Yengo, L., Jacques, J., Biernacki, C. & Canouil, M. (2016). Variable clustering in high-dimensional linear regression: The R package `clere`. R Journal 8(1), 92-106. (Fused-sparse PLS variants have no widely installable R / Python port.)) ## Description Fused sparse PLS (§7) From the `pls4all.sklearn.FusedSparsePLSRegression` docstring: > Fused-sparse PLS — L1 + adjacent-coef smoothing. ### Parameters | Name | Type | Default | Notes | |------|------|---------|-------| | `n_components` | `int` | `2` | Number of latent components extracted (k). | | `l1_lambda` | `float` | `0.05` | L1 sparsity penalty for fused-sparse PLS. | | `fusion_lambda` | `float` | `0.05` | Fused-lasso penalty enforcing smoothness between adjacent coefficients. | ## Explanations ### Bibliographic source Tibshirani, R., Saunders, M., Rosset, S., Zhu, J. & Knight, K. (2005). *Sparsity and smoothness via the fused lasso*. JRSS B 67(1), 91–108. — generalised to PLS loadings. ### Mathematical principle The fused lasso penalty combines L1 sparsity with an L1 penalty on first differences of neighbouring coefficients: $\mathcal{P}(\mathbf{w}) = \lambda_1 \|\mathbf{w}\|_1 + \lambda_2 \sum_{j=1}^{p-1}|w_{j+1} - w_j|$. For spectroscopic data this is uniquely useful: the second term *encourages neighbouring wavelengths to share a coefficient*, which produces piecewise-constant loadings reflecting the underlying band structure of the spectrum rather than the choppy sample-noise pattern that plain L1 alone produces. Tuning two penalties simultaneously is non-trivial: $\lambda_1$ controls overall sparsity, $\lambda_2$ controls within-band smoothness. Plotting the selected wavelength bands as a function of $\lambda_2$ at fixed sparsity is a common diagnostic. Computationally the fused-lasso step is a 1-D total variation denoising problem with a closed-form taut-string solution in $O(p)$; pls4all uses Condat's algorithm. ### Implementation `n4m_estimators_fused_sparse_pls_fit`. No widely installable reference library — `sgPLS` covers group-sparse but not fused-sparse. Documented as `paper_only` in the registry. R roxygen note (`methods_extra.R::fused_sparse_pls_fit`): > Fused-sparse PLS (L1 + adjacent-coef smoothing). > @param n_components Integer. Number of latent components. > @param l1_lambda Method-specific parameter. See the underlying `*_fit()` function for the exact semantics. > @param fusion_lambda 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/fused_sparse_pls.m`): ```text pls4all.fused_sparse_pls Fused-sparse PLS (L1 + adjacent-coef smoothing). ``` ### 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_fused_sparse_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 fused_sparse_pls_fit with pls4all.Context() as ctx, pls4all.Config() as cfg: res = fused_sparse_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 FusedSparsePLSRegression mdl = FusedSparsePLSRegression(n_components=2, l1_lambda=0.05, fusion_lambda=0.05) 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("fused_sparse_pls", X, y, n_components = 4L, params = list(l1_lambda = 0.05, fusion_lambda = 0.1)) # 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 <- fused_sparse_pls_fit(X, Y, n_components, l1_lambda = 0.05, fusion_lambda = 0.05) yhat <- pls4all_predict(res, X_test) ``` ::: :::{tab-item} MATLAB · pls4all (MEX) :sync: matlab-mex :class-label: lang-matlab ```matlab res = pls4all.fused_sparse_pls(X, y, 4); % see header of bindings/matlab/+pls4all/fused_sparse_pls.m for full % parameter surface: % res = fused_sparse_pls(X, Y, n_components, l1_lambda, fusion_lambda) yhat = predict(res, Xtest); ``` ::: :::{tab-item} MATLAB · pls4all (classdef) :sync: matlab-classdef :class-label: lang-matlab _No idiomatic classdef wrapper — invoke `pls4all.fit("fused_sparse_pls", X, y, …)` directly from the unified MEX factory._ ::: :::: **Registry parity references** 📐 :::{card} :class-card: external-refs - 📜 **Paper-only** — no executable parity reference; the `pls4all` implementation is verified by a smoke fit only. Canonical citation: Yengo, L., Jacques, J., Biernacki, C. & Canouil, M. (2016). Variable clustering in high-dimensional linear regression: The R package `clere`. R Journal 8(1), 92-106. (Fused-sparse PLS variants have no widely installable R / Python port.) ::: ### 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 🏆. ::::{tab-set} :class: parity-tabs :::{tab-item} 1 thread :sync: threads-1
BackendParity200×50 (ms)
Python · pls4all
pls4all.python✓ bind1.95 ms🏆
pls4all.sklearn✓ bind2.13 ms
R · pls4all
pls4all.R✓ bind6.13 ms
pls4all.R.formula✓ bind7.70 ms
pls4all.R.mdatools✓ bind7.55 ms
pls4all.R.pls✓ bind7.80 ms
::: :::{tab-item} 3 threads :sync: threads-3
BackendParity200×50 (ms)
Python · pls4all
pls4all.python✓ bind1.88 ms🏆
pls4all.sklearn✓ bind2.09 ms
R · pls4all
pls4all.R✓ bind6.14 ms
pls4all.R.formula✓ bind7.73 ms
pls4all.R.mdatools✓ bind8.04 ms
pls4all.R.pls✓ bind7.53 ms
::: :::{tab-item} 10 threads :sync: threads-10
BackendParity200×50 (ms)
Python · pls4all
pls4all.python✓ bind1.97 ms🏆
pls4all.sklearn✓ bind2.02 ms
R · pls4all
pls4all.R✓ bind6.56 ms
pls4all.R.formula✓ bind8.64 ms
pls4all.R.mdatools✓ bind8.07 ms
pls4all.R.pls✓ bind8.65 ms
::: :::: --- _See also_: [benchmark overview](../benchmarks/overview.md) · [methods index](index.md) · [interactive dashboard](../landing/dashboard.md)