# Cheap and Powerful Tests for Supervised Subspaces: Per-Component Inference for PLS

arXiv: https://arxiv.org/abs/2609.36307 · HTML: https://mlsanity.com/paper/2609.36307/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 58 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 12 of 12)

- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Unsupervised Cross-lingual Representation Learning at Scale](https://mlsanity.com/paper/1911.02116/) (2019)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Interpretability Beyond Feature Attribution: Quantitative Testing with Concept Activation Vectors (TCAV)](https://mlsanity.com/paper/1711.11279/) (2017)
- [Concept Bottleneck Models](https://mlsanity.com/paper/2007.04612/) (2020)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings](https://mlsanity.com/paper/1909.00512/) (2019)
- [All-but-the-Top: Simple and Effective Postprocessing for Word Representations](https://mlsanity.com/paper/1702.01417/) (2017)
- [Scaling Monosemanticity: Extracting Interpretable Features from Claude 3 Sonnet](https://mlsanity.com/paper/2605.29358/) (2026)
- [Projecting Assumptions: The Duality Between Sparse Autoencoders and Concept Geometry](https://mlsanity.com/paper/2503.01822/) (2025)
- [Probing Classifiers are Unreliable for Concept Removal and Detection](https://mlsanity.com/paper/2207.04153/) (2022)
- [Can parametric statistical methods be trusted for fMRI based group studies?](https://mlsanity.com/paper/1511.01863/) (2015)

## Abstract (arXiv)

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.