  We describe, in the detection of multi-sample aligned sparse signals, the
critical boundary separating detectable from nondetectable signals, and
construct tests that achieve optimal detectability: penalized versions of the
Berk-Jones and the higher-criticism test statistics evaluated over pooled
scans, and an average likelihood ratio over the critical boundary. We show in
our results an inter-play between the scale of the sequence length to signal
length ratio, and the sparseness of the signals. In particular the difficulty
of the detection problem is not noticeably affected unless this ratio grows
exponentially with the number of sequences. We also recover the multiscale and
sparse mixture testing problems as illustrative special cases.
