Kähler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold

We study landscapes for complex-parameterized networks. Our approach is motivated with an information-theoretic manifold perspective of the parameter and via classical optimization guarantees although of complex geometric variety such as through Dolbeault asymptotics. The descent path admits a Kähler information metric under a cross-entropy via the Wirtinger Hessian on the log-likelihood potential. We restrict attention to a descent update rule with natural gradient descent via a differentiated loss scaled by the inverse metric, so the descent path remains in the holomorphic tangent bundle. We emphasize Calabi-Yau information manifolds which profane theoretical guarantees via an ill-curvature-conditioned landscape. Under a Calabi-Yau metric, specifically in a non-compact setting with a global potential so defined geometrically rather than invoking the topological requirements of the Calabi conjecture, a wedged nowhere-vanishing holomorphic form is the top exterior product of the Kähler form up to constants, yielding a constant determinant condition. Under a fixed determinant, a metric almost low rank up to an eigenvalue tolerance implies a blow-up effect. Moreover, it has been discovered that negative curvature subverts the loss landscape, specifically sectional curvature, so we expand on this and draw interconnections to negative-definite Ricci curvature. Our arguments primarily exist in a geometric analytic modality, although we establish roots in deep learning theory such as through asymptotics at initialization and connections through failure modes of neural network guarantees under vanishing and negative Ricci curvature.

Physics-Aware Complex-Valued State Space Model with Scattering-Prior Feature Modulation for PolSAR Image Classification

Polarimetric synthetic aperture radar (PolSAR) image classification is a representative task for physics-aware GeoAI, where land-cover semantics are closely coupled with electromagnetic scattering mechanisms. Many existing complex-valued networks can preserve amplitude-phase information, but they are often limited in long-range spatial dependency modeling and usually incorporate polarimetric priors only as input-level or shallow auxiliary features. As a result, physical knowledge is insufficiently used to guide deep feature evolution. To address this issue, this paper proposes CV-SSMNet, a physics-aware complex-valued state-space network with scattering-aware feature modulation for PolSAR image classification. The proposed method builds a complex-valued state-space model (CV-SSM) in the original complex domain to capture long-range spatial dependencies while preserving polarimetric amplitude-phase coupling. Meanwhile, seven physically meaningful scattering priors, are encoded as FiLM-style modulation signals to adaptively recalibrate complex-valued representations during feature evolution. CV-SSMNet further integrates multi-scale complex convolutions, branch-wise CV-SSM encoding, prior-guided recalibration, and lightweight global context aggregation, enabling physically guided representation learning from local scattering structures to global spatial context. Experiments on three L-band benchmark datasets and an additional P-band BIOMASS evaluation demonstrate that CV-SSMNet achieves competitive accuracy, improved regional consistency, and better boundary preservation, supporting the effectiveness of embedding polarimetric scattering mechanisms into complex-valued long-range GeoAI representation learning.

Unified Complex-valued Neural Network: A Magnitude-Phase Computational Model for Event-Driven Neuromorphic Learning

Artificial neural networks (ANN) provide accurate continuous-valued representation, whereas spiking neural networks (SNN) offer event-driven temporal processing, yet both paradigms face limitations when value encoding and timing dynamics must be learned within a single computational structure. This paper introduces a network based on Unified Complex-valued Neuron (UCN), a new neural computational model that integrates continuous activation and phase-driven event generation through an asymmetric complex-valued state. In the UCN, magnitude encodes signal strength while phase governs intrinsic temporal evolution and valued spike emission. A foundational training framework combining backpropagation (BP) and backpropagation through time (BPTT) is first developed to optimize magnitude and phase pathways in a unified way. To reduce computational complexity, an event-driven adaptive phase learning (EAPL) rule is then introduced as a more efficient alternative. The proposed model is evaluated through object tracking and Lorenz attractor learning. Results demonstrate that UCN-based Network (UCNN) provides accurate, stable, and interpretable spatiotemporal learning while preserving sparse event-driven computation for neuromorphic and edge-AI applications.

Why SWAVE May Not Be All You Need:A Concept-Evolution Retrospective on Complex-Valued Recurrent Language Models

SWave is a complex-valued recurrent language model (169.26M parameters, D=384, L=16, T=2048) trained on FineWeb-Edu using 2xH100 NVL. It was designed around three founding premises: that representing language as complex waves rather than real-valued numbers enables richer information encoding; that a Cayley-parameterised unitary transition provides a mathematical guarantee against state decay or explosion; and that a hidden state which rotates rather than shrinks preserves signal integrity over arbitrarily long contexts. The core of SWave evolved substantially across three development phases. The Resonance Head was found to structurally admit imaginary-channel collapse as a global loss minimum (a failure mode we term cos-domination collapse) and was superseded by an untied head with independent real and imaginary embedding tables from the Phase-Associative Memory (PAM) architecture. This resolved the degenerate minimum and enabled stable 200,000-step training (best-step PPL 22.0 at step 89,861). ComplexNorm and the Wave Propagation Scan proved load-bearing throughout all three phases and were retained to the final architecture. ProtectGatedScan was reframed as a structural prior rather than a learned behaviour. The four multi-scale retention concepts showed no measurable improvement under controlled evaluation and were found non-load-bearing. The ComplexGatedUnit was superseded by a real-valued squared-ReLU channel mixer with fewer parameters. The auxiliary training objectives showed no benefit once structural constraints were resolved. The investigation yields a formal characterisation of cos-domination collapse, a parallel scan with a log-space backward pass for numerical stability, six transferable engineering principles for complex-valued recurrent training, and a plan-to-code traceability methodology for catching structural divergences that conventional test suites miss.

Product units in gated recurrent units improve nuclear-mass prediction

The prediction of masses of atomic nuclei using machine learning can complement theoretical models and advance the exploration of poorly known domains of the nuclear chart. We propose a machine learning technique based on gated recurrent units (GRU), which have demonstrated competitive performance in nuclear-mass prediction by exploiting long-term dependencies. By integrating multiplicative interactions and product-unit transformations within recurrent units, we report significant improvements in nuclear-mass prediction. Computations are performed in the complex domain to jointly capture amplitude and phase dynamics. For interpolation and temporal-extrapolation tasks based on the atomic mass evaluation (AME2016 and AME2020), the complex additive-multiplicative product-unit gated recurrent unit (AM-PU-GRU) model consistently achieves the lowest prediction errors, with an interpolation RMSE of 0.227 $\pm$ 0.004 MeV and an extrapolation RMSE of 0.179 $\pm$ 0.015 MeV. These results surpass other state-of-the-art machine learning models and also outperform the real-valued GRU baseline and product-unit ablation variants, while remaining robust to different theoretical priors, including WS4 and SEMF. Our findings establish complex-valued product-unit recurrent networks as a new benchmark for sequence-based nuclear-mass prediction.

Shortcomings and capacities of real-constrained neural networks in complex spaces

We find the asymptotic ratio between the storage capacities when enforcing real pre-activations in a complex hypothesis class as opposed to complex ones in the same class. Our methods depend on Gardner volume comparisons at critical capacity. Our proof relies on an application of the Harish-Chandra-Itzykson-Zuber (HCIZ) formula, nonstandard in literature. With the HCIZ formula, we may obtain a more robust approximation for the final asymptotic ratio. This strategy is applicable to our work specifically since we integrate over the unitary and orthogonal compact manifolds, facilitated via the Weyl integration formula and the Haar measure.