Information tradeoffs in sensing and sampling

Organisms sense the world through arrays of receptor cells. In some cases, such as the compound eyes of insects, these arrays are nearly crystalline. In other cases, including the human retina, sampling is much less regular. While ordered sampling typically gathers more information, this comes at the cost of specifying the positions of all the cells. We explore this tradeoff, asking about the maximum entropy of the sampling lattice that is consistent with gathering a certain amount of information from a Gaussian random signal; bits of sensory information are traded against bits of positional information. This problem maps to an equilibrium statistical mechanics problem for the positions of the receptor cells, with interactions that depend on the correlation structure of the input signal. In some limits we find that the information cost of disorder is surprisingly small. In other limits there are transitions where an ordered sampling lattice melts as we change the parameters of the sensory environment.