  Prior distributions of binarized natural images are learned by using a
Boltzmann machine. According the results of this study, there emerges a
structure with two sublattices in the interactions, and the nearest-neighbor
and next-nearest-neighbor interactions correspondingly take two discriminative
values, which reflects the individual characteristics of the three sets of
pictures that we process. Meanwhile, in a longer spatial scale, a longer-range,
although still rapidly decaying, ferromagnetic interaction commonly appears in
all cases. The characteristic length scale of the interactions is universally
up to approximately four lattice spacings $\xi \approx 4$. These results are
derived by using the mean-field method, which effectively reduces the
computational time required in a Boltzmann machine. An improved mean-field
method called the Bethe approximation also gives the same results, as well as
the Monte Carlo method does for small size images. These reinforce the validity
of our analysis and findings. Relations to criticality, frustration, and
simple-cell receptive fields are also discussed.
