# Search Dimension in Unlabeled Projection Pursuit: A Scaling Law for Subspace Restriction

arXiv: https://arxiv.org/abs/2609.37917 · HTML: https://mlsanity.com/paper/2609.37917/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 14 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 5 of 5)

- [Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression](https://mlsanity.com/paper/1804.02605/) (2018)
- [Likelihood-informed dimension reduction for nonlinear inverse problems](https://mlsanity.com/paper/1403.4680/) (2014)
- [Certified dimension reduction in nonlinear Bayesian inverse problems](https://mlsanity.com/paper/1807.03712/) (2018)
- [Wasserstein Projection Pursuit of Non-Gaussian Signals](https://mlsanity.com/paper/2302.12693/) (2023)
- [Large-Sample Properties of Blind Estimation of the Linear Discriminant Using Projection Pursuit](https://mlsanity.com/paper/2103.04678/) (2021)

## Abstract (arXiv)

Projection pursuit searches for a direction along which the data look least Gaussian. When the observation space contains a large Gaussian complement, the empirical objective can be minimized by a direction that carries no signal, with empirical kurtosis as low as at the truth. Sample splitting exposes rather than repairs this failure. Appending coordinates independent of the latent regime degrades the search while leaving Bayes recoverability unchanged. Restricting the search to the column space of a known forward operator removes the failure exactly on the negative-kurtosis branch. Estimating a principal subspace from the data is the alternative. In a controlled two-component model, the leading sufficient scalings differ in the gain with which the operator transmits the discriminant: varsigma-4 for covariance-spike estimation and varsigma-8 for fourth-moment search. At fixed search dimension, the measured threshold ratio collapses onto n/p2 with exponent 0.156, close to the predicted 1/8. This is an empirically supported scaling motivated by sufficient bounds, not a proved asymptotically tight law. When the search dimension is varied, the measured exponent is 0.325, substantially larger than 1/8, and the tested range does not identify its functional form. The crossing location also depends on calibration and model configuration. Under a downstream excess-error criterion, the scaling largely disappears.