  Consider a discrete memoryless multiple source with $m$ components of which
$k \leq m$ possibly different sources are sampled at each time instant and
jointly compressed in order to reconstruct all the $m$ sources under a given
distortion criterion. A new notion of sampling rate distortion function is
introduced, and is characterized first for the case of fixed-set sampling.
Next, for independent random sampling performed without knowledge of the source
outputs, it is shown that the sampling rate distortion function is the same
regardless of whether or not the decoder is informed of the sequence of sampled
sets. Furthermore, memoryless random sampling is considered with the sampler
depending on the source outputs and with an informed decoder. It is shown that
deterministic sampling, characterized by a conditional point-mass, is optimal
and suffices to achieve the sampling rate distortion function. For memoryless
random sampling with an uninformed decoder, an upper bound for the sampling
rate distortion function is seen to possess a similar property of conditional
point-mass optimality. It is shown by example that memoryless sampling with an
informed decoder can outperform strictly any independent random sampler, and
that memoryless sampling can do strictly better with an informed decoder than
without.
