  Deep neural networks can be trained in reciprocal space, by acting on the
eigenvalues and eigenvectors of suitable transfer operators in direct space.
Adjusting the eigenvalues, while freezing the eigenvectors, yields a
substantial compression of the parameter space. This latter scales by
definition with the number of computing neurons. The classification scores, as
measured by the displayed accuracy, are however inferior to those attained when
the learning is carried in direct space, for an identical architecture and by
employing the full set of trainable parameters (with a quadratic dependence on
the size of neighbor layers). In this Letter, we propose a variant of the
spectral learning method as appeared in Giambagli et al {Nat. Comm.} 2021,
which leverages on two sets of eigenvalues, for each mapping between adjacent
layers. The eigenvalues act as veritable knobs which can be freely tuned so as
to (i) enhance, or alternatively silence, the contribution of the input nodes,
(ii) modulate the excitability of the receiving nodes with a mechanism which we
interpret as the artificial analogue of the homeostatic plasticity. The number
of trainable parameters is still a linear function of the network size, but the
performances of the trained device gets much closer to those obtained via
conventional algorithms, these latter requiring however a considerably heavier
computational cost. The residual gap between conventional and spectral
trainings can be eventually filled by employing a suitable decomposition for
the non trivial block of the eigenvectors matrix. Each spectral parameter
reflects back on the whole set of inter-nodes weights, an attribute which we
shall effectively exploit to yield sparse networks with stunning classification
abilities, as compared to their homologues trained with conventional means.
