Optimal Distributed Quantizer Design for Binary Classification of Conditionally Independent Vector Sources

This work addresses the scenario where two dis-tributed sensor nodes encode the input vectors and send their messages to a server node where a joint decoder outputs one of two possible class labels. We assume that the vector sources are discrete and conditionally independent given the class label. The problem is to design the two encoders and the joint decoder such that the probability of classification error is minimized. Up to our knowledge, the only known globally optimal solution to this prob-lem is an exhaustive search, which requires $N^{K_{1}+K_{2}}(N+K_{1}K_{2})$ operations, where $N$ is the size of the largest alphabet of the input vectors and $K_{k}$ is the number of quantizer regions of the encoder at node $k$, for $k=1,2$. We propose a considerably faster globally optimal solution with time complexity $O(K_{1}K_{2}N^{3})$. To achieve this, we first convert the problem to a distributed scalar quantizer design problem in a transformed domain related to the likelihood ratio domain. Next, we prove that the problem is equivalent to a constrained minimum weight path problem in a certain weighted directed acyclic graph with $O(N^{3})$ vertices and $O(N^{4})$ edges. Further, we show that the dynamic programming solution algorithm to the latter problem can be accelerated by leveraging a fast matrix search technique in matrices with the Monge property.