Beyond Coordinates: Meta-Equivariance in Statistical Inference

2025-04-14

AI Summary

This paper addresses the sensitivity of statistical inference to parameterization—specifically, how optimization-based decisions depend on arbitrary coordinate choices—and establishes intrinsic geometric invariance of optimal solutions. We introduce *meta-equivariance*: under strictly convex, differentiable scalar objectives, the optimal matrix-valued solution transforms covariantly under affine reparameterizations. Unlike classical equivariance rooted in data symmetries, meta-equivariance arises fundamentally from convex optimization and affine differential geometry, ensuring optimality is independent of parameterization. Leveraging asymptotic normality theory and trace-weighted asymptotic mean-squared error (AMSE) risk analysis, we prove that the optimal linear combination of two asymptotically normal estimators remains invariant under any affine reparameterization. Our results establish statistical optimality as a coordinate-free, intrinsic geometric property—providing a rigorous foundation for robust, geometry-aware statistical inference.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Optimal statistical decisions should transcend the language used to describe them. Yet, how do we guarantee that the choice of coordinates - the parameterisation of an optimisation problem - does not subtly dictate the solution? This paper reveals a fundamental geometric invariance principle. We first analyse the optimal combination of two asymptotically normal estimators under a strictly convex trace-AMSE risk. While methods for finding optimal weights are known, we prove that the resulting optimal estimator is invariant under direct affine reparameterisations of the weighting scheme. This exemplifies a broader principle we term meta-equivariance: the unique minimiser of any strictly convex, differentiable scalar objective over a matrix space transforms covariantly under any invertible affine reparameterisation of that space. Distinct from classical statistical equivariance tied to data symmetries, meta-equivariance arises from the immutable geometry of convex optimisation itself. It guarantees that optimality, in these settings, is not an artefact of representation but an intrinsic, coordinate-free truth.

Problem

• Ensures estimator invariance under affine reparameterizations
• Introduces meta-equivariance in convex optimization geometry
• Guarantees coordinate-free optimality in statistical decisions

Innovation

• Affine reparameterisation invariant optimal estimator
• Meta-equivariance in convex matrix optimisation
• Coordinate-free geometric invariance principle