An algebraic-geometric approach for linear regression without correspondences

TL;DR

This paper introduces an algebraic-geometric method for linear regression without known correspondences, leveraging symmetric polynomials to handle permutation ambiguities and noise, enabling efficient solutions even with fully shuffled data.

Contribution

It proposes a novel algebraic geometric approach using symmetric polynomials to solve permutation-invariant linear regression problems, addressing limitations of existing methods.

Findings

Handles thousands of noisy, fully shuffled observations in milliseconds

Provides polynomial system with at most n! complex roots for generic samples

First efficient solution for small n with high data corruption

Abstract

Linear regression without correspondences is the problem of performing a linear regression fit to a dataset for which the correspondences between the independent samples and the observations are unknown. Such a problem naturally arises in diverse domains such as computer vision, data mining, communications and biology. In its simplest form, it is tantamount to solving a linear system of equations, for which the entries of the right hand side vector have been permuted. This type of data corruption renders the linear regression task considerably harder, even in the absence of other corruptions, such as noise, outliers or missing entries. Existing methods are either applicable only to noiseless data or they are very sensitive to initialization or they work only for partially shuffled data. In this paper we address these issues via an algebraic geometric approach, which uses symmetric polynomials to extract permutation-invariant constraints that the parameters $\xi^* \in \Re^n$ of the linear regression model must satisfy. This naturally leads to a polynomial system of $n$ equations in $n$ unknowns, which contains $\xi^*$ in its root locus. Using the machinery of algebraic geometry we prove that as long as the independent samples are generic, this polynomial system is always consistent with at most $n!$ complex roots, regardless of any type of corruption inflicted on the observations. The algorithmic implication of this fact is that one can always solve this polynomial system and use its most suitable root as initialization to the Expectation Maximization algorithm. To the best of our knowledge, the resulting method is the first working solution for small values of $n$ able to handle thousands of fully shuffled noisy observations in milliseconds.