Solving a Mixture of Many Random Linear Equations by Tensor Decomposition and Alternating Minimization

TL;DR

This paper introduces a new algorithm combining tensor decomposition and alternating minimization to efficiently solve mixed linear equations with multiple components, achieving linear sample complexity in dimension and polynomial in the number of components.

Contribution

The paper presents a tractable algorithm for mixed linear equations that guarantees exact recovery with improved sample complexity, using tensor methods for initialization and alternating minimization for convergence.

Findings

Algorithm guarantees exact recovery under certain conditions.

Sample complexity is linear in dimension and polynomial in number of components.

Convergence to the global optimum is linear after tensor-based initialization.

Abstract

We consider the problem of solving mixed random linear equations with $k$ components. This is the noiseless setting of mixed linear regression. The goal is to estimate multiple linear models from mixed samples in the case where the labels (which sample corresponds to which model) are not observed. We give a tractable algorithm for the mixed linear equation problem, and show that under some technical conditions, our algorithm is guaranteed to solve the problem exactly with sample complexity linear in the dimension, and polynomial in $k$, the number of components. Previous approaches have required either exponential dependence on $k$, or super-linear dependence on the dimension. The proposed algorithm is a combination of tensor decomposition and alternating minimization. Our analysis involves proving that the initialization provided by the tensor method allows alternating minimization, which is equivalent to EM in our setting, to converge to the global optimum at a linear rate.