Improved Kernel Partial Least Squares (IKPLS) and Fast Cross-Validation

The ikpls software package provides fast and efficient tools for PLS (Partial Least Squares) modeling. This package is designed to help researchers and practitioners handle PLS modeling faster than previously possible - particularly on large datasets.

Citation

If you use the ikpls software package for your work, please cite this Journal of Open Source Software article. If you use the fast cross-validation algorithm implemented in ikpls.fast_cross_validation, please also cite this Journal of Chemometrics article.

Unlock the Power of Fast and Stable Partial Least Squares Modeling with IKPLS

Dive into cutting-edge Python implementations of the IKPLS (Improved Kernel Partial Least Squares) Algorithms #1 and #2 [1] for CPUs, GPUs, and TPUs. IKPLS is both fast [2] and numerically stable [3] making it optimal for PLS modeling.

• Use our NumPy [4] based CPU implementations for fast PLS on the CPU, and our scikit-learn-conformant ikpls.sklearn.PLS wrapper for seamless integration with scikit-learn's [5] ecosystem of machine learning algorithms and pipelines. As the wrapper conforms to the scikit-learn estimator API, it can be used with scikit-learn's cross_validate.
• Use our JAX [6] implementations on CPUs or leverage powerful GPUs and TPUs for PLS modelling. Our JAX implementations are end-to-end differentiable allowing gradient propagation when using PLS as a layer in a deep learning model.
• Use our combination of NumPy and JAX IKPLS with Engstrøm's and Jensen's unbelievably fast cross-validation algorithm [7] to quickly determine the optimal combination of preprocessing and number of PLS components.
• Use any of the above in combination with sample-weighted PLS [8].
• Use our NumPy or JAX implementations for dimensionality reduction to score space with their respective transform methods.
• Use our NumPy or JAX implementations for reconstruction of original space from score space with their respective inverse_transform methods.

The documentation is available at https://ikpls.readthedocs.io/en/latest/; examples can be found at https://github.com/Sm00thix/IKPLS/tree/main/examples.

Fast Cross-Validation

In addition to the standalone IKPLS implementations, this package contains an implementation of IKPLS combined with the novel, fast cross-validation algorithm by Engstrøm and Jensen [7]. The fast cross-validation algorithm benefit both IKPLS Algorithms and especially Algorithm #2. The fast cross-validation algorithm is mathematically equivalent to the classical cross-validation algorithm. Still, it is much quicker. The fast cross-validation algorithm correctly handles (column-wise) centering and scaling of the $\mathbf{X}$ and $\mathbf{Y}$ input matrices using training set means and standard deviations to avoid data leakage from the validation set. This centering and scaling can be enabled or disabled independently from eachother and for $\mathbf{X}$ and $\mathbf{Y}$ by setting the parameters center_X, center_Y, scale_X, and scale_Y, respectively. In addition to correctly handling (column-wise) centering and scaling, the fast cross-validation algorithm correctly handles row-wise preprocessing that operates independently on each sample such as (row-wise) centering and scaling of the $\mathbf{X}$ and $\mathbf{Y}$ input matrices, convolution, or other preprocessing. Row-wise preprocessing can safely be applied before passing the data to the fast cross-validation algorithm.

Installation

• Install the package for Python3 using the following command:

  pip3 install ikpls

• Now you can import the NumPy and sklearn implementations with:

  from ikpls.numpy import PLS as NpPLS
  from ikpls.fast_cross_validation.numpy import PLS as NpPLS_FastCV
  
  from ikpls.sklearn import PLS as SkPLS

• You can also install the optional JAX dependency to get JAX implementations of IKPLS

  pip3 install "ikpls[jax]"

• Now, you can import the JAX implementations with:

  # The JAX PLS takes an `algorithm` argument (1 or 2), like the NumPy PLS,
  # e.g. JAXPLS(algorithm=1) or JAXPLS(algorithm=2).
  from ikpls.jax import PLS as JAXPLS
  from ikpls.fast_cross_validation.jax import PLS as JAXPLS_FastCV

Prerequisites for JAX

The JAX implementations support running on both CPU, GPU, and TPU.

• To enable NVIDIA GPU execution, install JAX and CUDA with:

  pip3 install -U "jax[cuda13]"

• To enable Google Cloud TPU execution, install JAX with:

  pip3 install -U "jax[tpu]"

These are typical installation instructions that will be what most users are looking for. For customized installations, follow the instructions from the JAX Installation Guide.

To ensure that JAX implementations use float64, set the environment variable JAX_ENABLE_X64=True as per the Common Gotchas. Alternatively, float64 can be enabled with the following function call:

import jax
jax.config.update("jax_enable_x64", True)

Quick Start

Use the ikpls package for PLS modeling

import numpy as np

from ikpls.sklearn import PLS


N = 100  # Number of samples.
K = 50  # Number of features.
M = 10  # Number of targets.
A = 20  # Number of latent variables (PLS components).

