Recovery of sparse linear classifiers from mixture of responses
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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 belongs 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.
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