  We extend the framework of Boltzmann machines to a network of complex-valued
neurons with variable amplitudes, referred to as Complex Amplitude-Phase
Boltzmann machine (CAP-BM). The model is capable of performing unsupervised
learning on the amplitude and relative phase distribution in complex data. The
sampling rule of the Gibbs distribution and the learning rules of the model are
presented. Learning in a Complex Amplitude-Phase restricted Boltzmann machine
(CAP-RBM) is demonstrated on synthetic complex-valued images, and handwritten
MNIST digits transformed by a complex wavelet transform. Specifically, we show
the necessity of a new amplitude-amplitude coupling term in our model. The
proposed model is potentially valuable for machine learning tasks involving
complex-valued data with amplitude variation, and for developing algorithms for
novel computation hardware, such as coupled oscillators and neuromorphic
hardware, on which Boltzmann sampling can be executed in the complex domain.
