The Sufficient Maximum

Two sensors at distant locations need to determine whether their signals share a common source. The communication link between them carries only k bits. With a k-bit channel, you cannot send the full signal — you must compress. The question is what to compress.

Weiss and Lancho (arXiv:2603.27733, March 2026) prove that the optimal strategy is absurdly simple: send the index of your largest sample. Each sensor takes N observations, identifies which observation had the maximum value, encodes its position as log₂(N) bits, and transmits. The receiving sensor performs a scalar threshold test on the two indices. If the maxima occurred at the same (or nearby) time indices, the signals are correlated. If not, they are independent.

This works because the location of the maximum is a near-sufficient statistic for correlation detection. Under the null hypothesis — independent signals — the maximum position is uniformly distributed. Under the alternative — shared source — the maximum positions cluster. Nearly all the information needed to distinguish these two cases is concentrated in a single number: where the peak occurred. The rest of the signal is redundant for this task.

The authors prove non-asymptotic detection guarantees despite the massive information loss. An entire signal sequence is reduced to a single index, yet the detection power degrades gracefully with the compression ratio. The proof establishes that the extremum carries a disproportionate share of the discriminative information — not because it is the largest value, but because its location encodes the temporal structure of the underlying process.

The structural observation: for the specific question "are these signals correlated?", almost everything about the signal is irrelevant. The task selects for a particular feature — the temporal position of the extremum — that is nearly sufficient, and everything else is noise. The apparent wastefulness of discarding all but one number is actually optimal because the question being asked is simpler than the signal. When the question is simple, the answer is concentrated in a small feature of the data, and the rest of the measurement is overhead.