X = np.random.uniform(size=(N, K))  # Predictor variables
Y = np.random.uniform(size=(N, M))  # Target variables
w = np.random.uniform(size=(N,))  # Sample weights (optional)

# ikpls.sklearn.PLS is a scikit-learn-conformant estimator wrapping
# ikpls.numpy.PLS, so it plugs straight into Pipeline, GridSearchCV, and
# cross_validate. The number of components is set on the constructor, and
# fit(X, y) follows the scikit-learn API. algorithm=1 or 2 selects the
# Improved Kernel PLS algorithm.
pls = PLS(n_components=A, algorithm=1)
pls.fit(X, Y)  # Pass sample_weight=w for weighted PLS.

# --- Prediction -------------------------------------------------------------

# predict() uses n_components (= A) components. Has shape (N, M) = (100, 10).
Y_pred = pls.predict(X)

# predict_all_components() keeps ikpls's vectorized feature of predicting with
# every number of components 1..A in a single call. Shape (A, N, M) =
# (20, 100, 10); the last slice uses all A components.
Y_pred_all = pls.predict_all_components(X)
(Y_pred_all[A - 1] == Y_pred).all()  # True

# --- Fitted attributes (scikit-learn PLSRegression conventions) -------------

# X weights matrix of shape (K, A) = (50, 20).
pls.x_weights_

# Y weights matrix of shape (M, A) = (10, 20). In Improved Kernel PLS this
# equals y_loadings_, matching sklearn.cross_decomposition.PLSRegression.
pls.y_weights_

# X loadings matrix of shape (K, A) = (50, 20).
pls.x_loadings_

# Y loadings matrix of shape (M, A) = (10, 20).
pls.y_loadings_

# X rotations matrix of shape (K, A) = (50, 20).
pls.x_rotations_

# Y rotations matrix of shape (M, A) = (10, 20). Lazily computed and cached on
# first access (it needs a pseudo-inverse that fit() does not otherwise pay for).
pls.y_rotations_

# Regression coefficients of shape (M, K) = (10, 50) and intercept of shape
# (M,) = (10,). Following scikit-learn, predict effectively centers X, so
# predict(X) == (X - X_mean) @ coef_.T + intercept_.
pls.coef_
pls.intercept_

# --- Transform to score space and reconstruct -------------------------------

# Project X onto the latent space. X scores have shape (N, A) = (100, 20).
x_scores = pls.transform(X)

# Passing Y as well returns both X scores and Y scores, each (N, A) = (100, 20).
x_scores, y_scores = pls.transform(X, Y)

# inverse_transform maps scores back to the original space: X_reconstructed has
# shape (N, K) = (100, 50) and Y_reconstructed has shape (N, M) = (100, 10). The
# round trip is exact only when n_components equals n_features / n_targets, so
# with A < K this is an approximate (dimensionality-reduced) reconstruction.
X_reconstructed = pls.inverse_transform(x_scores)
X_reconstructed, Y_reconstructed = pls.inverse_transform(x_scores, y_scores)

Examples

In examples, you will find:

• Fit and Predict with NumPy.
• Fit and Predict with JAX.
• Fit and Predict on many datasets at once with JAX and jax.vmap (e.g. 10,000 independent fits and predicts).
• Cross-validate with NumPy.
• Cross-validate with scikit-learn.
• Cross-validate with NumPy and fast cross-validation.
• Cross-validate with NumPy and weighted fast cross-validation.
• Cross-validate with JAX.
• Cross-validate with JAX and fast cross-validation (jax.vmap over folds on CPU/GPU/TPU).
• Compute the gradient of a preprocessing convolution filter with respect to the RMSE between the target value and the value predicted by PLS after fitting with JAX.
• Weighted Fit and Predict with NumPy.
• Weighted Fit and Predict with JAX.
• Weighted cross-validation with NumPy.
• Weighted cross-validation with JAX.
• Weighted cross-validation with scikit-learn.
• Fit, transform to score space, and inverse transform with NumPy.
• Fit, transform to score space, and inverse transform with JAX.

Changelog

See CHANGELOG.md for version history and release notes.

Contribute

To contribute, please read the Contribution Guidelines.

References

1. Dayal, B. S. and MacGregor, J. F. (1997). Improved PLS algorithms. Journal of Chemometrics, 11(1), 73-85.
2. Alin, A. (2009). Comparison of PLS algorithms when the number of objects is much larger than the number of variables. Statistical Papers, 50, 711-720.
3. Andersson, M. (2009). A comparison of nine PLS1 algorithms. Journal of Chemometrics, 23(10), 518-529.
4. NumPy
5. scikit-learn
6. JAX
7. Engstrøm, O.-C. G. and Jensen, M. H. (2025). Fast Partition-Based Cross-Validation With Centering and Scaling for $\mathbf{X}^\mathbf{T}\mathbf{X}$ and $\mathbf{X}^\mathbf{T}\mathbf{Y}$
8. Becker and Ismail (2016). Accounting for sampling weights in PLS path modeling: Simulations and empirical examples. European Management Journal, 34(6), 606-617.

Funding

• Up until May 31st 2025, this work has been carried out as part of an industrial PhD project receiving funding from FOSS Analytical A/S and The Innovation Fund Denmark. Grant number 1044-00108B.
• From June 1st 2025 and onward, this work is sponsored by FOSS Analytical A/S.