Recovery of sparse linear classifiers from mixture of responses

TL;DR

This paper investigates the query complexity for recovering multiple sparse linear classifiers from binary responses, extending 1-bit compressed sensing to multiple hyperplanes and providing rigorous bounds and algorithms.

Contribution

It introduces the first analysis of query complexity for recovering multiple sparse hyperplanes from binary responses, generalizing 1-bit compressed sensing.

Findings

Established upper bounds on the number of queries needed for recovery.

Developed efficient algorithms for hyperplane identification.

Extended 1-bit compressed sensing to multiple hyperplanes.

Abstract

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.