Bounded Guaranteed Algorithms for Concave Impurity Minimization Via Maximum Likelihood

Partitioning algorithms play a key role in many scientific and engineering disciplines. A partitioning algorithm divides a set into a number of disjoint subsets or partitions. Often, the quality of the resulted partitions is measured by the amount of impurity in each partition, the smaller impurity the higher quality of the partitions. In general, for a given impurity measure specified by a function of the partitions, finding the minimum impurity partitions is an NP-hard problem. Let $M$ be the number of $N$-dimensional elements in a set and $K$ be the number of desired partitions, then an exhaustive search over all the possible partitions to find a minimum partition has the complexity of $O(K^M)$ which quickly becomes impractical for many applications with modest values of $K$ and $M$. Thus, many approximate algorithms with polynomial time complexity have been proposed, but few provide bounded guarantee. In this paper, an upper bound and a lower bound for a class of impurity functions are constructed. Based on these bounds, we propose a low-complexity partitioning algorithm with bounded guarantee based on the maximum likelihood principle. The theoretical analyses on the bounded guarantee of the algorithms are given for two well-known impurity functions Gini index and entropy. When $K \geq N$, the proposed algorithm achieves state-of-the-art results in terms of lowest approximations and polynomial time complexity $O(NM)$. In addition, a heuristic greedy-merge algorithm having the time complexity of $O((N-K)N^2+NM)$ is proposed for $K<N$. Although the greedy-merge algorithm does not provide a bounded guarantee, its performance is comparable to that of the state-of-the-art methods. Our results also generalize some well-known information-theoretic bounds such as Fano's inequality and Boyd-Chiang's bound.