Symmetric cones, such as the space of positive-definite matrices or Lorentz cones, are fundamental to statistical modeling in fields ranging from neuroimaging to radar. Standard approaches, however, face a dilemma: intrinsic Riemannian methods require costly iterative optimization, while Euclidean approximations sacrifice desirable equivariance properties. We introduce a log-extrinsic framework that resolves this trade-off, utilizing the orbital decomposition of the cone to treat the determinant and the determinant-one surface independently. This approach yields an equivariant, closed-form mean that requires only elementary linear algebra, rendering it highly scalable to high-dimensional problems. We prove that under natural symmetry assumptions, this log-extrinsic mean coincides with the Riemannian Fréchet mean. Finally, we define an exponential family of log-extrinsic Gaussian distributions naturally paired with this mean. The result is a coherent statistical toolbox that preserves key geometric symmetries while remaining computationally tractable.
The Hessian structure is a geometric structure consisting of a pair (∇ ,g) of an affine connection ∇ and a Riemannian metric g satisfying certain compatibility conditions. In information geometry, it is also known as a dually flat structure, and the tangent bundle of a Hessian manifold is known to admit a natural Kähler structure. However, the Kähler structure does not necessarily distinguish the underlying Hessian manifolds. In this article, we introduce a richer geometric structure, called a strongly integrable Born structure, on the tangent bundle of a Hessian manifold. We prove that the strongly integrable Born structure can distinguish the underlying Hessian manifolds.
Recent experiments have uncovered that several high-dimensional datasets have a high probability of forming binary clusters after projecting the data on a randomly chosen one-dimensional subspace. This paper presents a probability model for the high-dimensional data that could explain this phenomenon. The model consists of several Bernoulli random variables which describe a parallelotope frame. The parallelotope forms a skeleton on which noise can be added. While clusters in high dimension are difficult to observe or may not exist at all, the groupings of sample points drawn from such a distribution can be easily identified. Such groupings can be used in place of clustering, especially when the dataset is small, and the statistical significance of these groups can be tested. The existence of such structure in the underlying probability model of a dataset allows for semantic grouping of datapoints based on different criteria. More generally, it provides a simple, binary representation for data points where each bit represents the group membership for each criterion, which could potentially be used for data compression.
The use of neural networks has shown significant potential to reduce the computational costs associated with the dynamics of industrial computational fluids. Weak adversarial networks (WAN) leverage weak solution theory to transform the problem of solving PDEs into a Min-Max optimization problem, which is then solved by training a generative adversarial network. Although this method has been successfully applied to two-dimensional (2D) Navier–Stokes (NS) equations, previous work says nothing about the Navier–Stokes-Brinkman (NSB) equations. In this study, we first leverage the stream function to introduce the biharmonic formulation of NSB equations. Then, we extend the WAN framework to solve NS equations in porous media (WAN2DNSB) and provide free surface flow as a numerical experiment. Our results demonstrate the stability and accuracy of the proposed method, highlighting its advantages over the traditional Physics-Informed Neural Networks (PINNs) and Physics-Informed Neural Operators (PINOs). This work contributes to the growing research on AI-driven numerical methods for complex fluid dynamics problems, offering a promising approach for industrial applications.
