Communication Complexity of Distributed High Dimensional Correlation Testing

Two parties observe independent copies of a $d$-dimensional vector and a scalar. They seek to test if their data is correlated or not, namely they seek to test if the norm $|\rho|_2$ of the correlation vector $\rho$ between their observations exceeds $\tau$ or is it $0$. To that end, they communicate interactively and declare the output of the test. We show that roughly order $d/\tau^2$ bits of communication are sufficient and necessary for resolving the distributed correlation testing problem above. Furthermore, we establish a lower bound of roughly $d^2/\tau2$ bits for communication needed for distributed correlation estimation, rendering the estimate-and-test approach suboptimal in communication required for distributed correlation testing. For the one-dimensional case with one-way communication, our bounds are tight even in the constant and provide a precise dependence of communication complexity on the probabilities of error of two types.