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A flag is a nested sequence of vector spaces. The type of the flag encodes the sequence of dimensions of the vector spaces making up the flag. A flag manifold is a manifold whose points parameterize all flags of a fixed type in a fixed vector space. This paper provides the mathematical framework necessary for implementing self-organizing mappings on flag manifolds. Flags arise implicitly in many data analysis contexts including wavelet, Fourier, and singular value decompositions. The proposed geometric framework in this paper enables the computation of distances between flags, the computation of geodesics between flags, and the ability to move one flag a prescribed distance in the direction of another flag. Using these operations as building blocks, we implement the SOM algorithm on a flag manifold. The basic algorithm is applied to the problem of parameterizing a set of flags of a fixed type. 

Latent space Energy-Based Models (EBMs), also known as energy-based priors, have drawn growing interests in generative modeling. Fueled by its flexibility in the formulation and strong modeling power of the latent space, recent works built upon it have made interesting attempts aiming at the interpretability of text modeling. However, latent space EBMs also inherit some flaws from EBMs in data space; the degenerate MCMC sampling quality in practice can lead to poor generation quality and instability in training, especially on data with complex latent structures. Inspired by the recent efforts that leverage diffusion recovery likelihood learning as a cure for the sampling issue, we introduce a novel symbiosis between the diffusion models and latent space EBMs in a variational learning framework, coined as the latent diffusion energy-based model. We develop a geometric clustering-based regularization jointly with the information bottleneck to further improve the quality of the learned latent space. Experiments on several challenging tasks demonstrate the superior performance of our model on interpretable text modeling over strong counterparts. 

The Kelvin-Helmholtz instability (KHI) and its effects relating to the transfer of energy and mass from the solar wind into the magnetosphere remain an important focus of magnetospheric physics. One such effect is the generation of Pc4-Pc5 ultra low frequency (ULF) waves (periods of 45–600 s). On July 3, 2007 at ∼ 0500 magnetic local time the Cluster space mission encountered Pc4 frequency Kelvin-Helmholtz waves (KHWs) at the high latitude magnetopause with signatures of persistent vortices. Such signatures included bipolar fluctuations of the magnetic field normal component associated with a total pressure increase and rapid change in density at vortex edges; oscillations of magnetosheath and magnetospheric plasma populations; existence of fast-moving, low-density, mixed plasma; quasi-periodic oscillations of the boundary normal and an anti-phase relation between the normal and parallel components of the boundary velocity. The event occurred during a period of southward polarity of the interplanetary magnetic field according to the OMNI data and THEMIS observations at the subsolar point. Several of the KHI vortices were associated with reconnection indicated by the Walén relation, the presence of deHoffman-Teller frames, field-aligned ion beams observed together with bipolar fluctuations in the normal magnetic field component, and crescent ion distributions. Global magnetohydrodynamic simulation of the event also resulted in KHWs at the magnetopause. The observed KHWs associated with reconnection coincided with recorded ULF waves at the ground whose properties suggest that they were driven by those waves. Such properties were the location of Cluster’s magnetic foot point, the Pc4 frequency, and the solar wind conditions.