Related papers: The topography of multivariate normal mixtures

Detecting the dimension of a hidden manifold from a point sample has become an important problem in the current data-driven era. Indeed, estimating the shape dimension is often the first step in studying the processes or phenomena…

We study the problem of detecting multiple change points in the mean vectors of an independent sequence of high-dimensional observations. We propose a family of ridge-regularized CUSUM statistics built upon the adaptable ridge-regularized…

Based on the manifold hypothesis, real-world data often lie on a low-dimensional manifold, while normalizing flows as a likelihood-based generative model are incapable of finding this manifold due to their structural constraints. So, one…

Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data --- for example, to approximate a point cloud by a low-dimensional…

Real-world graphs have inherently complex and diverse topological patterns, known as topological heterogeneity. Most existing works learn graph representation in a single constant curvature space that is insufficient to match the complex…

Phenomenological (P-type) bifurcations are qualitative changes in stochastic dynamical systems whereby the stationary probability density function (PDF) changes its topology. The current state of the art for detecting these bifurcations…

Many natural, engineered, and social systems can be represented using the framework of a layered network, where each layer captures a different type of interaction between the same set of nodes. The study of such multiplex networks is a…

Persistence diagrams are efficient descriptors of the topology of a point cloud. As they do not naturally belong to a Hilbert space, standard statistical methods cannot be directly applied to them. Instead, feature maps (or representations)…

We investigate properties of the Hodge metric of a mixed period domain. In particular, we calculate its curvature and the curvature of the Hodge bundles. We also consider when the pull back metric via a period map is K\"ahler. Several…

Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics of processes executed on the network. The analysis, discrimination, and synthesis of…

Normalizing flows are generative models that provide tractable density estimation via an invertible transformation from a simple base distribution to a complex target distribution. However, this technique cannot directly model data…

This article extends the literature on copulas with discrete or continuous marginals to the case where some of the marginals are a mixture of discrete and continuous components. We do so by carefully defining the likelihood as the density…

A scale mixture of normals is a distribution formed by mixing a collection of normal distributions with fixed mean but different variances. A generalized gamma scale mixture draws the variances from a generalized gamma distribution.…

We prove that the support of an $ m $ dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered $ \mathscr{H}^{m} $ almost everywhere by a countable union of $m$…

In a multiple linear regression model, the algebraic formula of the decomposition theorem explains the relationship between the univariate regression coefficient and partial regression coefficient using geometry. It was found that…

It is argued that in the case of a smooth transition across a (dilaton-driven) curvature bounce the growing mode of the vector fluctuations matches continuously with a decaying mode at later times. Analytical examples of this observation…

On a smooth manifold with distributions ${\cal D}_1$ and ${\cal D}_2$ having trivial intersection, we consider the integral of their mutual curvature, as a functional of Riemannian metrics that make the distributions orthogonal. The mutual…

We consider a finite mixture model with varying mixing probabilities. Linear regression models are assumed for observed variables with coefficients depending on the mixture component the observed subject belongs to. A modification of the…

We observe that the maximal open set of constant curvature k in a Riemannian manifold with curvature bounded below or above by k has a convexity type property, which we call "two-convexity". This statement is used to prove a number of…

The continuity, in a suitable topology, of algebraic and geometric operations on real analytic manifolds and vector bundles is proved. This is carried out using recently arrived at seminorms for the real analytic topology. A new…