Minimizing Weighted Concave Impurity Partition Under Constraints

Set partitioning is a key component of many algorithms in machine learning, signal processing, and communications. In general, the problem of finding a partition that minimizes a given impurity (loss function) is NP-hard. As such, there exists a wealth of literature on approximate algorithms and theoretical analysis for the partitioning problem under different settings. In this paper, we formulate and solve a variant of the partition problem called the minimum weighted concave impurity partition under constraint (MIPUC). MIPUC finds an optimal partition that minimizes a given weighted concave loss function under a given concave constraint. MIPUC generalizes the recently proposed Deterministic Information Bottleneck problem which finds an optimal partition that maximizes the mutual information between the input and partitioned output while minimizing the partitioned output entropy. Our proposed algorithm is based on an optimality condition, which allows us to find a locally optimal solution efficiently. We also show that the optimal partitions are separated by some hyperplanes in the space of posterior probability mass functions.