In this paper we present the framework of symmetry in nonparametric regression. This generalises the framework of covariate sparsity, where the regression function depends only on at most s < d of the covariates, which is a special case of translation symmetry with linear orbits. In general this extends to other types of functions that capture lower dimensional behavior even when these structures are non-linear. We show both that known symmetries of regression functions can be exploited to give similarly faster rates, and that unknown symmetries with Lipschitz actions can be estimated sufficiently quickly to obtain the same rates. This is done by explicit constructions of partial symmetrisation operators that are then applied to usual estimators, and with a two step M-estimator of the maximal symmetry of the regression function. We also demonstrate the finite sample performance of these estimators on synthetic data.
@article{christie2025symmetry,
author = {Christie, Louis Goldwater and Aston, John A. D.},
title = {Symmetry: A general structure in nonparametric regression},
journal = {The Annals of Statistics},
volume = {53},
number = {5},
pages = {2040--2076},
year = {2025},
doi = {10.1214/25-AOS2529}
}
We present a method for estimating the maximal symmetry of a continuous regression function. Knowledge of such a symmetry can be used to significantly improve modelling by removing the modes of variation resulting from the symmetries. Symmetry estimation is carried out using hypothesis testing for invariance strategically over the subgroup lattice of a search group 𝇛 acting on the feature space. We show that the estimation of the unique maximal invariant subgroup of 𝇛 generalises useful tools from linear dimension reduction to a non linear context. We show that the estimation is consistent when the subgroup lattice chosen is finite, even when some of the subgroups themselves are infinite. We demonstrate the performance of this estimator in synthetic settings and apply the methods to two data sets: satellite measurements of the earth's magnetic field intensity; and the distribution of sunspots.
@article{christie2025estimating,
author = {Christie, Louis G. and Aston, John A. D.},
title = {Estimating maximal symmetries of regression functions via subgroup lattices},
journal = {Journal of the Royal Statistical Society Series B: Statistical Methodology},
volume = {87},
number = {5},
pages = {1576--1618},
year = {2025},
doi = {10.1093/jrsssb/qkaf031}
}
Invariant and equivariant models incorporate the symmetry of an object to be estimated (here non-parametric regression functions). These models perform better (with respect to L² loss) and are increasingly being used in practice, but encounter problems when the symmetry is falsely assumed. In this paper we present a framework for testing for G-equivariance for any semi-group G. This will give confidence to the use of such models when the symmetry is not known a priori. These tests are independent of the model and are computationally quick, so can be easily used before model fitting to test their validity.
@misc{christie2022testing,
author = {Christie, Louis G. and Aston, John A. D.},
title = {Testing for geometric invariance and equivariance},
year = {2022},
eprint = {2205.15280},
archivePrefix = {arXiv},
primaryClass = {stat.ML}
}
We investigate the strategic behavior of firms in a Hotelling spatial setting. The innovation is to combine two important features that are ubiquitous in real markets: (1) the location space is two-dimensional, often with physical restrictions on where firms can locate; (2) consumers with some probability shop at firms other than the nearest. We characterise convergent Nash equilibria (CNE), in which all firms cluster at one point, for several alternative markets. In the benchmark case of a square convex market, we provide a new direct geometric proof of a result by Cox (1987) that CNE can arise in a sufficiently central part of the market. The convexity of the square space is of restricted realism, however, and we proceed to investigate grids, which more faithfully represent a stylised city's streets, characterising CNE that exhibit several new phenomena.
@article{cahan2021spatial,
author = {Cahan, Dodge and Chen, Hongjia H. and Christie, Louis G. and Slinko, Arkadii},
title = {Spatial competition on 2-dimensional markets and networks when consumers don't always go to the closest firm},
journal = {International Journal of Game Theory},
volume = {50},
number = {4},
pages = {945--970},
year = {2021},
doi = {10.1007/s00182-021-00776-y}
}
A vanishing sum of roots of unity is called minimal if no proper, nonempty sub-sum of it vanishes. This paper classifies all minimal vanishing sums of roots of unity of weight at most 16 by hand, thereby uncovering new phenomena beyond the earlier 1998 classification of Poonen and Rubinstein (SIAM J. Discrete Math.) that went up to weight 12. The paper also develops an algorithm to explore higher weights up to 21, yielding a conjectural extension of the classification.
@misc{christie2020classifying,
author = {Christie, Louis and Dykema, Kenneth J. and Klep, Igor},
title = {Classifying minimal vanishing sums of roots of unity},
year = {2020},
eprint = {2008.11268},
archivePrefix = {arXiv},
primaryClass = {math.NT}
}