Abstract
Partial Least Squares (PLS) regression extracts a few outcome-aligned directions in a high-dimensional X and is widely used across applied science, but inference on the resulting fit is either expensive, biased and discouraged, or absent. We reduce inference to held-out OLS refits of the supervised subspace, a primitive shared by PLS, supervised PCA, and linear probes, and supply two tests using held-out correlations: a Nadeau-Bengio corrected asymptotic t-test as a fast approximation, and a permutation test with comparable power, finite-sample valid under outcome-predictor independence and iid rows. Held-out predictions are unchanged under any orthogonal rebasing of the supervised span, so an interpretable basis such as varimax inherits the joint claim but not a per-axis p-value; per-component claims come from a fixed-sequence test on the PLS extraction order. We validate on synthetic geometries, two NIR chemometric datasets, and cross-lingual word-embedding regressions; the exact test also transfers to supervised PCA and a ridge probe. The proposed tests have more power than CV-permutation-Q^2, at a fraction of its cost. A pre-run check on n and the spectrum of X says when the approximation is safe. We release a Rust library with Python, R, and Julia bindings, plus a Python text pipeline.
Keywords
Subject
Publication details
- Journal
- Not available
- Open access
- Green open access
Cite this article
APA 7
Lenartowicz, P., & Plisiecki, H. (2026). Cheap and Powerful Tests for Supervised Subspaces: Per-Component Inference for PLS. https://omanscience.com/en/articles/cheap-and-powerful-tests-for-supervised-subspaces-per-component-inference-for-pls
MLA 9
Lenartowicz, Paweł, and Hubert Plisiecki. "Cheap and Powerful Tests for Supervised Subspaces: Per-Component Inference for PLS." https://omanscience.com/en/articles/cheap-and-powerful-tests-for-supervised-subspaces-per-component-inference-for-pls.
Chicago (author–date)
Lenartowicz, Paweł, and Hubert Plisiecki. 2026. "Cheap and Powerful Tests for Supervised Subspaces: Per-Component Inference for PLS." https://omanscience.com/en/articles/cheap-and-powerful-tests-for-supervised-subspaces-per-component-inference-for-pls.
Harvard
Lenartowicz, P. and Plisiecki, H. (2026) 'Cheap and Powerful Tests for Supervised Subspaces: Per-Component Inference for PLS', Available at: https://omanscience.com/en/articles/cheap-and-powerful-tests-for-supervised-subspaces-per-component-inference-for-pls.
Vancouver
Lenartowicz P, Plisiecki H. Cheap and Powerful Tests for Supervised Subspaces: Per-Component Inference for PLS. https://omanscience.com/en/articles/cheap-and-powerful-tests-for-supervised-subspaces-per-component-inference-for-pls
IEEE
P. Lenartowicz, and H. Plisiecki, "Cheap and Powerful Tests for Supervised Subspaces: Per-Component Inference for PLS," https://omanscience.com/en/articles/cheap-and-powerful-tests-for-supervised-subspaces-per-component-inference-for-pls.