Complex Diffusion Maps with ω-Parameterized Kernels Revealing Inherent Harmonic Representations

Stochastic Gradient Descent (SGD) is a known stochastic iterative method popular for large-scale convex optimization problems due to its simple implementation and scalability. Some objectives, such as those found in complex-valued neural networks, benefit from updates like in SGD and Gradient Descent (GD) with a newly defined ``gradient'' that allows for complex parameters. This complex variant of the SGD/GD methods has already been proposed, but convergence guarantees without analyticity constraints have not yet been provided. We propose a variant of SGD (complex SGD) that allows for complex parameters, and we provide convergence guarantees under assumptions that parallel those from the real setting. Notably, these results extend to GD as well, and with the same set of assumptions, we confirm that some directional bias results extend from the real to the complex setting for kernel regression problems. We provide empirical results demonstrating the efficacy of the complex SGD in kernel regression problems utilizing complex reproducing kernel Hilbert spaces. In particular, we demonstrate we may recover superoscillation functions and Blaschke products from the Fock Space and Hardy Space, respectively, as the optimal functions for a particular choice of a loss function.

Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy $Re=20,000$ lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails. These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.

We develop a data-driven approximation of the projected ambient connection Laplacian acting on differential forms over smooth Riemannian manifolds sampled by point clouds. The proposed construction extends the classical framework of diffusion maps and Vector Diffusion Maps from scalar functions and tangent vector fields to differential forms of arbitrary degree. Our approach is based on a novel representation of differential forms as alternating differential arrays obtained through an extension of the classical musical isomorphism. This representation enables the construction of a matrix-valued diffusion operator that approximates the projected ambient connection Laplacian directly from point cloud data without requiring a mesh or simplicial complex. The proposed discretization admits the asymptotically optimal kernel bandwidth scaling inherited from diffusion maps, leading to sharper convergence guarantees than previous data-driven approximations of the Hodge Laplacian. Building upon this operator, we derive a fully data-driven explicit Euler scheme for the heat equation on differential forms and validate the proposed methodology through numerical experiments on the unit sphere. The experiments confirm the predicted decay of the analytical solution and demonstrate the effectiveness of the proposed discretization. The proposed framework provides a natural generalization of Vector Diffusion Maps to differential forms of arbitrary degree and establishes a practical foundation for the numerical approximation of geometric partial differential equations directly from point cloud data.