# PolarBM: Complex-valued Boltzmann Machine for Modeling Audio Signals in Polar and Log-polar Coordinates

arXiv: https://arxiv.org/abs/2607.12417 · HTML: https://mlsanity.com/paper/2607.12417/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 30 references that are in the map.
- First posted on arXiv 14 Jul 2026.

## Builds on (most cited first, 6 of 6)

- [HiFi-GAN: Generative Adversarial Networks for Efficient and High Fidelity Speech Synthesis](https://mlsanity.com/paper/2010.05646/) (2020)
- [UTMOS: UTokyo-SaruLab System for VoiceMOS Challenge 2022](https://mlsanity.com/paper/2204.02152/) (2022)
- [A Survey of Complex-Valued Neural Networks](https://mlsanity.com/paper/2101.12249/) (2021)
- [WARP-Q: Quality Prediction For Generative Neural Speech Codecs](https://mlsanity.com/paper/2102.10449/) (2021)
- [Complex Amplitude-Phase Boltzmann Machines](https://mlsanity.com/paper/2005.01862/) (2020)
- [A Power-Weighted Noncentral Complex Gaussian Distribution](https://mlsanity.com/paper/2603.26344/) (2026)

## Abstract (arXiv)

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.