PolarBM: Complex Boltzmann Machine for Audio Signals

Many types of data, particularly audio signal spectra, are inherently represented using complex numbers. However, conventional machine learning often simplifies these problems by converting them to real-valued variables, which can lead to a loss of crucial structural information, especially the relationship between amplitude and phase.A new Boltzmann machine, named PolarBM, has been developed to directly handle complex-valued variables in polar coordinates (amplitude-phase representation). This model explicitly defines a probability density function where phase depends on amplitude, thereby preserving the physically significant relationships within complex signals. An extension, LogPolarBM, further enhances this by modeling amplitude on a logarithmic scale, aligning with human auditory perception.LogPolarBM introduces a flexible conditional probability density function, the power-weighted noncentral complex Gaussian (PW-NCCG) distribution, which encompasses several known distributions. Restricted variants, PolarRBM and LogPolarRBM, also demonstrate superior modeling accuracy compared to traditional deep neural networks by effectively capturing the amplitude-phase dependency. While tested on audio, the utility of these models extends to other complex-valued data in science and engineering.