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In this paper, we study the problem of sparse mixed linear regression on anunlabeled dataset that is generated from linear measurements from two differentregression parameter vectors. Since the data is unlabeled, our task is not onlyto figure out a good approximation of the regression parameter vectors but alsoto label the dataset correctly. In its original form, this problem is NP-hard.The most popular algorithms to solve this problem (such asExpectation-Maximization) have a tendency to stuck at local minima. We providea novel invex relaxation for this intractable problem which leads to a solutionwith provable theoretical guarantees. This relaxation enables exact recovery ofdata labels. Furthermore, we recover a close approximation of the regressionparameter vectors which match the true parameter vectors in support and sign.Our formulation uses a carefully constructed primal dual witnesses frameworkfor the invex problem. Furthermore, we show that the sample complexity of ourmethod is only logarithmic in terms of the dimension of the regressionparameter vectors.

Sparse Mixed Linear Regression with Guarantees: Taming an Intractable Problem with Invex Relaxation