Recovery of sparse linear classifiers from mixture of responses

In the problem of learning a mixture of linear classifiers, the aim is to learn a collection of hyperplanes from a sequence of binary responses. Each response is a result of querying with a vector and indicates the side of a randomly chosen hyperplane from the collection the query vector belong to. This model provides a rich representation of heterogeneous data with categorical labels and has only been studied in some special settings. We look at a hitherto unstudied problem of query complexity upper bound of recovering all the hyperplanes, especially for the case when the hyperplanes are sparse. This setting is a natural generalization of the extreme quantization problem known as 1-bit compressed sensing. Suppose we have a set of l unknown k-sparse vectors. We can query the set with another vector a, to obtain the sign of the inner product of a and a randomly chosen vector from the l-set. How many queries are sufficient to identify all the l unknown vectors? This question is significantly more challenging than both the basic 1-bit compressed sensing problem (i.e., l = 1 case) and the analogous regression problem (where the value instead of the sign is provided). We provide rigorous query complexity results (with efficient algorithms) for this problem.