Machine learning in spectral domain

Abstract: Deep neural networks are usually trained in the space of the nodes, by adjusting the weights of existing links via suitable optimization protocols. We here propose a radically new approach which anchors the learning process to reciprocal space. Specifically, the training acts on the spectral domain and seeks to modify the eigenvalues and eigenvectors of transfer operators in direct space. The proposed method is ductile and can be tailored to return either linear or non-linear classifiers. Adjusting the eigenvalues, when freezing the eigenvectors entries, yields performances which are superior to those attained with standard methods {\it restricted} to a operate with an identical number of free parameters. Tuning the eigenvalues correspond in fact to performing a global training of the neural network, a procedure which promotes (resp. inhibits) collective modes on which an effective information processing relies. This is at variance with the usual approach to learning which implements instead a local modulation of the weights associated to pairwise links. Interestingly, spectral learning limited to the eigenvalues returns a distribution of the predicted weights which is close to that obtained when training the neural network in direct space, with no restrictions on the parameters to be tuned. Based on the above, it is surmised that spectral learning bound to the eigenvalues could be also employed for pre-training of deep neural networks, in conjunction with conventional machine-learning schemes. Changing the eigenvectors to a different non-orthogonal basis alters the topology of the network in direct space and thus allows to export the spectral learning strategy to other frameworks, as e.g. reservoir computing.