Compressed Hypothesis Testing: to Mix or Not to Mix?

In this paper, we study the hypothesis testing problem of, among $n$ random variables, determining $k$ random variables which have different probability distributions from the rest $(n-k)$ random variables. Instead of using separate measurements of each individual random variable, we propose to use mixed measurements which are functions of multiple random variables. It is demonstrated that $O({\displaystyle \frac{k \log(n)}{\min_{P_i, P_j} C(P_i, P_j)}})$ observations are sufficient for correctly identifying the $k$ anomalous random variables with high probability, where $C(P_i, P_j)$ is the Chernoff information between two possible distributions $P_i$ and $P_j$ for the proposed mixed observations. We characterized the Chernoff information respectively under fixed time-invariant mixed observations, random time-varying mixed observations, and deterministic time-varying mixed observations; in our derivations, we introduced the \emph{inner and outer conditional Chernoff information} for time-varying measurements. It is demonstrated that mixed observations can strictly improve the error exponent of hypothesis testing, over separate observations of individual random variables. We also characterized the optimal mixed observations maximizing the error exponent, and derived an explicit construction of the optimal mixed observations for the case of Gaussian random variables. These results imply that mixed observations of random variables can reduce the number of required samples in hypothesis testing applications. Compared with compressed sensing problems, this paper considers random variables which are allowed to dramatically change values in different measurements.