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

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.

Broad Learning System (BLS) is an efficient alternative to deep architectures due to its fast training, analytical learning, and strong generalization under limited data. However, existing BLS variants are confined to real-valued representations, restricting their ability to capture nonlinear interactions and second-order statistical dependencies inherent in real-world data. Notably, no prior BLS model fully exploits the complete second-order statistics that naturally emerge when data are embedded in the complex domain. To address this limitation, this paper introduces the first complex augmented Broad Learning System (CA-BLS), which transforms real-valued inputs into phase-encoded complex representations and adopts widely linear modeling to jointly leverage covariance and pseudo-covariance information via complex conjugate augmentation. This enables effective modeling of latent nonlinearities, coherence structures, and second-order dependencies inaccessible to conventional BLS formulations. To mitigate the additional computational cost of complex augmentation, an Efficient Complex Augmented BLS (ECA-BLS) is further developed, reformulating CA-BLS entirely in the real domain while preserving its exact decision function, achieving up to 75% fewer multiplications and over 60% fewer additions. A rigorous theoretical analysis proves the mathematical equivalence between CA-BLS and ECA-BLS, ensuring zero theoretical loss. Extensive experiments on 26 benchmark datasets from the UCI and KEEL repositories demonstrate that ECA-BLS consistently outperforms classical BLS and recent state-of-the-art randomized neural networks in accuracy, average rank, and statistical significance, establishing augmented second-order modeling as a critical and previously missing dimension of BLS research.

Designing effective feature extractors is critical for blind signal analysis tasks such as automatic modulation recognition (AMR), signal scheme recognition (SSR), and \color{black} signal structure parsing (SSP). In this work, we propose dual-channel neural network (DualNN) that efficiently exploits complex-valued signals through parameter sharing across IQ channels. Unlike traditional real-valued or complex-valued models, DualNN is a groundbreaking framework which shares the network parameters for processing the real and imaginary parts of the complex-valued signals, and is theoretically shown to reduce generalization error while preserving expressive capacity. Specifically, we propose a novel Transformer-based architecture to implement DualNN, called Dualformer. The Dualformer segments input signals into patch-level tokens and captures multi-granularity features, enabling robust performance across diverse signal analysis tasks. Furthermore, we conduct extensive experiments comparing Dualformer with three Transformer-based baselines and four conventional DL-based approaches. Results demonstrate consistent performance improvements on AMR, SSR, and SSP tasks. Besides, the modular design of DualNN allows it to generalize well to blind signal processing tasks such as blind source separation and low-SNR spectrum sensing. This work paves the way for a broader application of DualNN architectures in unsupervised and weakly supervised complex-valued signal analysis scenarios.