Solving a Mixture of Many Random Linear Equations by Tensor Decomposition and Alternating Minimization
This provides a more efficient solution for mixed linear regression, a problem in machine learning and statistics with applications in data analysis, though it is incremental in improving computational bounds.
The paper tackles the problem of solving mixed random linear equations (noiseless mixed linear regression) to estimate multiple linear models from unlabeled samples, achieving exact recovery with sample complexity linear in dimension and polynomial in the number of components, improving upon previous exponential or super-linear dependencies.
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.