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Recovery of support of a sparse vector from simple measurements is awidely-studied problem, considered under the frameworks of compressed sensing,1-bit compressed sensing, and more general single index models. We considergeneralizations of this problem: mixtures of linear regressions, and mixturesof linear classifiers, where the goal is to recover supports of multiple sparsevectors using only a small number of possibly noisy linear, and 1-bitmeasurements respectively. The key challenge is that the measurements fromdifferent vectors are randomly mixed. Both of these problems have also receivedattention recently. In mixtures of linear classifiers, the observationscorrespond to the side of queried hyperplane a random unknown vector lies in,whereas in mixtures of linear regressions we observe the projection of a randomunknown vector on the queried hyperplane. The primary step in recovering theunknown vectors from the mixture is to first identify the support of all theindividual component vectors. In this work, we study the number of measurementssufficient for recovering the supports of all the component vectors in amixture in both these models. We provide algorithms that use a number ofmeasurements polynomial in $k, \log n$ and quasi-polynomial in $\ell$, torecover the support of all the $\ell$ unknown vectors in the mixture with highprobability when each individual component is a $k$-sparse $n$-dimensionalvector.

Support Recovery of Sparse Signals from a Mixture of Linear Measurements