A Power-Weighted Noncentral Complex Gaussian Distribution

Although vast amounts of data, such as audio signal spectra, are naturally represented using complex numbers, conventional machine learning methods often simplify complex-domain problems by employing frameworks designed for real-valued variables. While this simplification offers computational benefits, it discards structural information regarding the inherent relationship between amplitude and phase. In this paper, we propose a novel Boltzmann machine (BM), named PolarBM, capable of naturally handling complex-valued variables in the polar coordinate (i.e., an amplitude-phase representation). PolarBM defines a probability density function for complex variables in which the phase explicitly depends on the amplitude, thereby capturing the physically important relationships of complex-valued signals. Furthermore, to process audio signals in accordance with human auditory perception, we propose LogPolarBM, which models amplitude on a logarithmic scale. This extension yields a flexible conditional probability density function, a power-weighted noncentral complex Gaussian (PW-NCCG) distribution, whose marginal amplitude distribution encompasses the Rice, Nakagami, and noncentral chi distributions as special cases. For practical applications, we also introduce the restricted variants of these proposed models: PolarRBM and LogPolarRBM. Experimental results demonstrate that by explicitly modeling the dependency between amplitude and phase, the proposed RBMs achieve superior modeling accuracy compared to conventional models, including deep neural networks. Although our experiments focus on audio signals, the utility of the proposed BMs is not limited to audio applications; their potential extends widely across various fields of science and engineering that involve complex-valued data, such as wireless communications and quantum mechanics.

Complex-valued Neural Networks (CVNNs) are often motivated by domains where information is naturally encoded in magnitude and phase. Yet complex-valued inputs alone do not determine when complex arithmetic improves learning: the label signal may lie in amplitude, phase, their coupling, or a symmetry that real-valued models can also represent under suitable coordinates. We study this through a representation-first evaluation of CVNNs against Cartesian real, polar, phase-only, magnitude-only, parameter-matched real, and FLOP-matched real baselines. Across synthetic RF tasks, complex representations are useful but not universally superior. PSK-only tasks favor phase-aware and complex-valued models, QAM-only tasks favor magnitude-based models, mixed PSK+QAM gives only a small complex-valued advantage, and unseen carrier-phase rotations break coordinate-dependent models without augmentation. Similar patterns appear beyond RF: in quantum-wavefunction prediction, momentum is invisible to $|ψ|$ but recoverable from phase, while EEG analytic-signal experiments show that phase locking, amplitude bursts, and phase-amplitude coupling each favor different coordinate views. We also identify a benchmarking artifact on RadioML 2018.01A. Under matched-shared-trial selection, a CReLU complex model exceeds the best real baseline by 22.94 PP; under independent per-family tuning on the same data and 16-trial search space, the gap collapses to 2.46 PP. Gradient analysis traces the inflated gap to high-learning-rate first-step instability in real baselines, while complex parameter coupling distributes the loss signal more robustly. A learning-rate $\times$ activation factorial confirms the failure is primarily hyperparameter-driven. Overall, CVNNs are best viewed as structured inductive biases whose gains depend on representation, symmetry, and optimization, not as universally superior architectures.

Conditional density estimation is complicated by multimodality, heteroscedasticity, and strong non-Gaussianity. Gaussian processes (GPs) provide a principled nonparametric framework with calibrated uncertainty, but standard GP regression is limited by its unimodal Gaussian predictive form. We introduce the Generalized Gaussian Mixture Process (GGMP), a GP-based method for multimodal conditional density estimation in settings where each input may be associated with a complex output distribution rather than a single scalar response. GGMP combines local Gaussian mixture fitting, cross-input component alignment and per-component heteroscedastic GP training to produce a closed-form Gaussian mixture predictive density. The method is tractable, compatible with standard GP solvers and scalable methods, and avoids the exponentially large latent-assignment structure of naive multimodal GP formulations. Empirically, GGMPs improve distributional approximation on synthetic and real-world datasets with pronounced non-Gaussianity and multimodality.