  We consider a three-layer Sejnowski machine and show that features learnt via
contrastive divergence have a dual representation as patterns in a dense
associative memory of order P=4. The latter is known to be able to
Hebbian-store an amount of patterns scaling as N^{P-1}, where N denotes the
number of constituting binary neurons interacting P-wisely. We also prove that,
by keeping the dense associative network far from the saturation regime
(namely, allowing for a number of patterns scaling only linearly with N, while
P>2) such a system is able to perform pattern recognition far below the
standard signal-to-noise threshold. In particular, a network with P=4 is able
to retrieve information whose intensity is O(1) even in the presence of a noise
O(\sqrt{N}) in the large N limit. This striking skill stems from a redundancy
representation of patterns -- which is afforded given the (relatively) low-load
information storage -- and it contributes to explain the impressive abilities
in pattern recognition exhibited by new-generation neural networks. The whole
theory is developed rigorously, at the replica symmetric level of
approximation, and corroborated by signal-to-noise analysis and Monte Carlo
simulations.
