# Linear Regression Beyond Covariance: Latent Slope Spectra from Higher-Order Cumulants

arXiv: https://arxiv.org/abs/2610.17772 · HTML: https://mlsanity.com/paper/2610.17772/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 17 references that are in the map.
- First posted on arXiv 13 Oct 2026.

## Builds on (most cited first, 7 of 7)

- [The Blessings of Multiple Causes](https://mlsanity.com/paper/1805.06826/) (2018)
- [Causal Discovery with Unobserved Confounding and non-Gaussian Data](https://mlsanity.com/paper/2007.11131/) (2020)
- [Causal Discovery with Latent Confounders Based on Higher-Order Cumulants](https://mlsanity.com/paper/2305.19582/) (2023)
- [Spectral Deconfounding via Perturbed Sparse Linear Models](https://mlsanity.com/paper/1811.05352/) (2018)
- [Higher-order least squares: assessing partial goodness of fit of linear causal models](https://mlsanity.com/paper/2109.14544/) (2021)
- [Goodness-of-Fit Tests for Linear Non-Gaussian Structural Equation Models](https://mlsanity.com/paper/2311.04585/) (2023)
- [Causal Effect Identification in lvLiNGAM from Higher-Order Cumulants](https://mlsanity.com/paper/2506.05202/) (2025)

## Abstract (arXiv)

Least squares summarizes the joint distribution of a response and regressor through second-order information. When an observed regressor is composed of latent components carrying different relationships with the response, this summary can average over distinct linear effects. We study the additional linear information contained in higher-order mixed cumulants and formalize it through the latent slope spectrum. Ordinary least squares is a variance-weighted barycenter of this spectrum, whereas higher-order cumulants can separate its components when the spectrum is sufficiently parsimonious. In the scalar one-confounder model, fourth-order cumulants identify the unordered pair β,β+γ through a 2×2 linear system and a quadratic equation. We develop estimation and root-n inference under regular separation and show that detection and estimation diverge near degeneracy: slope multiplicity remains detectable on an n-1/4 local scale even when the individual slopes are not uniformly regularly estimable. In multiple regression, a quadratic tensor recurrence yields a test of one versus multiple effective coefficient vectors that does not require the vectors themselves to be identified; multiplicity can therefore be consistently detectable even when the recurrence operator is noninjective. The same quadratic diagnostic can retain signal in directions where a first-order higher-cumulant restriction cancels exactly. The results distinguish representation, identification, detection, and estimation of latent linear heterogeneity, while making explicit the limits of structural and causal interpretation from observational data alone.