Shuffled Linear Regression via Spectral Matching

TL;DR

This work tackles shuffled linear regression under unknown measurement permutations by introducing a spectral matching approach that leverages the eigenstructure of covariance matrices in the large-sample regime (t ≫ m). By aligning eigenvectors of the observed data covariance with those of the latent feature covariance, the method yields a computationally efficient permutation estimate via a linear assignment, followed by LS or LASSO recovery of latent features. The authors establish permutation-estimation bounds that decay as O(1/√t) and demonstrate that shuffled LS and shuffled LASSO estimates achieve asymptotically optimal error rates, with LASSO benefiting from sparsity. The framework extends to 3D image registration through a translation-robust, non-iterative spectral-matching pipeline, and extensive synthetic and real-data experiments show superior estimation and registration performance with favorable complexity compared to state-of-the-art baselines.

Abstract

Shuffled linear regression (SLR) seeks to estimate latent features through a linear transformation, complicated by unknown permutations in the measurement dimensions. This problem extends traditional least-squares (LS) and Least Absolute Shrinkage and Selection Operator (LASSO) approaches by jointly estimating the permutation, resulting in shuffled LS and shuffled LASSO formulations. Existing methods, constrained by the combinatorial complexity of permutation recovery, often address small-scale cases with limited measurements. In contrast, we focus on large-scale SLR, particularly suited for environments with abundant measurement samples. We propose a spectral matching method that efficiently resolves permutations by aligning spectral components of the measurement and feature covariances. Rigorous theoretical analyses demonstrate that our method achieves accurate estimates in both shuffled LS and shuffled LASSO settings, given a sufficient number of samples. Furthermore, we extend our approach to address simultaneous pose and correspondence estimation in image registration tasks. Experiments on synthetic datasets and real-world image registration scenarios show that our method outperforms existing algorithms in both estimation accuracy and registration performance.

Figures (11)

• Figure 1: Schematic view of the spectral matching method.
• Figure 2: Shuffled LS performance with varying SNR values.
• Figure 3: Shuffled LS accuracy versus the fraction of permuted rows $p_e$ in the observations with $\text{SNR}=20$ dB.
• Figure 4: Shuffled LS accuracy versus the sample size $t$ with $p_e=0.5$ and $\text{SNR}=20$ dB.
• Figure 5: Shuffled LS accuracy versus $n$ with $m=200$, $\text{SNR}=20$ dB, and $p_e=0.5$.