  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 $\ell$ unknown
$k$-sparse vectors. We can query the set with another vector $\boldsymbol{a}$,
to obtain the sign of the inner product of $\boldsymbol{a}$ and a randomly
chosen vector from the $\ell$-set. How many queries are sufficient to identify
all the $\ell$ unknown vectors? This question is significantly more challenging
than both the basic 1-bit compressed sensing problem (i.e., $\ell=1$ 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.
